发布时间： 2023-06-16

**Description**

**Organizing committee of Beijing-Moscow Mathematics Colloquium **

(1) Huijun Fan (SMS PKU, symplectic geometry and mathematical physics, geometric analysis)

(2) Sergey Gorchinskiy (MI RAS, algebra and geometry: algebraic geometry, K-theory)

(3) Hailiang Li (SMS CNU, fluid mechanics, partial differential equations, analysis)

(4) Jinsong Liu (AMSS, algebraic geometry: singularity theory)

(5) Yi Liu (BICMR, Topology of 3-manifolds, hyperbolic geometry)

(6) Denis Osipov (MI RAS, algebraic geometry, number theory, integrable system)

(7) Ye Tian (UCAS, AMSS, Number Theory, Arithmetic Geometry, Iwasawa Theory)

(8) Alexey Tuzhlin (MSU, geometry: Riemannian and metric geometry)

(9) Yue Yang (CE PKU, computation mathematics and mechanics)

(10) Ping Zhang (AMSS, P. D. E.: fluid equation and semi-classical analysis)

(11) Alexander Zheglov (MSU, geometry: algebraic geometry, integrable system)

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The lecture announcements will be continually updated. The arrangement of the upcoming lectures is as follows:

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**Lecture Series 55 ****—— June 16, 2023 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/A43E4AC9258D4E2F3D8240E7E4CBB058

Valid Until: 2027-07-31 23:59

**Lecture 1****——Some mathematical problems of gravitational collapse in general relativity**

**Speaker:** Junbin Li (Sun Yat-Sen University)

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** I will give a brief review on the mathematical theories of gravitational collapse in general relativity, and talk about some problems and recent developments, including my works on the formation of trapped surfaces, black holes and instability of naked singularities.

**Bio:** Junbin Li is currently a professor at Sun Yat-Sen University. He obtained his doctoral degree also at Sun Yat-Sen University in 2014. The research interest of Junbin Li is geometric analysis, in particular the mathematical theories of general relativity.

**Lecture 2****——Self-interlocking structures in $R^2$ and $R^3$**

**Speaker:** Alexei Kanel-Belov (Bar-Ilan University, Moscow State University, Moscow Institute of Physics and Technology)

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Consider a set of contacting convex figures in $R^2$. It can be proven that one of these figures can be moved out of the set by translation without disturbing others. Therefore, any set of planar figures can be disassembled by moving all figures one by one. However, attempts to generalize it to $R^3$ have been unsuccessful and quite unexpectedly interlocking structures of convex bodies were found. Author proposed the following mechanical use of this effect. In a small grain there is no room for cracks, and crack propagation should be arrested on the boundary of the grain. On the other hand, grains keep each other. So it is possible to get "materials without crack propagation" and get new use of sparse materials, say ceramics. Quite unexpectedly, such structures can be assembled with any type of platonic polyhedra, and they have a geometric beauty. Some patents were obtained

https://www.elibrary.ru/item.asp?id=47260049

https://www.elibrary.ru/item.asp?id=47259870

https://www.elibrary.ru/item.asp?id=46607120

The talk is devoted to the different structures. The talk is devoted to the theory of self-interlocking structures and to the recent progress in it by Manturov: a) There exist two-dimensional self-interlocking structures in 3-dimensional space; b) One can construct self-interlocking 2-dimensional structures which are rigid once two polygons are fixed. Vassily O. Manturov, Alexei Kanel-Belov, Seongjeong Kim, Two-dimensional self-interlocking structures in three-space, 2021 (Published online) , 21 pp., arXiv: 2109.06426. Kanel-Belov, A.J., A.V. Dyskin, Y. Estrin, E. Pasternak and I.A.Ivanov. 2010. Interlocking of convex polyhedra: towards a geometric theoryof fragmented solids. Moscow Mathematical Journal, arXiv:0812.5089v1. Dyskin, A.V., Y.Estrin, A.J.Kanel–Belov and E.Pasternak,“Interlocking properties of buckyballs.”, Physics Letters A, 319 (2003),373–378 jumas, L., Simon, G.P., Estrin, Y. et al. Deformation mechanicsof non-planar topologically interlocked assemblies with structuralhierarchy and varying geometry. Naure, SciRep 7, 11844 (2017).https://doi.org/10.1038/s41598-017-12147-3

In recent works, some interlocking structures were constructed. These structures were based on periodic tailings in a plane and were nearplanar. In addition, they were unstable in the following sense. After removing a finite set of elements, they can de disassembled one by one. Here we present spatial structures that are interlocking in several planes simultaneously. There is also another spatial effect. If a plane is divided into convex figures, there always one such that it has common edges with no more then 6 its neighbours. However, in 3D space it is not so. For any $n$ there is a division of space into congruent convex bodies such that any one of them has a common face with more than n its neighbours. This construction allows us to have interlocking in any number of layers. Many structures were generated during Sirius Math. camp. We discuss a way to generate interlocking structures and their possible properties. Finally, we present a review of different interlocking structures and formulate some open problems.

**Bio:** Alexei Kanel-Belov is a Professor at Bar-Ilan University, Moscow State University and Moscow Institute of Physics and Technology. He studies various problems related to the finite basis property of systems of identities, automorphisms of algebraic varieties (together with M.Kontsevich), combinatorics of words and symbolic dynamical systems, particularly IET.

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**Lecture Series 54 ****—— June 2, 2023 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/5B9CB00F9249BA5A27BACFBF7366AD1D

Valid Until: 2027-07-31 23:59

**Lecture 1****——Fundamental Group and Heegaard Floer Homology**

**Speaker:** Jiajun Wang (Peking University)

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** Heegaard Floer homology and Monopole Floer Homology play an important role in the study of low dimensional topology. A three-manifold is essentially determined by its fundamental group. Ozsvath and Szabo asked for a concrete relation between the fundamental group and Heegaard Floer homology. We will talk about such a concrete relation for (1,1)-knots. This is joint work with Xiliu Yang.

**Bio:** Jiajun Wang is a Professor and Head of the Department of Geometry and Topology at the School of Mathematical Sciences, Peking University. He received bachelor’s degree from Peking University in 2001 and PhD from UC Berkeley in 2007. Prof. Wang’s research interests are low-dimensional topology and gauge field.

**Lecture 2****——Local Conditions of Crystal Structures**

**Speaker:** Nikolay Dolbilin (Steklov Mathematical Institute of RAS)

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** We will give an overview of the Local Theory of regular systems and also its connection with studies of quasicrystals and of arbitrary Delone/Delaunay set. The Local Theory of regular systems relates to the foundations of Geometric Crystallography.

The mathematical model of an ideal crystal (its atomic structure) is a discrete subset X in a finite-dimensional Euclidean space that is invariant with respect to some crystallographic group G of isometries of the Euclidean space. In other words, a crystal X is the union of a finite number of G-orbits.

A single-point orbit is termed a regular system. Our attention will be focused on the lower and upper bounds for the regularity radius, which is the minimum size of clusters whose pairwise equivalence at all points of a Delone set provides the regularity of the set.

**Bio:** Nikolay Dolbilin is a leading researcher at the Steklov Mathematical Institute of the Russian Academy of Sciences and a part-time professor at Moscow State University. He also had long-term professor positions in mathematical centers and universities in Hungary, Germany, Japan, USA.

Among scientific results are the following: the proof of a local criterion of regularity and development of the local theory of regular systems, the first complete proof of the celebrated Kac-Ward formula for the statistical sum for the Ising model, studies on the theory of parallelohedra (the concepts of the contact face and its index, a theorem on the sum of indices of contact faces).

Besides, Nikolay Dolbilin paid much attention to the issues of mathematical education. He was an invited speaker at ICME-2000 at Tokyo/Makuhari, a member of the International Programm Committee of ICME-2004 in Copenhagen, a member of EC ICMI in 2003-2007.******

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**Lecture Series 53 ****—— May 19, 2023 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/69AF81979252F9687C2BA1087CB71CAE

Valid Until: 2027-06-30 23:59

**Lecture 1****——Space-time energy spectra for turbulence shear flows**

**Speaker:** Guowei He (Institute of Mechanics of CAS)

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** A space-time (frequency-wavenumber) energy spectrum describes the energy distribution of velocity fluctuations over a broad range of spatial and temporal length scales. It not only characterizes dynamic coupling between spatial and temporal scales in turbulent flows but also plays a key role in turbulence-generated noise. In this lecture, our recent work is introduced on space-time energy spectra and its application to turbulence-generated noise. The Taylor, Kraichnan-Tennekes, and elliptic approximation (EA) models are re-examined in terms of the picture of turbulent passage, which is proposed by Taylor’s frozen-flow hypothesis and the Kraichnan-Tennekes random sweeping hypothesis; The composite resolvent operators are developed for space-time energy spectra; the large-eddy simulation for frequency spectra is used to study the noise radiated by turbulent flows around an axisymmetric body of revolution.

Reference:

1. He, Jin and Yang, Annu. Rev. Fluid Mech. 49 51-70. 2017

2. Wu and He, Phys. Rev. Fluids Vol. 6 100504 2021 (APS invited talk)

**Bio:** Dr. Guowei He is a professor and the academic director of the Institute of Mechanics, the Chinese Academy of Sciences. He is an elected Academician of the Chinese Academy of Sciences and a fellow of the American Physical Society. He is the associated editor of the APS journal “Phys Rev. Fluids”. His research interests include turbulence statistical theory and computational modeling, large eddy simulation of turbulence-generated noise and machine learning.

**Lecture 2****——Folding in fluids**

**Speaker:** E.A. Kuznetsov (P.N. Lebedev Physical Institute of RAS)

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** The formation of the coherent vortical structures in the form of thin pancakes for three-dimensional flows is studied at the high Reynolds regime when, in the leading order, the development of such structures can be described within the Euler equations for ideal incompressible fluids. Numerically and analytically on the base of the vortex line representation [1, 2] we show that compression of such structures and respectively increase of their amplitudes are possible due to the compressibility of the vorticity ω in the 3D case [3]. It is demonstrated that this growth has an exponential behavior and can be considered as folding (analog of breaking) for the divergence-free fields of vorticity. At high amplitudes this process in 3D has a self-similar behavior connected the maximal vorticity and the pancake width by the relation of the universal type [4]: ω_{max} ∝ l ^{−2/3}

[1] E.A. Kuznetsov, V.P. Ruban, Hamiltonian dynamics of vortex lines for systems of the hydrodynamic type, Pis’ma ZhETF , 76, 1015 (1998) [JETP Letters, 67, 1076-1081 (1998)].

[2] E.A. Kuznetsov, Vortex line representation for flows of ideal and viscous fluids , Pis’ma v ZHETF, 76, 406-410 (2002) [JETP Letters, 76, 346-350 (2002)].

[3] D.S. Agafontsev, E.A. Kuznetsov, A.A. Mailybaev, and E.V. Sereshchenko, Compressible vortical structures and their role in the hydrodynamical turbulence onset, UFN 192, 205-225 (2022) [Physics Uspekhi, 65 189 - 208 (2022)].

[4] D.S. Agafontsev, E.A. Kuznetsov and A.A. Mailybaev, Development of high vorticity structures and geometrical properties of the vortex line representation, Phys. Fluids 30, 095104-13 (2018); Stability of tangential discontinuity for the vortex pancakes, Pisma ZHETF, 114, 67-71 (2021) [JETP Letters, 2021, 114, 71–75 (2021)]

**Bio:** Evgeny A. Kuznetsov is currently a Principle Research Fellow at Lebedev Physical Institute of RAS and Landau Institute for Theoretical Physics. He graduated from Novosibirsk State University in 1969 and received PhD from the Institute for Nuclear Physics in 1973. In 1981, he obtained the Doctor of Sciences degree at Space Research Institute. Kuznetsov is an Academician of RAS since 2016. His research interests include developed turbulence in plasma physics, hydrodynamics, magnetohydrodynamics, and stability problems of nonlinear waves and solitons.

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**Lecture Series 52 ****—— May 5, 2023 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/8FBEB1B4EA1EF5DCCCBFE7DD969C3285

Valid Until: 2027-06-30 23:59

**Lecture 1****——On the superposition principle for probability solutions to Fokker-Planck-Kolmogorov equations**

**Speaker:** Stanislav Shaposhnikov (Lomonosov Moscow State University)

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** The superposition principle expresses a deep connection between the solutions of the martingale problem and the probability solutions of the Fokker-Planck-Kolmogorov equation. It has been extensively studied in recent years and the best-known results were obtained by L. Ambrosio, A. Figalli, and D. Trevisan. We will present a generalization of the superposition principle in the case of unbounded coefficients and arbitrary domain, demonstrate several counterexamples and formulate open problems.

**Bio:** Stanislav Shaposhnikov is a Professor at the Department of Mathematical Analysis of Lomonosov Moscow State University. He graduated from MSU in 2006, obtained PhD degree in 2009 and Doctor of Sciences degree in 2011. His research focuses on the Fokker-Planck-Kolmogorov equations. Prof. Shaposhnikov was awarded the Shuvalov Prize of MSU and the Kolmogorov Prize of the Russian Academy of Sciences.

**Lecture 2****——Mapping class group of 3-dimensional complete intersections**

**Speaker:** Yang Su (AMSS CAS)

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** In the first part of the talk, I will briefly recall some knowledge of mapping class group of high dimensional manifolds. Then I will present our computation of the mapping class group of certain complex 3-dimensional algebraic varieties --- complete intersections. Similar to the classical mapping class group of surfaces, the results will be helpful in understanding the moduli space of structures on these manifolds. This is a joint work with Matthias Kreck from the University of Bonn.

**Bio:** Yang Su is an Associate Professor at the Academy of Mathematics and System Sciences, the Chinese Academy of Sciences. He graduated from Peking University in 1999 and received PhD at Heidelberg University in 2007. His research interest is the topology of manifolds in dimension 5 and higher.

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**Lecture Series 51 ****—— April 21, 2023 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/1D4C3B7471B2508408D7267536758314

Valid Until: 2026-05-31 23:59

**Lecture 1****——The geometric Bombieri-Lang conjecture for varieties of maximal Albanese dimension**

**Speaker:** Junyi Xie (Beijing International Center for Mathematical Research)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** This is a joint work with Xinyi Yuan. Let K=k(B) the function field a variety B over a field k of characteristic 0. Let X be a projective variety over K. Assume that there is a finite morphism from X to an abelian variety A with trivial trace. We show that X(K) is contained in the algebraic special subset. In particular, if further X is of general type, then X(K) is not Zariski dense.

**Bio:** Junyi Xie is a Professor at Beijing International Center for Mathematical Research, Peking University. He received Licence 3 and Master from École Normale Supérieure and Paris 7 in 2011, and PhD from Centre de mathématiques Laurent Schwartz de École Polytechnique. He was a full researcher at CNRS from 2016 to 2021. Also, he had postdoc experiences at the University of Rennes 1 and the Institute of Mathematics of Toulouse. The main research interests of Junyi Xie lie in arithmetic dynamics and related questions in algebraic geometry.

**Lecture 2****——Coregularity of smooth Fano threefolds**

**Speaker:** Konstantin Loginov (Steklov Mathematical Insitute of RAS)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Fano varieties are an important class of varieties studied in birational geometry. A natural way to study Fano varieties is by looking at its (pluri-)anti-canonical divisors. Coregularity measures how singular such divisors could be. We explain how to compute the coregularity of smooth Fano varieties of dimension 3.

**Bio: **Konstantin Loginov graduated from the Faculty of Mathematics and Mechanics of Lomonosov Moscow State University in 2015. He obtained PhD in the Department of Mathematics of the Higher School of Economics in Moscow in 2020. He was a scientific researcher in the Laboratory of Algebraic Geometry of the Higher School of Economics. Now he is a scientific researcher in the Department of Algebraic Geometry of Steklov Mathematical Institute and in the Moscow Institute of Physics and Technology. His research interests include algebraic geometry, especially birational geometry.

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**Lecture Series 50 ****—— April 7, 2023 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/6333F9FBDF221B6B354F73F952FB7EE8

Valid Until: 2027-05-31 23:59

**Lecture 1****——The Mathematical Theory of Neural Network-based Machine Learning**

**Speaker:** Weinan E (Peking University)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** The task of supervised learning is to approximate a function using a given set of data. In low dimensions, its mathematical theory has been established in classical numerical analysis and approximation theory in which the function spaces of interest (the Sobolev or Besov spaces), the order of the error and the convergence rate of the gradient-based algorithms are all well-understood. Direct extension of such a theory to high dimensions leads to estimates that suffer from the curse of dimensionality as well as degeneracy in the over-parametrized regime.

In this talk, we attempt to put forward a unified mathematical framework for analyzing neural network-based machine learning in high dimension (and the over-parametrized regime). We illustrate this framework using kernel methods, shallow network models and deep network models. For each of these methods, we identify the right function spaces (for which the optimal complexity estimates and direct and inverse approximation theorems hold), prove optimal generalization error estimates and study the behavior of gradient decent dynamics.

**Bio:** Weinan E is a professor in the School of Mathematical Sciences and the Center for Machine Learning Research (CMLR) at Peking University. He is also a professor at the Department of Mathematics and Program in Applied and Computational Mathematics at Princeton University. His main research interest is numerical algorithms, machine learning and multi-scale modeling, with applications to chemistry, material sciences and fluid mechanics.

Weinan E was awarded the ICIAM Collatz Prize in 2003, the SIAM Kleinman Prize in 2009 and the SIAM von Karman Prize in 2014, the SIAM-ETH Peter Henrici Prize in 2019, and the ACM Gordon-Bell Prize in 2020. He is a member of the Chinese Academy of Sciences, and a fellow of SIAM, AMS and IOP. Weinan E is an invited plenary speaker at ICM 2022, an invited speaker at ICM 2002 in Beijing, ICIAM 2007 as well as the AMS National Meeting in 2003. He has also been an invited speaker at APS, ACS, AIChe annual meetings and the American Conference of Theoretical Chemistry.

**Lecture 2****——Reflection algebras and conservativity spectra of theories**

**Speaker:** Lev Beklemishev (Steklov Mathematical Insitute of RAS)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Turing introduced progressions of theories obtained by iterating the process of extension of a theory by its consistency assertion. Generalized Turing progressions can be used to characterize the theorems of a given arithmetical theory of quantifier complexity level $\Pi^0_n$, for any specific $n$. Such characterizations reveal a lot of information about a theory, in particular, yield consistency proofs, bounds on provable transfinite induction and provably recursive functions.

The conservativity spectrum of an arithmetical theory is a sequence of ordinals characterizing its theorems of quantifier complexity levels $\Pi_1$, $\Pi_2$, \etc. by iterated reflection principles. We describe the class of all such sequences and show that it bears a natural structure of an algebraic model of a strictly positive modal logic - reflection calculus with conservativity modalities.

**Bio:** Lev Beklemishev is a Principal researcher at Steklov Mathematical Institute of the Russian Academy of Sciences, Head of the Department of Mathematical Logic, and also an Academician of the Russian Academy of Sciences (2019). He graduated from the Faculty of Mathematics and Mechanics of Lomonosov Moscow State University in 1989 and defended Ph.D. thesis in 1992. He was awarded the Moscow Mathematical Society prize in 1994 and Alexander von Humboldt Fellowship (Germany) in 1998. His research interests are in mathematical logic, in particular in proof theory, formal arithmetic, provability logic, and modal logic.

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**Lecture Series 49 ****—— March 24, 2023 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/7C0EB99D9485550897133D7A4A329D82

Valid Until: 2027-04-30 23:59

**Lecture 1****——Practical challenges in non-convex optimization.**

**Speaker:** Ivan Oseledets (Skoltech, AIRI, INM RAS)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** In this talk, I will discuss several topics. First, is the optimization over low-rank matrix and tensor manifolds, which often appear in applications. Low-rank approximation of matrices is one of the rare examples when a non-convex problem can be solved in a numerically exact way by using singular value decomposition (SVD). There also exists a large class of methods for solving optimization with low-constraints.

In the second part of the talk (if time permits), I will discuss the peculiarities of optimization with deep neural networks. The theory of such optimization is still a big mystery, with a lot of empirical results and theoretical results under unrealistic assumptions. Here I plan to highlight the main points and research directions.

**Bio:** Ivan Oseledets is a Director of the Center for Artificial Intelligence Technology, Head of the Laboratory of Computational Intelligence, Skoltech.

Ivan’s research covers a broad range of topics. He proposed a new decomposition of high-dimensional arrays (tensors) – tensor-train decomposition and developed many efficient algorithms for solving high-dimensional problems. These algorithms are used in different areas of chemistry, biology, data analysis and machine learning. His current research focuses on the development of new algorithms in machine learning and artificial intelligence such as the construction of adversarial examples, the theory of generative adversarial networks and the compression of neural networks.

Ivan Oseledets got several awards for his research and industrial cooperation, including two gold medals from the Russian Academy of Sciences (for students in 2005 and young researchers in 2009), the SIAM Outstanding Paper Prize (2018), the Russian President Award for young researchers in science and innovation (2018), Moscow Government Prize for Young Scientists (2023), Best Professor award from Skoltech (2019), the best cooperation project leader award from Huawei (2015, 2017).

**Lecture 2****——Optimization with Least Constraint Violation**

**Speaker:** Yu-Hong Dai (AMSS CAS)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** A study about theory and algorithms for nonlinear programming usually assumes the feasibility of the problem. However, there are many important practical nonlinear programming problems whose feasible regions are not known to be nonempty or not. This leads to a class of problems called optimization with least constraint violation.

Firstly, the optimization problem with least constraint violation is proved to be a Lipschitz equality constrained optimization problem and an elegant necessary optimality condition, named as L-stationary condition, is established. Properties of the classical penalty method for this Lipschitz minimization problem are developed and the proximal gradient method for the penalized problem is studied.

Secondly, the optimization problem with least constraint violation is reformulated as an MPCC problem and a local minimizer of the MPCC problem is proved to an M-stationary point. The smoothing Fischer-Burmeister function method is constructed and analyzed for solving the related MPCC problem.

Thirdly, the solvability of the dual of the optimization problem with least constraint violation is investigated. The optimality conditions for the problem with least constraint violation are established in terms of the augmented Lagrangian. Moreover, it is proved that the augmented Lagrangian method can find an approximate solution to the optimization problem with least constraint violation and has a linear rate of convergence under an error-bound condition.

Finally, the constrained convex optimization problem with the least constraint violation is considered and analyzed under a general measure function. Several other related works on the optimization problem with least constraint violation will also be mentioned.

**Bio:** Yu-Hong Dai (http://lsec.cc.ac.cn/~dyh/) is a Professor of Mathematical Optimization at the Academy of Mathematics and Systems Science (AMSS) of the Chinese Academy of Sciences (CAS). Currently, he is the President of the Association of Asia-pacific Operational Research Societies (APORS), President of the Operations Research Society of China, as well as Director of the Center for Optimization and Applications of AMSS of CAS. His research interests include continuous optimization, integer programming and applied optimization. Particularly, he is known for the Dai-Yuan nonlinear conjugate gradient method and the perfect non-convergence example for the BFGS quasi-Newton method. He is also interested in building software and attacking practical optimization problems. He received many honors including the Shiing-Shen Chern Mathematics Award, the Keng Kang Prize of Scientific Computing and the Xiao Shutie Applied Mathematics Award. He is also the invited speaker of ICM 2022.

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**Lecture Series 48 ****—— March 10, 2023 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/75E4F78ABA835368BBC9736011785A8E

Valid Until: 2027-04-30 23:59

**Lecture 1****——Energy-stable parametric finite element methods (PFEM) for geometric PDEs and applications**

**Speaker:** Weizhu Bao (National University of Singapore)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** In this talk, I begin with a review of different geometric flows (PDEs) including mean curvature (curve shortening) flow, surface diffusion flow, Willmore flow, etc., which arise from materials science, interface dynamics in multi-phase flows, biology membrane, computer graphics, geometry, etc. Different mathematical formulations and numerical methods for mean curvature flow are then discussed. In particular, an energy-stable semi-implicit parametric finite element method (PFEM) is presented in detail. Then the PFEM is extended to surface diffusion flow and anisotropic surface diffusion flow, and a structure-preserving implicit PFEM is proposed. Finally, sharp interface models and their PFEM approximations are presented for solid-state dewetting. This talk is based on joint works with Harald Garcke, Wei Jiang, Yifei Li, Robert Nuernberg, Yan Wang and Quan Zhao.

**Bio:** Weizhu BAO is a Professor at the Department of Mathematics, National University of Singapore (NUS). He got his PhD from Tsinghua University in 1995 and afterwards he had postdoc and faculty positions at Tsinghua University, Imperial College, Georgia Institute of Technology and the University of Wisconsin at Madison. His research interests include numerical methods for partial differential equations, scientific computing/numerical analysis, analysis and computation for problems from physics, chemistry, biology and engineering sciences. He has made significant contributions in modeling and simulation of Bose-Einstein condensation, solid-state dewetting and geometric PDEs; and in multiscale methods and analysis for highly oscillatory PDEs. He had been on the Editorial Board of SIAM Journal on Scientific Computing during 2009--2014 and is currently on the Editorial Board of SIAM Journal of Numerical Analysis. He was awarded the Feng Kang Prize in Scientific Computing by the Chinese Computational Mathematics Society in 2013. Weizhu Bao was an invited speaker at the ICM 2014 in Seoul. He is a Fellow of the American Mathematical Society, the Society of Industrial and Applied Mathematics and the Singapore National Academy of Science.

**Lecture 2****——Kolmogorov's theory of turbulence, the Landau criticism and their rigorous 1d versions.**

**Speaker:** Sergei Kuksin (Steklov Mathematical Institute of RAS)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Kolmogorov's theory of turbulence contains two celebrated heuristic laws, related to the second and third moments of increments of the fluid's velocity field u(t,x+r)-u(t,x) for "small but not too small r". Kolmogorov claimed that the moments as functions of r have the form (universal pre-factor) x (certain exponent of |r|). Landau's criticism was that the pre-factor indeed may be universal for the third moment, but not for the second. In my talk, I will explain that the two heuristic laws allow rigorous versions, related to a fictitious one-dimensional fluid, described by the Burgers equation. There - indeed - the pre-factor in the law for the third moment is explicit and universal, but that for the second moment cannot be such. I will explain the difference between the two moments which leads to this effect.

**Bio:** Sergey Kuksin is a Leading Scientific Researcher at Steklov Mathematical Institute of RAS, a Senior Researcher at the Mathematics Institute of Jussieu–Paris Rive Gauche and a Professor at Heriot-Watt University. He became Doctor of Sciences from Lomonosov Moscow State University in 1981. Kuksin’s research deals with KAM theory in PDE. He was an invited speaker of ICM 1998 in Berlin. In 2016, he received Lyapunov Award from the Russian Academy of Sciences.

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**Lecture Series 47 ****—— February 24, 2023 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/ECECDE7A00BF43BF844D2C0D0FA302EB

Valid Until: 2027-03-31 23:59

**Lecture 1****——Congruence of modular forms and arithmetic of Shimura varieties**

**Speaker:** Yifeng Liu (Zhejiang University)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** The congruence of modular forms is an important phenomenon in the arithmetic study of modular forms, or more generally, automorphic forms. For classical modular forms, many results have been obtained by Serre, Ribet, et al, for more than thirty years. In particular, Ribet used the arithmetic geometry of modular curves to find such congruence relation, also known as level raising. We recall as follows: Fix a prime l; consider a weight-2 level-N newform f satisfying the mod l level-raising condition at a prime p coprime to Nl. Ribet proved that the first Galois cohomology of the mod l Galois representation of Q_p associated with f can be realized as the Abel-Jacobi image of the supersingular locus of the level-N modular curve over F_p.

In ongoing joint work with Yichao Tian (MCM) and Liang Xiao (PKU), we generalize this phenomenon to higher-dimensional unitary Shimura varieties at inert places (which remains a conjecture in general), and its relation with a certain Ihara type lemma for such varieties. In the talk, I will explain cases for which we have confirmed such conjecture; and if time permits, we will mention its number-theoretical implications.

**Bio:** Yifeng Liu is a Professor at the Institute for Advanced Study in Mathematics, Zhejiang University. His primary areas of interest span algebraic geometry, automorphic representations, and number theory. He obtained his B.S. degree from Peking University in 2007 and PhD from Columbia University in 2012. He was a professor at Yale before joining Zhejiang University in 2021. Yifeng Liu received a Sloan Research Fellowship in 2017 and was awarded the SASTRA-Ramanujan Prize in 2018 for his contribution to number theory and arithmetic geometry.

**Lecture 2****——45 years of derived categories of rational homogeneous varieties**

**Speaker:** Anton Fonarev (Steklov Mathematical Institute of RAS)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** In 1978, Beilinson published a celebrated paper in which he showed that the bounded derived category of a projective space admits a full exceptional collection. Since then mathematicians have been trying to prove that the same holds for all rational homogeneous varieties. We will give a gentle introduction to this problem and give an overview of the progress that has been made.

**Bio:** Anton Fonarev is a Senior Scientific Researcher in the Department of Algebraic Geometry of Steklov Mathematical Institute of RAS. He graduated from the Faculty of Mechanics and Mathematics of Lomonosov Moscow State University in 2011 and obtained his PhD at Steklov Mathematical Institute in 2014. Then he was a scientific researcher in the Laboratory of Algebraic Geometry and in the Laboratory of Mirror Symmetry and Automorphic Forms of Higher School of Economics in Moscow. His research interests include algebraic geometry and homological algebra.

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**Lecture Series 46 ****—— January 13, 2023 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/A07BDF7E34AC9C1BD87EB37BF6B9E789

Valid Until: 2027-02-28 23:59

**Lecture 1****——Trace polynomials for closed curves on the enlarged modular torus: Monotonicity, positivity, and log-concavity**

**Speaker:** Ying Zhang (Soochow University)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** The modular torus is a once-cusped hyperbolic torus with the maximal order of symmetry. The traces of the simple closed geodesics on the modular torus give the geometric version of the classical Markoff numbers. We study trace polynomials for the closed curves on the enlarged modular torus, with the single variable the enlargement factor. We obtain partial ordering of the polynomials for certain simple closed curves. We obtain positivity of the polynomials for all closed curves. We have observed log-concavity for these polynomials and confirmed log-concavity of the polynomials for the simple closed curves. We also mention the problem to enlarge an arbitrary complete one-holed hyperbolic torus. This is joint work with Xiangfei Li.

**Bio:** Ying Zhang obtained his Ph.D. from the National University of Singapore. He is currently a professor of mathematics at Soochow University. His research interests include hyperbolic geometry and low-dimensional topology. He has contributed to the geometry of hyperbolic cone-surfaces, generalizations of McShane identity, dynamics and topology of the character varieties of surfaces, trigonometry of 4-dimensional hyperbolic space etc.

**Lecture 2****——Diagrammatic approach to the classification of knots and contact topology**

**Speaker:** Ivan Dynnikov (Steklov Mathematical Institute of RAS)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** About two decades ago I discovered that the problem of recognizing the unknot could be solved using a monotonic simplification procedure for certain type knot diagrams, which I called rectangular. They are also now known as grid diagrams. For arbitrary knots, the procedure does not work as well as it does for the unknot, since there may be many non-equivalent rectangular diagrams representing the same knot type that are not subject to any simplification by elementary moves. However, for many knot types, the set of non-simplifiable diagrams is finite, and it does not seem impossible that this is true for all knots. The classification of non-simplifiable rectangular diagrams turns out to be deeply related to the classification of Legendrian knots. This relation has been recently studied in our joint works with Maxim Prasolov and Vladimir Shastin. As a by-product of this study, we obtained an algorithm for comparing Legendrian knots, which has not been known before. In my talk, I will overview the most recent results on the subject.

The present work is supported by the Russian Science Foundation under grant~22-11-00299.

**Bio:** Ivan A. Dynnikov is a Professor at the Department of Higher Geometry and Topology, Lomonosov Moscow State University and also a Leading Scientific Researcher at Steklov Mathematical Institute of RAS. He obtained his PhD from Lomonosov Moscow State University in 1996 and became Doctor of Sciences in 2007. Dynnikov's research interests include low-dimensional topology and its applications in mathematical physics, knot theory, discrete Schrödinger operators, and the theory of dynamical systems. He was an invited speaker at the 2nd European Congress of Mathematicians (Budapest, 1996). In 2000, I.A. Dynnikov was awarded the prize of the Moscow Mathematical Society for young scientists. I.A. Dynnikov is the Scientific Secretary of the Moscow Mathematical Society and the author of more than 35 scientific publications.

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**Lecture Series 45 ****—— December 23, 2022 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/9E311B9B60E51416E32FC149EA494B27

Valid Until: 2027-01-31 23:59

**Lecture 1****——Frobenius structures from Calabi-Yau categories**

**Speaker:** Junwu Tu (ShanghaiTech University)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** Primitive forms were introduced by K. Saito in his construction of period mapping in the unfolding space of singularities. The Hodge theoretic structure involved in this construction is known as the semi-infinite Hodge structure introduced by Barannikov and Kontsevich. Following Kontsevich’s proposal in his 1994 ICM address, we shall discuss the appearance of such structures in the categorical contexts, as well as a few open problems in this direction.

**Bio:** Junwu Tu is a Professor at the Insitute of Mathematical Sciences of ShanghaiTech University. He got his bachelor’s degree from Nanjing University in 2005 and Ph.D. from the University of Wisconsin-Madison in 2011. His research centers around homological algebra and its applications in algebraic geometry, symplectic geometry, homological mirror symmetry and data sciences. Recently, he has been working on defining and understanding categorical Gromov-Witten invariants.

**Lecture 2****——Non-commutative analogue of the Berglund-Hübsch-Henningson duality and symmetries of orbifold invariants of singularities**

**Speaker:** Sabir M. Gusein-Zade (Lomonosov Moscow State University)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** The first regular construction of (conjecturally) mirror symmetric orbifolds belongs to Berglund, Hübsch and Henningson. The Berglund-Hübsch-Henningson- (BHH- for short) duality is a duality on the set of pairs (f,G) consisting of an invertible polynomial group and a subgroup G of diagonal symmetries of f. Symmetries of (orbifold) invariants of BHH-dual pairs are related to mirror symmetry. There were prooved symmetries for the orbifold Euler characteristic, orbifold monodromy zeta-function, and orbifold E-function. One has a method to extend the BBH-duality to the set of pairs (f,G^, where G^ is the semidirect product of a group G of diagonal symmetries of f and a group S of permutations of the coordinates preserving f. The construction is based on ideas of A.Takahashi and therefore is called the Berglund-Hübsch-Henningson-Takahashi- (BHHT-) duality. Invariants of BHHT-dual pairs have symmetries similar to mirror ones only under some restrictions on the group S: the so-called parity condition (PC). Under the PC-condition it is possible to prove some symmetries of the orbifold invariants of BHHT-dual pairs.

The talk is based on joint results with W.Ebeling.

**Bio:** Sabir Medzhidovich Gusein-Zade is a Professor at the Department of Higher Geometry and Topology of Lomonosov Moscow State University. He graduated from the Faculty of Mechanics and Mathematics of MSU in 1974 and defended his PhD thesis under the supervision of Sergei Novikov in 1975. In 1991, Gusein-Zade became Doctor of Physical and Mathematical Sciences. Gusein-Zade has been the faculty of the Department of Higher Geometry and Topology since 1996. His scientific interests include the theory of singularities and the topology of algebraic spaces. Prof. Gusein-Zade is the author of more than 120 publications on pure and applied mathematics (including 4 monographs). He is also the editor of the Moscow Mathematical Journal and has been the Secretary of the Moscow Mathematical Society since 1996.

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**Lecture Series 44 —— December 9, 2022 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

**To Join Zoom Meeting:**

**https://us02web.zoom.us/j/81923909450?pwd=VnBQWUNlRXdheThmM0Zwd3FqR3VsUT09**

**Meeting ID:** 819 2390 9450

**Password:** 987654

**Lecture 1****——Third-order structure function beyond the inertial range in homogeneous isotropic turbulence**

**Speaker:** Jin-Han Xie (Peking University)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** In 1941, Kolmogorov proposed the 4/5 law for the inertial range in three-dimensional isotropic turbulence, which links measurable third-order structure functions with energy transfer. The inertial-range theories are applied to quantify the direction and magnitude of energy transfer in natural turbulence. However, the applicability of these theories is limited due to the effects of forcing scale and bidirectional energy transfer. Thus, expressions for structure functions beyond the inertial range are required. We derive a forcing-scale-resolving global expression that captures the bidirectional energy transfer. We apply this expression to analyse the drifter data in the Gulf of Mexico and provide evidence for bidirectional energy transfer in ocean turbulence. Also, this new expression implies a conjugate regime to Kolmogorov's theory for the scales larger than the forcing scale in three-dimensional homogeneous isotropic turbulence. This new regime points out the importance of the energy injection even in the large scales, which was believed to be described by absolute equilibrium because of zero averaged energy flux across scales, and potentially provides a foundation for superresolution.

**Bio:** Jin-Han Xie is an Assistant Professor in the Department of Mechanics and Engineering Science, College of Engineering at Peking University. He was a Courant Instructor at the Courant Institute of Mathematical Sciences, New York University. Before that, he was a postdoctoral researcher at the physics department of the University of California, Berkeley. He obtained his PhD degree in Applied and Computational Mathematics at the School of Mathematics of the University of Edinburgh in 2015. He got his Bachelor’s degree in Theoretical and Applied Mechanics at the College of Engineering, Peking University in 2011.

Jin-Han Xie's research focuses on fluid dynamics, in particular, geophysical fluid dynamics, turbulence, magnetohydrodynamics and wave-mean flow interaction.

**Lecture 2****——Fluid mechanics and its application for describing basic processes in nature, technology and society**

**Speaker:** Nikolaj N. Smirnov (Lomonosov Moscow State University)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** The lecture provides some examples of successful application of fluid mechanics methods and approaches for predictive mathematical modeling of basic processes in nature, technology and social aspects, in particular:

1. Global processes in the Universe: formation of supernovae, interactions of molecular clouds;

2. Ecology of near-Earth space, ensuring flight safety;

3. Modeling of processes in power plants and engines of various types;

4. Modeling the effectiveness of fire and explosion safety measures;

5. Predictive modeling of traffic flows, evaluation of the effectiveness of various control strategies.

**Bio:** Nikolaj N. Smirnov is a professor at the Department of Gas and Wave Dynamics of Lomonosov Moscow State University since 1994. He graduated from the Faculty of Mechanics and Mathematics of MSU in 1976 and became Doctor of Physical and Mathematical Sciences in 1990. Prof. Smirnov is the Corresponding Member of the Russian Academy of Natural Sciences (1998), a member of the Combustion Institute (1992), a member of the Russian National Committee on Theoretical and Applied Mechanics (2001), a member of the International Academy of Astronautics (2005). He was awarded the M.V.Keldysh Medal and the S.P.Korolev Medal (Federation of Astronautics of Russia).

Research interests: dynamics of physically and chemically transforming multiphase media, fundamental problems of ecology, mathematical modeling of the evolution of space debris, and traffic flow. He has published over 200 scientific papers. 15 Ph.D. and 2 doctoral dissertations were defended under his supervision.

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**Lecture Series 43 ****—— November 25, 2022 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/C291CDF17D52919F78338D54E4AE1325

Valid Until: 2026-12-31 23:59

**Lecture 1****——Closed geodesics on flat surfaces**

**Speaker:** Weixu Su (Sun Yat-Sen University)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** Any holomorphic quadratic differentials on a compact Riemann surface induces a flat metric with conical singularities. Each regular closed geodesic on the flat metric is contained in a maximal flat cylinder. In this talk, I will survey some of our recent research on distribution of flat cylinders.

**Bio:** Weixu Su is currently a Professor at the School of Mathematics at Sun Yat-Sen University. He got his PhD at Sun Yat-Sen University in 2011 and then started to work at Fudan University where he became a professor in 2021. He specializes in Teichmüller space theory and has published relevant results on Math. Annalen, Adv. Math. etc.

**Lecture 2****——Analytical Approach to CR Geometry**

**Speaker:** Valerii K. Beloshapka (Lomonosov Moscow State University)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** In the framework of analysis of several complex variables it is natural to identify biholomorphically equivalent geometrical objects. This is appropriate for everything: domains, its boundaries, singular subsets of boundaries (Shilov boundaries), orbits of holomorphic Lie group action, etc.

A germ of a real submanifold in complex space is a highly interesting object. There are three interrelated aspects of this interest: holomorphic automorphisms of the germ, its invariants and classification. These issues belong to CR geometry, which is a domain of interplay between different directions: complex analysis, differential geometry, Lie groups and algebras, theory of differential equations, algebraic geometry, invariant theory, and so on. CR geometry takes its origin in the seminal papers of H. Poincare and E. Cartan. Since then the two approaches in it have been crystallized: analytical, which develops the ideas of Poincare, and geometrical, developing that of Cartan. The author, working in the Poincare paradigm, is going to give a survey of the modern state of the analytical branch of CR geometry.

**Bio:** Professor Valerii K. Beloshapka graduated from the Department of Function Theory and Functional Analysis of the Faculty of Mechanics and Mathematics of Moscow State University in 1975. He defended PhD thesis in 1979 and his doctoral dissertation "Description of holomorphic automorphisms of real surfaces of high codimension" in 1991. He works at the Faculty of Mechanics and Mathematics since 1992 and became a Professor at the Department of Function Theory and Functional Analysis in 1996.

The main research area of Valerii Beloshapka is real submanifolds of complex spaces, their holomorphic automorphisms, classification and invariants. This problem organically combines the methods and approaches of multidimensional complex analysis, differential geometry and algebra.

Valerii Beloshapka was an invited speaker at many international conferences. Laureate of the Prize of Mathematics Department of USSR Academy of Sciences (1989). He is a fellow of the International Science Foundation (1993). In 1998 he got State Scientific Scholarship for Outstanding Scientists.

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**Lecture Series 42 ****—— November 11, 2022 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/01616C34D28516B018A9EFD7537DD4C6

Valid Until: 2026-12-31 23:59

**Lecture 1****——Kaehler-Ricci flow on Fano G-manifolds**

**Speaker:** **Xiaohua Zhu (Peking University)**

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** I will talk about a recent work jointly with Tian on Kaehler-Ricci flow on Fano G-manifolds. We prove that on a Fano G-manifold, the Gromov-Hausdorff limit of Kaehler-Ricci flow with initial metric in $2\pi c_1(M)$ must be a Q-Fano horosymmetric variety which admits a singular Keahler-Ricci soliton. Moreover, we show that the complex structure of limit variety can be constructed by a $C^*$-degeneration induced by an element in the Cartan torus Lie algebra of G. A similar result can be also proved for Kaehler-Ricci flows on any Fano horosymmetric manifolds.

**Bio:**** Xiaohua Zhu is a Professor at the Department of Mathematics at Peking University. He graduated from Hangzhou University in 1990 and received his PhD in 1995. Professor Zhu is mainly engaged in the research of differential geometry and geometric analysis. He has deep research work in complex geometry, Ricci flow, complex Monge-Ampere equation, Ricci solitons, ring-type algebraic cluster geometry, minimal small manifolds and so on. He has published over 50 papers in journals such as Acta Math., JAMS, Duke Math., GAFA, JEMS, etc. He was awarded the Outstanding Youth Prize by Hong Kong Qiushi Foundation in 2001, the Henry Fok Education Fund of the Ministry of Education of China in 2002, the National Natural Science Fund for Outstanding Youth in 2004, the ICTP Prize for young scientists in 2005, the Second Prize of the National Natural Science Award in 2014, and the 16th Shing S. Chern Mathematics Award in 2017. In 2009, he was elected Changjiang Distinguished Professor of the Ministry of Education of China. He is an invited speaker of ICM 2022 and the editorial board member of Acta Math. Sinica and Advances in Mathematics (China).**

**Lecture 2****——Regularity of solutions to stationary Fokker-Planck-Kolmogorov equations**

**Speaker:** Vladimir Bogachev (Lomonosov Moscow State University)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** We discuss the regularity of solutions to double divergence form equations of the form

the drift coefficient. The equation is understood in the sense of distributions, so the coefficients can be rather irregular. The key problems concern the existence of solution densities and their regularity properties, and also the existence and uniqueness of probability solutions. In particular, we discuss some recent results obtained jointly with Röckner and Shaposhnikov on Zvonkin’s transform of the drift coefficient, which enables one to improve the drift.

**Bio:** Vladimir Bogachev is a professor at the Chair of Function Theory and Functional Analysis of Lomonosov Moscow State University, and also the Corresponding Member of the Russian Academy of Science. He graduated from the Faculty of Mechanics and Mathematics of Moscow State University in 1983 and has been working there since 1986. He obtained the degree of Doctor of Physical and Mathematical Sciences in 1991. Bogachev is the author of more than 200 papers in scientific journals and 15 monographs. His scientific interests include measure theory, probability theory, diffusion processes, stochastic analysis, nonlinear infinite-dimensional analysis, and Fokker-Planck-Kolmogorov equations. Vladimir Bogachev was awarded a prize from the Presidium of the USSR Academy of Sciences for young scientists (1990), a prize from the Japan Society for the Promotion of Science (2000), and the Kolmogorov prize of the Russian Academy of Sciences (2016). He is a member of the editorial boards of the journals “Analysis Mathematica”, “European Journal of Mathematics”, “Infinite-Dimensional Analysis, Quantum Probability and Related Topics”, “Theory of Probability and Its Applications”, “Izvestia Mathematics”, “Functional Analysis and Its Applications”, “Moscow Mathematical Journal”.

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**Lecture Series 41 ****—— October 28, 2022 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/6A99ED6E46A7C530759124CF292BC5D2

Valid Until: 2026-11-30 23:59

**Lecture 1****——A dynamic homogenization framework predicting the spatiotemporal nonlocality and nonuniformity of field quantities in heterogeneous media**

**Speaker:** **Jianxiang Wang (Peking University)**

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** The homogenization of heterogeneous media has been attracting the attention of scientists for more than a century. The classical homogenization methods are focused on the prediction of the effective or overall properties of heterogeneous media such as the effective elastic tensors or potentials, and conductivity tensors. Recently, the topic becomes a focus of research in mathematics, mechanics, and materials science due to its importance in the modelling of advanced composite materials, in particular, metamaterials. However, the classical methodology produces local forms of governing equations and cannot describe complex dynamic responses of various heterogeneous media such as the dispersion and bandgaps of elastic waves. This lecture will present the recent work of the author’s group on the dynamic homogenization of heterogeneous media. Starting from conventional local linear elastic constituents, we develop a dynamic homogenization framework and derive the macroscopic governing equations for heterogeneous media. The governing equations can reflect spatiotemporal nonlocality and nonuniformity of the field quantities and can correspond to conventional local models and the well-known nonlocal models including the Mindlin equation, Willis formalism, Eringen constitutive relation, and peridynamic formulation. For heat conduction, the governing equation of the average temperature can correspond to the Jeffreys-type equation, Nunziato equation, Gurtin and Pipkin equation, peridynamic formulation, and dual-phase-lag (DPL) equation. All the parameters in the governing equations can be determined from the geometrical and physical parameters of the constituents; thus the framework also sheds light on the physical mechanisms of the relevant formulations.

**Bio:**** Jianxiang Wang is currently a Changjiang Scholar Professor of mechanics in the Department of Mechanics and Engineering Science of Peking University. He received his PhD from The University of Sydney in 1995. He joined Peking University in 1998, after doing post-doctoral research at Imperial College in 1996 and Aalborg University in 1997. Jianxiang Wang’s research interests cover fracture/failure analyses of composite materials, constitutive relations and transport properties of heterogeneous materials and nano-structured materials, surface effects in heterogeneous materials and nanomaterials, and the Eshelby formalism and Eshelby conjecture. He served as secretary-general of the Chinese Society of Theoretical and Applied Mechanics (2006-2010), secretary-general of the 23rd International Congress of Theoretical and Applied Mechanics (ICTAM2012) of the International Union of Theoretical and Applied Mechanics (IUTAM), and member of Congress Committee of the IUTAM (2014-2022). Jianxiang Wang was awarded Excellent Teacher of Beijing (2009), Changjiang Scholar Professor (2008), the Royal Society Visiting Fellowship (1999, 2004), and Honorary Visiting Professor of Cardiff University (2006-2016).**

**Lecture 2****——On the rational theory of continuum mechanics**

**Speaker:** George L. Brovko (Lomonosov Moscow State University)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Modern approaches to the construction and development of a rational theory of continuum mechanics according to D. Hilbert's sixth problem are considered.

An inspiring example of rational approaches in other fields of mechanics were the famous works of J. Lagrange and P. Appel. The fundamental basis for such views is the mathematical foundations of mechanics, created in the times of I. Newton and L. Euler, and the powerful development of mechanics of deformable media, provided in the 19th century by the works of A. Cauchy, his contemporaries and followers. The 20th century was marked by the creation of a unified theory of continuum mechanics, which united various sections of the science of deformable bodies and media, and by the creation of a general theory of constitutive relations in the works of A.A. Ilyushin as well as W. Noll together with C. Truesdell, which appeared as the most important discovery of mechanics in the 20th century. All this led to the search and development of rational approaches in the theory of continuum mechanics.

The paper focuses on the axiomatization of the general theory of deformable media mechanics and its topical problems. A modern system of axioms of general Newtonian mechanics is given. All kinds of tensor mechanical characteristics of different ranks are considered, a set of tensors obeying special rules when the reference frame is replaced, called objective tensors, are marked out and their classification is proposed. The mappings that connect such characteristics are studied, the mappings that do not depend on the reference frame are distinguished, and the criterion for such independence is established. The system of axioms is constructed and the theory of generalized tensor measures of stresses and strains is developed. Axioms of the new general theory of constitutive relations are introduced, taking into account additional parameters of mechanical processes, and the new general reduced system of constitutive relations is obtained. A generalization of the notions of the process image and the Ilyushin’s isotropy postulate to the range of finite deformations is considered.

**Bio:** Graduated from the Faculty of Mechanics and Mathematics of Moscow State University (1971). Doctor of Physics and Mathematics Sciences (1996). Professor of the Department of Elasticity Theory of the Faculty of Mechanics and Mathematics of Moscow State University (1999). George L. Brovko defended his Ph.D. thesis "Analysis of the formulation and methods for solving boundary value problems in the theory of elastoplastic processes of small curvature" in 1978. Subject of doctoral dissertation "Development of the mathematical apparatus and foundations of the general theory of constitutive relations in continuum mechanics" (1996).

In January 2015, G.L. Brovko was awarded the honorary title "Honored Professor of Moscow University".

Area of scientific interests:

1. Fundamentals of the mechanics of deformable media and structures: basic concepts and axioms (postulates, hypotheses), balance laws, general theory of constitutive relations of continuum mechanics.

2. Development of the tensor apparatus of continuum mechanics, the use of elements of the theory of invariants and the invariance properties of tensors and tensor relations. Objective kinematic and dynamic tensor processes in continuum mechanics, their mappings and equations of relations between them, independent of the frame of reference.

3. Modeling of properties of elastic and inelastic bodies under finite deformations (theoretical models, numerical simulation). Classical and non-classical models of microheterogeneous media: filled porous bodies, moment media (Cosserat continuum), nanostructures.

4. Application of the apparatus of functional analysis and modern theory of differential equations in problems of mechanics: analysis of formulations and methods for solving boundary value problems of mechanics of a deformable solid body (existence and uniqueness of solutions in functional spaces, theoretical substantiation of numerical methods).

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**Lecture Series 40 ****—— October 14, 2022 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/30CE439C8E7E695A13394DCD1E8D74AA

Valid Until: 2024-11-30 23:59

**Lecture 1****——Special unipotent representations of classical Lie groups**

**Speaker:** Binyong Sun (Zhejiang University)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** One fundamental problem in representation theory is the unitary dual problem, namely to construct and classify all irreducible unitary representations of a given Lie group G. An important principle is the orbit method introduced by A. A. Kirillov, and it seeks to describe irreducible unitary representations of G by its coadjoint orbits. The most mysterious ingredient of orbit method is to attach irreducible unitary representations to nilpotent coadjoint orbits. For classical Lie groups, we construct some irreducible unitary representations attached to nilpotent coadjoint orbits, by using the theory of local theta correspondence initiated by R. Howe. These are the special unipotent representations in the sense of Arthur and Barbasch-Vogan. This is a report on a recent joint work with Dan M. Barbarsch, Jia-Jun Ma and Chen-Bo Zhu.

**Bio:**** Binyong Sun received his bachelor's degree from Zhejiang University in 1999, and doctorate degree from the Hong Kong University of Science and Technology in 2004. After a short postdoctoral experience at the Swiss Federal Institute of Technology Zurich, he worked at the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences since 2005. In 2020, he then joined the Institute for Advanced Study in Mathematics (IASM) at Zhejiang University as a permanent member.**

Binyong Sun’s research interests include representation theory of Lie groups and the theory of automorphic forms. By proving some long-standing conjectures, he has established several deep and fundamental results for representations of classical groups. He received the Tan Kah Kee Young Scientist Award in 2014, the Outstanding Youth Science and Technology Talent Award in 2016, and the State Natural Science Award (second class) in 2018. In 2019, he was elected a member of the Chinese Academy of Sciences.

**Lecture 2****——Derived categories of complex manifolds, their DG-enhancement and Bott-Chern classes.**

**Speaker:** Alexey Bondal

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** I will describe the following series of results, partly obtained in collaboration with A. Rosly, about derived categories of coherent sheaves on complex manifolds. For a general complex torus X, D^b(Coh-X) has a semiorthogonal decomposition, and it is not equivalent to D^b_{coh}(O_X-mod). There is a twist-closed DG-enhancement of the latter category by dbar-superconnections for any smooth compact complex manifold. This DG-enhancement allows us to define Bott-Chern cohomology for any object of D^b_{coh}(O_X-mod), in particular, for a coherent sheaf. If time permits, I will describe the extension of the enhancement theorem to the case of non-compact complex manifolds and applications to constructing the moduli space of objects in the above category.

**Bio:** Prof. Bondal received his Ph.D. in Steklov Mathematical Institute of RAS in Moscow in 1989. He was an associate professor at Moscow State University from 1990 to 1994. Prof. Bondal joined Steklov Institute in 1994, where he was promoted to a Leading Researcher. He was an invited speaker at the ICM 2022 in Beijing. His research interests include Algebraic and Complex Geometry, Homological Algebra and Representation Theory. Currently, he is visiting the Institute for Physics and Mathematics of the Universe, Tokyo University, Japan.

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**Lecture Series 39 ****—— September 30, 2022 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/687DE2295D4131AA019483A54A43080B

Valid Until: 2026-11-30 23:59

**Lecture 1****——Approximation of long time statistical properties of large dissipative chaotic dynamical systems**

**Speaker:** Prof. Xiaoming Wang

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** It is well-known that physical laws for large chaotic systems are revealed statistically. We consider temporal and spatial approximations of stationary statistical properties of dissipative chaotic dynamical systems. We demonstrate that appropriate temporal/spatial discretization viewed as discrete dynamical system is able to capture asymptotically the stationary statistical properties of the underlying continuous dynamical system provided that appropriate Lax type criteria are satisfied.

We also show a general framework on when the long-time statistics of the system can be well-approximated by BDF2 based schemes.

Application to the infinite Prandtl number model for convection as well as the two-dimensional barotropic quasi-geostrophic equations will be discussed.

**Bio:**** Prof. Wang received his Ph.D. in Applied Mathematics from Indiana University - Bloomington in 1996. He was a postdoctoral fellow / Courant Instructor at the Courant Institute from 1996 to 1998. Dr. Wang joined Iowa State University in 1998 where he was promoted to Associate Professor with Tenure in 2001. He moved to Florida State University in 2003 where he was promoted to Tenured Professor and served as the Chair of the Math Department at Florida State University before he returned to his motherland in 2017. He is currently a Chair Professor of Mathematics and the Chair of the Department of Mathematics at Southern University of Science and Technology.**

Prof. Wang's current research focuses on modern applied mathematics, especially problems related to fluid dynamics, groundwater research, geophysical fluid dynamics and turbulence, and big data and machine learning. He develops and utilizes tools from Partial Differential Equations, Dynamical Systems, Stochastic Analysis, Numerical Analysis and Scientific Computing in his research. A distinctive feature of his work is the combination of rigorous mathematics with genuine physical applications.

**Lecture 2——Implicit finite difference schemes for viscous gas dynamics problems**

**Speaker:** Prof. Georgy Kobelkov (Lomonosov Moscow State University)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** In one-dimensional case implicit monotone finite difference schemes are proposed to approximate gas dynamics problems. It is proved that a solution to FDEs exists and is unique. The schemes proposed can be extended to many-dimensional case. Numerical solution of some gas dynamics problems is demonstrated.

**Bio:** Georgy Kobelkov graduated from high school with a gold medal. In 1970 he graduated from the Faculty of Mechanics and Mathematics of Lomonosov Moscow State University in the Department of Computational Mathematics. He completed his thesis work under the guidance of Academician A.N.Tikhonov. He defended his Ph.D. thesis in 1975 (supervisor - Academician N.S.Bakhvalov). Doctor of Physical and Mathematical Sciences since 1985. Professor since 1987. Since 1991 he has been the head of the laboratory of computer modeling at the Faculty of Mechanics and Mathematics of Moscow State University. Since 2007 he has been the head of the Department of Computational Mathematics, a leading researcher at the Marchuk Institute of Computational Mathematics of RAS.

Awards:

Laureate of the Prize of the Department of Mathematics of the USSR Academy of Sciences (1989)

Laureate of the Prize. M. V. Lomonosov (1998)

Laureate of the Prize. M. V. Lomonosov for pedagogical activity (2010)

******

**Lecture Series 38#2 —— September 23, 2022 (16:30-17:30 Beijing time, 11:30-12:30 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/06B5C5135448C373F00DD5B545AB29D3

Valid Until: 2026-10-31 23:59

**Lecture —— Normalized tangent bundle, pseudoeffective cone and varieties with small codegree**

**Speaker:** Baohua Fu

**Time: **16:30-17:30 Beijing time (11:30-12:30 Moscow time)

**Abstract:** We propose a conjectural list of Fano manifolds of Picard number one whose normalized tangent bundle is pseudoeffective and we prove it in various situations by relating it to the complete divisibility conjecture of Russo and Zak on varieties with small codegrees. The pseudoeffective cone of the projectivized tangent bundle of a rational homogeneous space of Picard number one is explicitly determined by studying the total dual VMRT and the geometry of stratified Mukai flops. This is a joint work with Jie LIU.

**Bio: **Baohua Fu is a professor in the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences. He got his bachelor's degree from Peking University in 1998 and then PhD from Nice University in 2003 under the tuition of A. Beauville. He was the charge de recherche at CNRS from 2004 to 2010. His research interests are focused on symplectic singularities and the geometry of Fano manifolds. He was awarded the National Science Fund for Distinguished Young Scholars in 2013. He received the Outstanding Youth Award of the Chinese Academy of Sciences in 2017 and the ICCM Silver Medal of Mathematics in 2019.

******

**Lecture Series 38#1 —— September 16, 2022 (16:00-17:00 Beijing time, 11:00-12:00 Moscow time)**

Recording: **https://disk.pku.edu.cn:443/link/9998D8D5ADA38FF0C8E46E3CEBE92E8C**

Valid Until: 2026-10-31 23:59

**Lecture —— Rationality problem for conic bundles**

**Speaker: **Yuri Prokhorov

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** A conic bundle is a flat morphism $f: X\to Z$ of smooth algebraic varieties whose fibers are plane conics. I will discuss the problem of rationality of algebraic varieties having conic bundle structures. First, I recall almost classical results on birational properties of surface conic bundles over non-closed fields. Then I concentrate on the three-dimensional case. The main focus will be on the conjectural criterion of rationality.

**Bio: **Yuri Prokhorov is a Chief Scientific Researcher of the Department of Algebraic Geometry of the Steklov Mathematical Institute of the Russian Academy of Sciences and a Professor at the Lomonosov Moscow State University. He is a Corresponding Member of the Russian Academy of Sciences. He was an invited speaker at the 8th European Congress of Mathematics in 2021. He was an invited speaker at the ICM 2022 in the section Algebraic and Complex Geometry. His research interests are in algebraic geometry (especially birational geometry).

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**Lecture Series 37 —— June 10, 2022 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/53480B83434362AFD4CB41F2AAB7FF7D

Valid Until: 2026-07-31 23:59

**Lecture 1——Some topics in quantum control**

**Speaker:** Prof. Alexander Pechen (Steklov Mathematical Institute of RAS)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** Quantum control, that is control of individual quantum systems, attracts now high attention both due to fundamental interest and various existing and prospective applications in quantum technologies. In practical applications, controlled systems typically interact with the external environment, so that they are open quantum systems. We will discuss incoherent control of open quantum systems which uses spectral density of the environment as a tool to manipulate the system's dynamics, including a recent analysis of controllability and finding of unreachable sets of states for a qubit interacting with the environment. The controllability of controlled system is the first main question which should be answered. The next question is how difficult or easy to find optimal controls. This question is related to the analysis of quantum control landscapes. We will discuss various results on the analysis of quantum control landscapes, including a recent finding of trap-free behavior for single qubit phase-shift gate generation. In the context of gradient optimization for quantum technologies, we will discuss a convenient parametrization of quantum channels by points of the complex Stiefel manifold. Another topic is the control by measurements, including by back-action of non-selective measurements and feedback control, with possible applications to energy transfer in quantum photosynthesis.

**Bio: **Alexander Pechen is a Professor of the Russian Academy of Sciences. He is the head of the Department of Mathematical Methods for Quantum Technologies at Steklov Mathematical Institute of the Russian Academy of Sciences. He graduated from the Physical Department of Moscow State University in 2001. He obtained Ph.D. degree in Mathematical Physics from Steklov Mathematical Institute in 2004. From 2005 to 2010, he worked at Princeton University. Alexander Pechen is a laureate of the Blavatnik Award for Young Scientists (USA, 2009), and the Award of the Moscow Government for Young Scientists in Mathematics, Mechanics, and Informatics for "outstanding results in the theory of quantum control" (2013). His research interests include diverse topics in mathematics of quantum technologies, dynamics and control of quantum systems.

**Lecture 2——On stochastic PDE control**

**Speaker:** Prof. Xu Zhang (Sichuan University)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** In this talk, I will give a short introduction to control theory for stochastic distributed parameter systems (governed by stochastic differential equations in infinite dimensions, typically by stochastic PDEs). I will explain the new phenomena and difficulties in the study of controllability and optimal control problems for these sorts of equations. In particular, I will show by some examples that both the formulation of corresponding stochastic control problems and the tools to solve them may differ considerably from their deterministic/finite-dimensional counterparts, and one has to develop new methods, say, the stochastic transposition method introduced in our previous works, to solve some problems in this field.

**Bio:** Xu Zhang is a professor at the School of Mathematics, Sichuan University, Chengdu, China. He was an invited speaker at ICM (Control Theory & Optimization Section, 2010). He is/was the editor in chief/corresponding editor/associate editor for several journals including Mathematical Control and Related Fields, ESAIM: Control, Optimisation and Calculus of Variations, SIAM Journal on Control and Optimization, Annual Reviews in Control, etc. His research interests include mathematical control theory, related partial differential equations and stochastic analysis.

**Slides:** **/pub/zesxzx2021/docs/20220613165040978280.pdf**

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**Lecture Series 36 ****—— May 27, 2022 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/297EE2DDEAF407BA14690A9EC8850B8E

Valid Until: 2026-06-30 23:59

**Lecture 1****——Product structure and regularity theorem for totally nonnegative flag varieties**

**Speaker:** Xuhua He (CUHK)

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this talk, we introduce a (new) $J$-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.

We show that the $J$-totally nonnegative flag variety has a cellular decomposition into totally positive $J$-Richardson varieties. Moreover, each totally positive $J$-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive $J$-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the $J$-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. Combined with the generalized Poincare conjecture established by Smale, Freedman and Perelman, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam.

This talk is based on a joint work with Huanchen Bao.

**Bio:** Dr. Xuhua He is currently the Choh-Ming Li Professor of Mathematics at the Chinese University of Hong Kong (CUHK). Prof. He received his B.S. degree in mathematics from Peking University in 2001, and Ph.D. degree from Massachusetts Institute of Technology in 2005. Before joining CUHK, he used to work at the State University of New York at Stony Brook (2006-2008), Hong Kong University of Science and Technology (2008-2014), and the University of Maryland (2014-2019). Moreover, he was a von Neumann Fellow at the Institute for Advanced Study for the academic year 2016–2017, and a Simons Visiting Professor at the Université Sorbonne Paris Nord (Paris 13 University) in 2017. For his outstanding achievements in the fields of algebraic groups, representation theory, and arithmetic geometry, he was an invited speaker at the ICM (2018) and was awarded the Morningside Gold Medal of Mathematics in 2013 and the AMS Chevalley Prize in Lie Theory in 2022.

**Lecture 2****——Group varieties and group structures**

**Speaker:** Vladimir Popov (Steklov Mathematical Institute of RAS)

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Since group operations of algebraic groups agree with the structure of their underlying varieties, there must be a dependence between them. A striking illustration of it is the classical theorem about commutativity of every connected algebraic group whose group variety is complete. In an explicit or implicit form, this problem was considered in the classical papers of A. Weil, C. Chevalley, A. Borel, A. Grothendieck, M. Rosenlicht, M. Lazard. This talk is aimed to discuss to what extent the group variety of a connected algebraic group or the group manifold of a connected real Lie group determines its group structure.

**Bio:**** Prof. Vladimir Popov is a Chief Scientific Researcher at Steklov Mathematical Institute of RAS. He is a Corresponding Member of the Russian Academy of Sciences. His research interests include algebraic transformation groups, invariant theory, algebraic groups, Lie groups, Lie algebras and their representations, algebraic geometry, automorphism groups of algebraic varieties, and discrete reflection groups.**

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**Lecture Series 35 ****—— May 13, 2022 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/C48549B953F682393E90162B22FBE108

Valid Until: 2026-06-30 23:59

**Lecture 1****——Evolution of the spatial heterogeneity of metallic glasses and its correlation with the macroscopic visco-plasticity**

**Speaker:** Xiaoding Wei (College of Engineering, Peking University)

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** Amorphous alloys (also known as metallic glasses) have no long-range order as crystalline counterparts, thus lacking traditional defects such as dislocations and grain boundaries to facilitate plastic deformation and strain hardening mechanisms. Instead, they have intrinsic hierarchical spatial heterogeneity inherited from the material preparation processes. More importantly, this spatial heterogeneity evolves with temperature and stress fields. In this study, we propose a chemo-mechanical constitutive law for metallic glasses that can describe the evolution of their spatial heterogeneity and establish the connection between this evolution with the macroscopic plastic deformation. Furthermore, our constitutive law reveals the underlying micro-mechanisms of metallic glasses under creep, relaxation, and fatigue.

**Bio:** Dr. Xiaoding Wei received his B.S. degree from the Department of Modern Mechanics at the University of Science and Technology of China in 2003, and Ph.D. degree from the Department of Mechanical Engineering at Columbia University in 2009. Then, Dr. Wei worked as a postdoc researcher at Northwestern University until he joined the department of Mechanics and Engineering Science at Peking University in 2016. His research interests include the fundamental mechanics of low-dimensional materials, crystalline and amorphous metals, and bio-inspired materials. His achievements include measuring the intrinsic strength of monolayer graphene for the first time in the world. Dr. Wei has published 55 peer-reviewed papers in Science, Nature Communications, Journal of the Mechanics and Physics of Solids, etc.

**Lecture 2****——****Stability of the aneurysm in a membrane tube with localized wall thinning filled with a fluid with a non-constant velocity profile**

**Speaker:** Andrej Il'ichev (Steklov Mathematical Institute of RAS)

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** We perform the stability analysis of bulging localized structures on the wall of a fluid-filled axisymmetric membrane elastic tube. The wall of the tube is assumed to be subjected to localized thinning. The problem has no translational invariance anymore, hence the stability of a bulging wave centered in the point of the localization of imperfection is essential, and not orbital stability up to a shift as in the case of translationally invariant governing equations. Localized bulging motionless wave solutions of the governing equations are called aneurysm solutions. We assume that the fluid is subjected to the power law for viscous friction of a non-Newtonian fluid, though the viscosity of the fluid does not play a significant role and can be neglected. The velocity profile remains not constant along the cross section of the tube (even in the absence of the viscosity) because no-slip boundary conditions are performed on the tube walls. Stability is established by demonstrating the non-existence of the unstable eigenvalues of the linearized problem with a positive real part. This is achieved by constructing the Evans function depending only on the spectral parameter, analytic in the right half of the complex plane Ω^{+} and which zeroes in Ω^{+} coincide with the unstable eigenvalues of the problem. The non-existence of the zeroes of the Evans function is performed using the argument principle from the analysis of complex variables. Finally, we discuss the possibility of applying the results of the present analysis to the aneurysm formation in damaged human vessels under the action of internal pressure.

**Bio: **Professor Andrej Il'ichev is currently a Leading Scientific Researcher at Steklov Mathematical Institute of RAS and a Professor at Bauman Moscow State Technical University. He graduated from the Faculty of Mechanics and Mathematics of Lomonosov Moscow State University in 1981 and got his Ph.D. degree there in 1986. In 1996, he became Doctor of Sciences. Professor Il'ichev’s research field includes nonlinear waves, dissipative and dispersive systems, Hamiltonian systems, dynamical stability of bound states, solitary waves, and qualitative theory of differential equations. Also, he has published more than 100 papers and 3 monographs.

******

**Lecture Series 34 ****—— April 29, 2022 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/21483613E7EBAD10312B51853959F8A0

Valid Until: 2026-06-30 23:59

**Lecture 1****——Finiteness problem for hyper-Kähler varieties**

**Speaker:** Zhiyuan **Li**

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** The classical Shafarevich conjecture is about the finiteness of isomorphism classes of curves of given genus defined over a number field with good reduction outside a finite collection of places. It plays an important role in Falting’s proof of the Mordell conjecture. Similar finiteness problems arise for higher dimensional varieties. In this talk, I will talk about finiteness problems for hyper-Kähler varieties in arithmetic geometry. This includes the unpolarized Shafarevich conjecture for hyper-Kähler varieties the cohomological generalization of the Shafarevich conjecture by replacing the good reduction condition with the unramifiedness of the cohomology. I will also explain how to generalize Orr and Skorobogatov’s finiteness result on K3 surfaces to hyper-Kähler varieties, i.e. the finiteness of geometric isomorphism classes of hyper-Kähler varieties of CM type in a given deformation type defined over a number field with bounded degree. This is a joint work with Lie Fu, Teppei Takamatsu and Haitao Zou.

**Bio:** Zhiyuan Li is an associated professor at Shanghai Center for Mathematical Sciences. His research interests are algebraic geometry and arithmetic geometry.

**Lecture 2——Automorphisms of algebraic surfaces**

**Speaker:** Сonstantin Shramov (Steklov Mathematical Institute of RAS)

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** I will discuss boundedness properties for finite subgroups in the groups of (birational) automorphisms of algebraic surfaces. The main focus will be on the Jordan property of such groups and its analogs suitable for fields of positive characteristics.

**Bio: **Constantin Shramov is a Leading Scientific Researcher at the Department of Algebraic Geometry of the Steklov Mathematical Institute of the Russian Academy of Sciences. His research interests are algebraic geometry, especially birational geometry.

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**Lecture Series 33 ****—— April 15, 2022 (16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/A9CD1CB87FAFA66E664594E1F9E4CA14

Valid Until: 2026-05-31 23:59

**Lecture 1****——****Fullerenes, geometric combinatorics and hyperbolic geometry.**

**Speaker:** **Victor Buchstaber (Steklov Mathematical Institute of RAS and Lomonosov Moscow State University)**

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** At present, the fullerene classification problem is well-known and is important due to the applications in chemistry, physics, biology and nanotechnology. A mathematical fullerene is a three-dimensional convex simple polytope with all 2-faces being pentagons and hexagons. The talk is devoted to fundamental connections between the mathematical theory of fullerenes and geometric combinatorics, graph theory, the four-color problem, Coxeter groups, Pogorelov polyhedra, and Lobachevsky geometry. We present applications to the fullerene classification problem and to the classical problem of three-dimensional hyperbolic manifolds.

**Bio:** **Prof. Victor Buchstaber is a Chief Scientific Researcher at Steklov Mathematical Institute of RAS and a Professor at Lomonosov Moscow State University. He is a Corresponding Member of the Russian Academy of Sciences. His research interests are geometry, topology and integrable systems**.

**Lecture 2****——Exotic phenomena on 4-manifolds that survive a stabilization**

**Speaker:**** Jianfeng Lin**

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Starting in dimension 4, there is a significant difference between the category of smooth manifolds and the category of topological manifolds. Such phenomena are called the "exotic phenomena". In dimension 4, there is an extra complication due to the failure of the h-cobordism theorem (in the smooth category). Stabilization on 4-manifolds means doing connected-sum with S2 cross S2. This operation naturally appears when one tries to adapt the proof of h-cobordism theorem in dimension 4. In the 1960s, Wall discovered an important principle: all exotic phenomena on orientable 4-manifolds will eventually disappear after sufficiently many stabilizations. Since then, it has been a fundamental problem to search for exotic phenomena that survive one stabilization. In this talk, we will discuss relevant backgrounds and show that such phenomena actually exist by proving the following two results (1) There exists a pair of diffeomorphisms on a 4-manifold that are topologically isotopic but not smoothly isotopic even after one stabilization. (2) There exists a pair of properly embedded surfaces in a 4-manifold with boundary which are topologically isotopic but not smoothly isotopic even after one stabilization (a part of the talk is based on the joint work with Anubhav Mukherjee).

**Bio: **Jianfeng Lin is an associate professor at Yau Mathematical Sciences Center, Tsinghua University. His research focuses on mathematical gauge theory, Floer homology and homotopy theory, and their applications in low dimensional topology.

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**Lecture Series 32 ****—— April 1, 2022 (****16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/EA0137B7FF3499884E6386E50694C639

Valid Until: 2026-05-31 23:59

**Lecture 1****——Quantum information related physics and mathematics**

**Speaker:** Shaoming Fei

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** We introduce quantum information processing and the related physics and mathematics, including quantum coherence, quantum correlations, information masking, quantum uncertainty relations, as well as tensor network compressed sensing and machine learning.

**Bio:** Prof. Shao-Ming FEI, School of Mathematical Sciences, Capital Normal University, Beijing. Current research areas: quantum information & computation, and the related fundamental problems in quantum physics.

**Lecture 2****——On classical capacity of quantum channels**

**Speaker:**** Grigory Amosov** **(Steklov Mathematical Institute of RAS)**

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** The quantum coding theorem proved in 1996 independently by A.S. Holevo, B. Schumacher and M.D. Westmoreland sets an upper bound on the number of states of a quantum system that can be used to encode classical information, so that the information can be asymptotically accurately restored after transmission. Such the quantity is known as a classical capacity of quantum channel. Due to the presence of entangled states in a composite quantum system that are not simple tensors, calculating the classical capacity turned out to be an extremely technically difficult task. An important example is given by channels generated by projective unitary representations of finite groups. This class includes Weyl channels, which, in turn, cover all unital qubit channels. Recently I found examples of channels belonging to this class for which the classical capacity can be calculated explicitly. The proof relies on the Karamata majorization method for probability distributions.

**Bio: **Dr. Grigory Amosov is a Leading Scientific Researcher at **Steklov Mathematical Institute of RAS. His research interests are **non-commutative probability theory and its applications in quantum theory of information and statistical decisions.

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**Lecture Series 31 —— March 18, 2022 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/16EF94A12F9984919BAED5B0FEF3F4C7

Valid Until：2026-04-30 23:59

**Lecture 1——Presymplectic gauge PDEs and Batalin-Vilkovisky formalism**

**Speaker:** Maxim Grigoriev

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** Gauge PDE is a geometrical object underlying what physicists call a local gauge field theory defined in terms of BV-BRST formalism. Although gauge PDE can be defined as a PDE equipped with extra structures, the generalization is not entirely straightforward as, for instance, two gauge PDEs can be equivalent even if the underlying PDEs are not. As far as Lagrangian gauge systems are concerned the powerful framework is provided by the Batalin-Vilkovisky (BV) formalism on jet-bundles. However, just like in the case of usual PDEs it is difficult to encode the BV extension of the Lagrangian in terms of the intrinsic geometry of the equation manifold while working on jet-bundles is often very restrictive, especially in analyzing boundary behavior, e.g., in the context of AdS/CFT correspondence. We show that BV Lagrangian (or its weaker analogs) can be encoded in the compatible graded presymplectic structure on the gauge PDE. In the case of genuine Lagrangian systems this presymplectic structure is related to a certain completion of the canonical BV symplectic structure. A presymplectic gauge PDE gives rise to the BV formulation through an appropriate generalization of the Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) sigma-model construction followed by taking the symplectic quotient. The construction is illustrated on the standard examples of gauge theories with particular emphasis on the Einstein gravity, where this naturally leads to an elegant presymplectic AKSZ representation of the BV formulation for the Cartan-Weyl Lagrangian.

**Bio:** Maxim Grigoriev is the deputy director of Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University. Maxim Grigoriev's scientific interests include mathematical methods for describing gauge systems (dynamical constraints and symmetry, Batalin-Vilkovisky quantization), higher spin gauge theories, holography, sigma models in superstring theory, and noncommutative theories. He proposed the so-called parent formulation of gauge theories, which systematically combines the Batalin-Vilkovisky and Hamiltonian BRST approaches into a single formalism having the structure of the Aleksandrov-Kontsevich-Schwartz-Zaboronsky (AKSZ) (generalized) sigma model.

**Lecture 2——Khovanov skein homology for links in the thickened torus**

**Speaker: **Yi Xie

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Khovanov homology is a powerful combinatorial invariant for links in the 3-sphere. Asaeda, Przytycki and Sikora defined a generalization of Khovanov homology for links in thickened compact surfaces. In this talk, we will review their definition and show that the Asaeda-Przytycki-Sikora homology detects the unlink and torus links in the thickened torus. This is joint work with Boyu Zhang.

**Bio:** Yi Xie is an assistant professor at Beijing International Center for Mathematical Research, Peking University. His research focuses on mathematical gauge theory and its application in low-dimensional topology.

**Slides: ../../docs/20220322160158893897.pdf**

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**Lecture Series 30 —— March 4, 2022 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/0951A197D2D64E8E3847297BA19E5C4B

Valid Until：2026-04-30 23:59

**Lecture 1——Algebraic semantics for modal logic with propositional quantifiers**

**Speaker:** Yifeng Ding

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** Algebraic semantics for modal logic with propositional quantifiers interpret formulas not just as true or false at some possible states, but directly as elements in a Boolean algebra, understood as an algebra of propositions. This general perspective allows us to study modal logics with propositional quantifiers that standard relational semantics based on states (possible worlds) cannot even define. In this talk, I will showcase the use of algebraic semantics in identifying conceptually significant modal logics with propositional quantifiers and also in proving mathematical/computational properties of such logics.

**Bio:** Yifeng Ding is an assistant professor at the Department of Philosophy at Peking University. He obtained his Ph.D. in Logic and the Methodology of Science from UC Berkeley in 2021. Before that, he received BA in philosophy and economics from Peking University in 2015.

He works mainly in modal logic, with serious interests also in decision theory and social choice theory. In modal logic, he has published works on logics for different kinds of knowledge, theories of non-normal modal logics, logics with propositional quantifiers, and comparative logics for probabilistic reasoning and set theory. In social choice, he is working on the axiomatization of several margin-graph-based voting methods.

**Lecture 2——Circular proofs for non-classical logics**

**Speaker:** Stepan Kuznetsov

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Usually, in a logical proof a cycle, that is, using the statement we wish to prove as an argument towards proving it, is considered incorrect (circulus vitiosus). There is, however, a vividly developing field of study of circular proofs, where such cycles are allowed to be used, under certain conditions, without losing logical validity. In this talk, we survey several applications of this approach in non-classical logics, namely, modal logics and substructural logics with Kleene star.

**Bio:** Stepan Kuznetsov is a senior scientific researcher at Steklov Mathematical Institute, RAS. He currently offers courses in mathematical logic at MSU; and also courses in computer sciences at HSE University.

******

**Lecture Series 29 —— January 21, 2022 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/297519F065FD06FE91B6405B19FE9693

Valid Until: 2026-06-30 23:59

**Lecture 1——Real algebraic and real pseudoholomorphic curves**

**Speaker:** Stephan Orevkov (Steklov Mathematical Institute of RAS; Paul Sabatier University, Toulouse)

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** According to Gromov's theory, smooth symplectic 2-surfaces in CP^2 share many properties with complex algebraic curves. The same phenomenon takes place in the real case. Namely, smooth symplectic surfaces invariant under the complex conjugation (we call them real pseudoholomorphic curves) have many common properties with plane projective real algebraic curves.

An open question (Symplectic Isotopy Problem): does each connected component of the space of symplectic surfaces contain an algebraic curve? The same question can be asked in the real case and a negative answer will be given in the talk. We shall prove certain inequalities for the complex orientations of plane real algebraic curves which are not satisfied by an infinite series of real pseudoholomorphic curves.

**Bio:** Stephan Orevkov, PhD (Phys&Math), is a senior researcher at Steklov Institute of Mathematics, a lead researcher at MIPT, and also a researcher at Université Toulouse III - Paul Sabatier, France. His academic interests include topology of flat real algebraic curves and surfaces, the theory of braids, complex surface mapping (as applicable to the Jacobian hypothesis).

**Lecture 2——On Ahlfors currents**

**Speaker:** Song-Yan Xie

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** We answer a basic question in Nevanlinna theory that Ahlfors currents associated to the same entire curve may be nonunique. Indeed, we will construct one exotic entire curve which produces infinitely many cohomologically different Ahlfors currents. Moreover, concerning Siu's decomposition, for an arbitrary positive integer k or k=infinity, some of the obtained Ahlfors currents have singular parts supported on k irreducible curves. In addition, they can have nonzero diffuse parts as well, which answers a question of Brunella. This is joint work with Dinh Tuan Huynh.

**Bio:** Song-Yan Xie got his Ph.D. from Paris-Sud (Orsay) University in 2016. In his thesis he proved an ampleness conjecture of Debarre—— the cotangent bundles of a large class of complete intersections are ample. He is currently an associate professor at the Academy of Mathematics and Systems Science. His research interest is complex geometry, especially complex hyperbolicity and Nevanlinna theory.

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**Lecture Series 28 —— December 17, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/63AB27F30444C926D8DADF2F74213506

Valid Until：2026-04-30 23:59

**Lecture 1——Quantization and Index Theory**

**Speaker:** Si Li

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** In the first part, we introduce some basic ideas and various recent mathematical developments about quantization that arises from quantum field theory and string theory. In the second part, we discuss several applications to geometry and topology. In particular, we present an effective quantization theory for 2d chiral field theories, and explain its connection with elliptic chiral homology of chiral algebras and index theory.

**Bio:** Si Li got his Ph.D. in mathematics from Harvard University in 2011. He is currently professor at Yau Mathematical Sciences Center (YMSC), Tsinghua University. He works on algebraic and geometric aspects of quantum field theory and string theory.

**Lecture 2——Korevaar–Schoen's energy on strongly rectifiable spaces**

**Speaker:** Alexander Tyulenev (Steklov Mathematical Institute of RAS)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** We extend Korevaar-Schoen’s theory of metric valued Sobolev maps to cover the case of the source space being an RCD-space. When the target space is CAT(0) we establish that the corresponding energy functional is convex, lower semicontinuous and admits a unique minimizer, in line with the smooth situation. The talk is based on the joined work: Nicola Gigli, Alexander Tyulenev, “Korevaar–Schoen's energy on strongly rectifiable spaces”, Calc. Var. Partial Differential Equations, 60 (2021), 235, 54 pp., arXiv: 2002.07440

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**Lecture Series 27 —— December 03, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/B1A478AB131AF84A7525B8A3083A54B1

Valid Until：2026-04-30 23:59

**Lecture 1——****Tikhonov's solution to a class of linear systems equivalent within perturbations**

**Speaker:** Eugene Tyrtyshnikov (INM RAS, Moscow)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **A standard approach to incorrect problems suggests that a problem of interest is reformulated with the knowledge of some additional a priori information. This can be done by several well-known regularization techniques. Many practical problems are successfully solved on this way. What does not still look as completely satisfactory is that the new reset problem seems to appear rather implicitly in the very process of its solution.

In 1980, A.N. Tikhonov proposed a reformulation that arises explicitly before the discussion of the solution methods. He suggested a notion of normal solution to a family of linear algebraic systems described by a given individual system and its vicinity comprising perturbed systems, under the assumption that there are compatible systems in the class notwithstanding the compatibility property of the given individual system. Tikhovov proved that the normal solution exists and unique. However, a natural queston about the correctness of the reset problem was not answered. In this talk we address a question of correctness of the reformulated incorrect problems that seems to have been missed in all previous considerations. The main result is the proof of correctness for Tikhonov's normal solution. Possible generalizations and diffculties will be also discussed.

**Bio: **Marchuk Institute of Numerical Mathematics of RAS,director, Academician of RAS.

**Lecture 2— Liouville Properties of the Incompressible Navier-Stokes Equations**

**Speaker: **Zhen Lei

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** In this talk we will report recent results on the Liouville properties of bounded ancient solutions to the three-dimensional incompressible Navier-Stokes equations. These properties are crucial for excluding potential type I singularities or understanding the structure of possible singulairites of local smooth solutions of the corresponding Cauchy problem.

**Bio:** Dr. Zhen Lei is a distinguished professor of School of Mathematical Sciences at Fudan University. His honors include: Second-prize Winner of the National Prize of Natural Sciences; Winner of Shanghai Peony Prize of Natural Science; National Science Foundation for Distinguished Young Scholars; Changjiang Distinguished Professor; National Special Support Program for Leading Talents in Science and Technology Innovation. He is the Vice President of China Society for Industry and Applied Mathematics. Professor Lei's research is focused on the theory of PDEs arising from fluid mechanics and methematical physics. He introduced the concept of strong null condition and proved the global well-posedness of classical solutions to the incompressible elastodynamics in 2D. He has also made significant contributions to the well-posedness theory and Liouville properties of the incompressible Navier-Stokesequations. Professor Lei holds the position of associate editor-in-chief of Chinese Annals of Mathematics; associate editor-in-chief of Journal of Fudan University (Natural Science). He also serves at the editorial board for several academic journals, such as Communications in Mathematical Sciences, Communications on Pure and Applied Mathematics, Fundamental Research, etc.

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**Lecture Series 26 —— November** **19, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/2DB90D3577355CD54F6DB04AAB1831AD

Valid Until：2026-04-30 23:59

**Lecture 1——Propagation of quasi-particles on singular spaces. Relation to the behavior of geodesics and to certain problems of analytic number theory**

**Speaker:** Andrey Shafarevich

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **We study propagation of semi-classical localized solutions of Schroedinger or wave equations (Gaussian beams) on a certain class of singular spaces. These spaces are obtained by connecting of a number of smooth manifolds by several segments. Laplacians on such spaces are defined with the help of extension theory an depend on boundary conditions in the points of gluing. Statistics of a number of Gaussian packets is governed by the behavior of geodesics on manifolds and is connected with certain problems of analytic number theory - in particular, with the problem of distribution of abstract primes.

**Bio:** Prof. A.I. Shafarevich is currently the Dean of the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University. He is also the Corresponding Member of the Russian Academy of Sciences.

The main scientific interests of A.I.Shafarevich lie in the field of mathematical physics, asymptotic and geometric theory of linear and nonlinear partial differential equations, quantum mechanics and hydrodynamics. He solved the problem posed by V.P. Maslov and widely discussed in the scientific literature on the multiphase asymptotics for the equations of hydrodynamics.

**Lecture 2——Toda equations and cyclic Higgs bundles over non-compact surfaces**

**Speaker: **Qiongling Li

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** For Higgs bundles over compact Kahler manifods, it is known by Hitchin and Simpson that the existence of Hermitian-Einstein metric is equivalent to the polystability of the Higgs bundle. There are some generalizations to non-compact cases. On a Riemann surface with a holomorphic r-differential, one can naturally define a Toda equation and a cyclic Higgs bundle with a grading. A solution of the Toda equation is equivalent to a Hermitian-Einstein metric of the Higgs bundle for which the grading is orthogonal. In this talk, we focus on a general non-compact Riemann surface with an r-differential which is not necessarily meromorphic at infinity. In particular, we discuss the Hermitian-Einstein metrics on the cyclic Higgs bundles determined by r-differentials. This is joint work with Takuro Mochizuki (Kyoto University).

**Bio: **Qiongling Li got her Ph.D. from Rice University in 2014. She is currently a research fellow at Chern Institute of Mathematics, Nankai University. Her main research fields are Higgs bundles, harmonic maps, and higher Teichmuller theory. Her recent works have been focused on understanding the non-abelian Hodge correspondence over Riemann surfaces.

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**Lecture Series 25 ——November 05, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/75414BFD1162CE3C9CCC5376E2FE94F0

Valid Until：2026-04-30 23:59

**Lecture 1——On isochronous dynamics.**

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**Lecture Series 24 —— October 22, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/6108285A02ADB53008FDB05BEB66F6D6

Valid Until: 2026-06-30 23:59

**Lecture 1——What we know and what we do not know about the zeros of Riemann zeta-function**

**Speaker: **Prof. Maxim Korolev

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** In the talk, we will discuss from different points of view the connection between quite "transcendental" objects, that is, between zeros of the Riemann zeta-function, and purely arithmetic objects, that is, prime numbers.

**Bio: **Maxim Korolev is a Professor of the Russian Academy of Sciences. He has received the Vinogradov Prize of the Russian Academy of Sciences in 2019.

**Lecture 2——Finiteness and Duality for the Cohomology of Prismatic Crystals**

**Speaker:** Yichao Tian

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Prismatic site of a p-adic formal scheme was introduced in the recent pioneer work of Bhatt—Scholze. It provides a uniform framework for various p-adic cohomology theories. Prismatic crystals are natural analogues of classical crystalline crystals on prismatic sites. In this talk, after reviewing some basic definitions of the prismatic site, I will discuss some basic properties of the cohomology of prismatic crystals on smooth p-adic formal schemes. The key ingredient is an explicit local description of (reduced) prismatic crystals in terms of Higgs modules.

**Bio: **Yichao Tian got his Ph. D. from University Paris in 2008. He is currently a professor in the Morningside Center of Mathematics at the Chinese Academy of Science. His main research fields are Arithmetic algebraic geometry: p-adic Hodge theory, Geometry of Shimura varieties in characteristic p > 0, p-divisible groups, and p-adic modular forms.

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**Lecture Series 23 —— October 08, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/A57739A4E5705369970A3175767FC017

Valid Until：2026-04-30 23:59

**Lecture 1——Weighted graphs, tetrahedron equation and loop quantum gravity**

**Speaker:** Prof. Dmitry Talalaev

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **Edge-weighted graphs are a combinatorial object parametrizing several archetypal models of statistical physics: the dimer model, the Ising model and the model of electrical networks. Each of these models describes some combinatorial problem, is associated with some solution of the tetrahedron equation, is associated with one of the versions of the completely positive Grassmannian. I will talk about these phenomena and how these problems are related to loop quantum gravity, namely spin foam evaluations.

**Bio: **Dmitry Talalaev Graduated from the Faculty of Mechanics and Mathematics of Moscow State University, received a doctorate in Physics and Mathematics at the Steklov Institute, specializes in classical and quantum integrable systems, including models of statistical physics, applications in low-dimensional topology and cluster algebras.

**Lecture 2——An invitation to categorical enumerative invariants.**

**Speaker: **Junwu Tu

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: **In this talk, we shall sketch the definition of categorical enumerative invariants following Costello (2004/2005) and Caldararu-T. (2020). Then we survey known calculations of these categorical invariants. In the end, we also discuss some research problems in this direction.

**Bio:** Junwu Tu got his Ph.D. from University of Wisconsin-Madison in 2011. He is currently an associate professor in the Institute of Mathematical Sciences (IMS) at ShanghaiTech University. His main research fields are homological algebras related to mirror symmetry. His recent works have been focused on understanding categorical enumerative invariants defined by Costello back in 2004.

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**Lecture Series 22 —— September 17, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/15C3899F5E087E67898778568B354DC0

Valid Until：2026-04-30 23:59

**Lecture 1——Liouville theorem for a class of semilinear elliptic equation on Heisenberg group**

**SPEAKER: **Prof. Xinan Ma, University of Science and Technology of China.

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **We obtain a Liouville type theorem for a class of semilinear subcritical elliptic equation on Heisenberg group. The proof is based on a'priori integral estimate from a generalized differential identity found by Jerison and Lee in 1988. We also get a point-wise estimate near the isolated singularity. This is a joint work with Qianzhong Ou.

**Bio: **Xinan Ma got Ph. D from Hangzhou University in 1996. He is now working at the School of Mathematical Sciences, University of Science and Technology of China. His main research fields are nonlinear elliptic equation and geometry analysis.

**Lecture 2 ——Symplectic and Contact Geometry of Monge–Ampère equation: Introduction and application.**

**Speaker: **Vladimir Rubtsov (Université d'Angers)

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: ** I am going to present an introduction into the geometric approach to Monge–Ampère operators and equations based on contact and symplectic structures of cotangent and the 1st jet bundles of a smooth manifold. This approach was developed by V. Lychagin and goes back to the ideas of E.Cartan and his successor T. Lepage. I shall try to make my talk self-contained. I also plan to discuss various applications and links with important geometric structures.

**Bio: **Vladimir (Volodya) Rubtsov graduated in Mathematics from the Moscow State University in 1974 with a MSc in Differential Geometry and Applications. He has a PhD in Higher Geometry and Topology (1983) and held research and teaching positions at various Mathematics and Applied Mathematics Laboratories in the former Soviet Union. Presently he is Professor at the Department of Mathematics, Université d'Angers, and a member of LAREMA (Anjou Research Mathematical Laboratory) of CNRS (France). Since 1993 he is a Senior Researcher at the Theory Division in the Alikhanov Institute for Theoretical and Experimental Physics (ITEP) in Moscow. He held visiting positions at Ecole Polytechnique (Palaiseau, France), Universities of Lyon, Lille and Strasbourg (France), University of Uppsala (Sweden), SISSA (Italy) and others. He was invited member at IHES, MPIM (Bonn), the Newton Institute (Cambridge, UK) and others. His research is in the area of Poisson geometry, quantum Groups, integrable systems, symplectic and contact geometric methods in non-linear differential equations and applications in hydrodynamics.

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**Lecture Series 21 —— June 4, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/5D3D940A96DAC4AAF631D25AC6CA0288

Valid Until：2026-04-30 23:59

**Lecture 1——Integrable systems with elliptic dependence on momenta and related topics**

**Speaker:** Andrei Zotov (Steklov Mathematical Institute of Russian Academy of Sciences)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **We discuss a family of integrable many-body systems of classical (and quantum) mechanics. Some interrelations (dualities) predict existence of integrable many-body systems with elliptic dependence on particles momenta – the most general representative of this family. We describe some recent results on this topic. Next, we discuss relations of the many-body systems to other families of integrable models including integrable tops and spin chains. Finally, some interesting open problems are formulated.

**Bio: **Andrei Zotov is a leading researcher at Steklov Mathematical Institute. Also, associative professor at Moscow Institute of Physics and Technology and researcher at ITEP and HSE – Skoltech International Laboratory of Representation Theory and Mathematical Physics. Main field of research is mathematical physics and integrable systems.

**Lecture 2 ——Off-diagonal Bethe ansatz approach to quantum integrable models **

**Speaker:** Wenli Yang

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: ** Applying the recent developed method-the off-diagonal Bethe ansatz method, we construct the exact solutions of the Heisenberg spin chain with various boundary conditions. The results allows us to calculate the boundary energy of the system in the thermo dynamic limit. The method used here can be generalized to study the thermodynamic properties and boundary energy of other high rank models with non-diagonal boundary fields.

**Bio: **Wenli Yang is a professor and PhD supervisor in the School of Physics at North-western University. He is currently the executive director of the Chinese Physical Society and a member of the National Committee on Condensed Matter Theory and Statistical Physics. He received his bachelor's degree from Xi'an Jiaotong University in 1990 and his Ph.D. degree from North-western University in 1996, and has worked at the University of Bonn, Germany, Kyoto University, Japan, and the University of Queensland, Australia. In 2009, he was selected as one of the first "Hundred Talents Plan" in Shaanxi Province, and in 2014, he was awarded the National Outstanding Youth Fund, and in 2015, he was selected as one of the Changjiang Distinguished Professors of the Ministry of Education. His achievements were awarded the Second Prize of Natural Science of the Ministry of Education in 2010 and the First Prize of Science and Technology of Shaanxi Province in 2012.

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**Lecture Series 20 —— May 21, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

**Lecture 1 ****—— On the application of the Ważewski method to the problem of global stabilization**

**Speaker:** Ivan Polekhin (Steklov Mathematical Institute of RAS).

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **In 2000, S.P. Bhat and D.S. Bernstein proved that if the configuration space of an autonomous control mechanical system is closed (compact without boundary), then the system cannot have a globally asymptotically stable equilibrium [1]. We will present a similar result for non-autonomous control systems defined on manifolds with non-empty boundaries. The talk is based on the paper [2].

[1] Bhat S.P., Bernstein D.S. A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon Systems Control Lett., 39 (1) (2000), pp. 63-70

[2] I. Polekhin, “On the application of the Ważewski method to the problem of global stabilization”, Systems & Control Letters, 153 (2021) Share Link: https://authors.elsevier.com/a/1d2Qoc8EXim67

**Bio: **Russian Academy of Sciences prize for young Russian scientists (2020)

**Lecture 2 ——High Fidelity Simulations of High-Pressure Turbines Cascades for Data-Driven Model Development**

**Speaker: **Yaomin Zhao, College of Engineering, Peking University

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Gas turbines (GT) are, and will continue to be, the backbone of aircraft propulsion, as well as power generation and mechanical drive. A small GT performance improvement is expected to have a fuel-spend advantage of the billion-dollar order, together with a significant CO2 emission benefit. Part of the possible performance improvements can be enabled by continuously advancing the understanding of the GT flow physics and thus further reducing the inaccuracy of current design tools based on computational fluid dynamics (CFD). By exploiting the capability of our high-fidelity CFD solver on leadership GPU-accelerated supercomputers, we have been able to perform state-of-the-art high fidelity simulations of turbomachinery flows. The generated data, therefore, can shed light on the detailed fundamental flow physics, in particularly the behavior of transitional and turbulent boundary layers affected by large-scale violent freestream turbulence, under strong pressure gradient and curvature. Furthermore, machine learning methods are applied to the high-fidelity data to develop low order models readily applicable to GT designs.

**Bio: **Dr. Yaomin Zhao is an Assistant Professor at Peking University, China. He obtained his B.Sc. in 2011 and then Ph.D. in 2017 both from Peking University. The Ph.D. thesis, Lagrangian investigation on transitional wall-bounded flows, was awarded the Outstanding PhD Thesis by Chinese Society of Theoretical and Applied Mechanics in 2018. From 2017 to late 2020, he was a post-doctocal research fellow at the University of Melboune, Australia. His research interests include high-fidelity simulations of turbomachinery flows, turbulence model development with machine learning methods, and boundary layer transition, etc.

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**Lecture Series XIX —— May 14, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/00B707ED52223278BF9974EA3B1ED9DA

Valid Until：2026-04-30 23:59

**Lecture 1 —— Lorentz geometry and contact topology**

**Speaker:** Stefan Nemirovski (MIAN).

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** Roger Penrose observed four decades ago that the space of light rays of a reasonable spacetime carries a natural contact structure and raised the problem of describing the causality relation of the spacetime in its terms. The talk will survey the progress made in this direction from the seminal work of Robert Low to the more recent applications of global contact rigidity.

**Bio: **Corresponding member of the Russian Academy of Sciences, winner of the European Mathematical Society prize (2000).

**Lecture 2 ——Gromov-Hausdorff limit of manifolds and some applications**

**Speaker:** Wenshuai Jiang(Zhejiang University)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: **Gromov-Hausdorff distance is a distance between two metric spaces, which was introduced by Gromov 1981. From Gromov’s compactness theorem, we knew that any sequence of manifolds with uniform lower Ricci curvature bounds has a converging subsequence in Gromov-Hausdorff topology to a limit metric space. The limit metric space in general may not be a manifold. The structure of such limit metric space has been studied by Cheeger, Colding, Tian, Naber and many others since 1990. It turns out that such theory has powerful application in geometry. In fact, the resolution of Yau-Tian-Donaldson conjecture was largely relied on the development of the study of the limit metric space.

In the first part of the talk, we will discuss some recent progress of the Gromov-Hausdorff limit of a sequence of manifolds with Ricci curvature bounds. In the second part, we will discuss some applications based on the study of Gromov-Hausdorff limits.

**Bio: **Wenshuai Jiang studied in the Department of mathematics of Nanjing University from 2007 to 2011 and obtained his bachelor's degree. From 2011 to 2016, he studied in school of Mathematical Sciences of Peking University and obtained a doctorate under the guidance of Professor Gang Tian. He has been working in Zhejiang University since 2016, and is currently an associate professor of Zhejiang University. His major research interest is geometric analysis.

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**Lecture Series XVIII —— April 23, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/00B707ED52223278BF9974EA3B1ED9DA

Valid Until：2026-04-30 23:59

**Lecture 1 —— VALUES OF PERMANENT AND POSITIVE SOLUTION OF WANG-KRÄUTER PROBLEM**

**Speaker: **A.E. GUTERMAN (RUSSIA, MSU)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:**

**Lecture 2 —— From Sphere Packings to Post-Quantum Cryptography **

**Speaker: **Chuanming Zong (Tianjin University)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: **

**Bio: **Chuanming Zong obtained his PhD from Vienna University of Technology in 1993. He was a professor at the Chinese Academy of Sciences and Peking University. Currently, he is a distinguished professor at Tianjin University. He mainly works in number theory. He has made important contribution in Hilbert’s 18th problem and tiling theory. He has been awarded a Conant Prize by Amer Math Soc in 2015, a National Science Prize by the Chinese government in 2009, and a S. S. Chern Prize by the Chinese Math Soc in 2007. He was a plenary speaker at Asiacrypt2012, a author of two solicited papers in Bull AMS and three books at Springer and Cambridge University Press.

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**Lecture Series XVII —— April 9, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/E6EB60DE09499782270BB3B51570DFA1

Valid Until：2026-04-30 23:59

**Lecture 1 —— Mathematical methods of quantum key distribution.**

**Speaker: **Anton Trushechkin (MIAN).

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **Quantum key distribution and, more generally, quantum cryptography is a modern branch of science where methods of secure communication based on principles of quantum mechanics are studied. The rigorous proof of the security of quantum key distribution gave rise to a complex and beautiful mathematical theory, which is based on methods of quantum information theory, namely, quantum entropic measures and entropic uncertainty relations. In particular, to estimate secret key rate, one needs to minimize the quantum relative entropy (a convex function) subject to linear constraints. The problem is, in general, infinite-dimensional, but symmetry properties of the problem reduces the dimensionality and allows one to solve this problem analytically. However, currently, an important task is to prove the security of quantum key distribution with imperfect (i.e., practical) devices. Imperfections introduce asymmetries and thus make the problem more complicated. In the talk, estimations for the secret key rate in the case of detection-efficiency mismatch will be presented. Using entropic uncertainty relations, an infinite-dimensional problem is reduced to a one-dimensional one.

**Bio：**Results of Anton Trushechkin in quantum cryptography were nominated as one of most important mathematical achievements of the Russian Academy of Sciences in 2020.

**Lecture 2 ——A quantum leap in security.**

**Speaker:** Prof. Feihu Xu, University of Science and Technology of China.

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Quantum cryptography or quantum key distribution (QKD) offers information-theoretic security based on the laws of physics. This is the technology at the basis of the quantum satellite “Mozi”, put in orbit by the Chinese Academy of Sciences in 2016. In practice, however, the imperfections of realistic devices might introduce deviations from the idealized models used in the security proofs of QKD. Can quantum code breakers successfully hack real systems by exploiting the side channels? Can quantum code makers design innovative countermeasures to foil quantum code breakers? In this talk, I will talk about the theoretical and experimental progress in the practical security aspects of quantum code making and quantum code breaking. After numerous attempts over the past decades, researchers now thoroughly understand and are able to manage the practical imperfections. Recent advances, such as the decoy-state, measurement-device-independent (MDI) and twin-field (TF) protocols, have closed critical side channels in the physical implementations in a rigorous and practical manner. Further readings in [Xu et al., Rev. Mod. Phys. 92, 025002 (2020)].

**Bio：**Feihu Xu has been a Professor at USTC since Oct. 2017. Before joining USTC, he was a Postdoctoral Associate at MIT in 2015-2017. He received an M.A.Sc and Ph.D from University of Toronto in 2011 and 2015. He works on quantum information science and has co-authored more than 70 journal papers. As the first/corresponding author, he has published more than 40 journal papers in Rev. Mod. Phys. (1), Nat. Photon. (4), Nat. Phys. (1), etc. He is the recipient of Early Career Award by NJP in 2020, 35 Innovators Under 35 of China (by MIT Technology Review) in 2019, Outstanding Dissertation Award (by OCPA) in 2015, and Best Paper Award of QCrypt in 2014.

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**Lecture Series XVI ——March 26, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/C95AD535339960AB095CCE3E7C231216

Valid Until：2026-04-30 23:59

**Lecture 1 —— Transference principle in Diophantine approximation**

**Speaker:** Oleg German (MSU)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** The talk will be devoted to one of the fundamental principles in Diophantine approximation called transference principle. It reflects the relation of duality between certain problems. This principle is usually formulated in terms of Diophantine exponents - they generalise to the multidimensional case the measure of irrationality of a real number. We plan to give an account on the existing relations Diophantine exponents satisfy and try to reveal the geometric nature of those relations. After having described some basic geometric constructions, we shall look from this perspective at the famous linear independence criterion that belongs to Nesterenko. It appears that our approach provides an alternative proof of this criterion, which bases on rather simple geometric considerations.

**Bio：**Oleg German graduated from Moscow State University in 2001, defended the Candidate thesis in 2005 at MSU and the Doctorate thesis in 2013 at Steklov Mathematical Institute. He works at the Department of Number Theory, Faculty of Mechanics and Mathematics, MSU. His research interests include geometry of numbers, Diophantine approximation, multidimensional continued fractions.

**Lecture 2 ——Introduction to p-adic Langlands program for GL_2**

**Speaker: **Hu Yongquan

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: **The p-adic and mod p Langlands program is an avatar of the classical Langlands program and has been first initiated by C. Breuil. In this colloquium talk, I will give a brief introduction to the program and survey some recent progress in the case of GL_2.

**Bio:**Yongquan Hu received PhD degree from University Paris-Sud in 2010. After that, he has worked at University of Rennes 1 (France) as a Maître de Conférence. Starting from 2015, he is a Professor at Morningside Center of Mathematics, Academy of Mathematics and Systems Science. His research interest lies in p-adic and mod p Langlands program.

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**Lecture Series XV —— March 12, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/35FAFAB495AAAB508AE966572F3DF218

Valid Until：2026-04-30 23:59

**Lecture 1 ——Mathematical problems in the theory of topological insulators**

**Speaker: **Armen Sergeev (Steklov Mathematical Institute, Moscow).

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

In our talk we consider the topological insulators invariant under time reversal. In the first part we present the physical basics of their theory while the second part deals with the mathematical aspects. These aspects are closely related to K-theory and non-commutative geometry.

**Bio：**Prof. Armen Glebovich Sergeev is a leading scientific researcher of the department of complex analysis in Steklov Mathematical Institute and a professor in Mechanical and Mathematical department of Moscow State University. He Got Ph. D in Moscow State University in 1975 and Doctor of Sciences in Steklov Mathematical Institute at Moscow in 1989. He is a foreign member of Armenian Academy of Sciences, member of the board of Moscow Mathematical Society and a member of Executive Committee of European Mathematical Society. He is the Chief-editor of many mathematical journals and published 106 papers and is the author of 10 books. His Principal fields of research include Pseudoconvex polyhedral, Invariant domains of holomorphy, Geometric quantization, Twistor quantization, Seiberg-Witten equations, Pseudoholomorphic curves and Vortex equations. He has been the scientific advisors of many doctors. Together with Prof. Xiangyu Zhou, they have organized a series of Sino-Russia Joint mathematical conferences for many years which has promoted greatly the mathematical cooperation between two countries.

**Abstract: ** The multiplier ideal sheaf plays an important role in several complex variables, complex geometry and algebraic geometry. The strong openness property for multiplier ideal sheaves was conjectured by Demailly and proved by Guan-Zhou. In this talk, we will recall some recent applications of the strong openness property on the restriction formula and subadditivity property related to multiplier ideal sheaves. This is joint work with Professor Xiangyu Zhou.

After graduation, he worked as a postdoctoral researcher in Beijing International Center for Mathematics Research for two years, and his co-advisor is Professor Xiaobo Liu.

In 2013, he joined the School of Mathematical Sciences of Peking University and is now a professor.

Qi'an Guan is mainly engaged in the study of several complex variables.

Qi'an Guan has won the "outstanding postdoctoral Award" (2013) of Peking University, the "Young Teacher Award" (2016) of Huo Yingdong education foundation, and the "Chang Jiang Scholars Program - Young Scholars" (2016) and the "Science Research Famous Achievement

Award in Higher Institution – Youth Science Award" (2017) of the Ministry of Education, the "Qiu Shi Outstanding Young Scholars Award " (2016), the " National Award for Youth in Science and Technology--Special Prize" (2019) of the Chinese Association for science and technology, “The Tan Kah Kee Young Scientist Award in Mathematics & Physics” (2020).

Qi'an Guan was supported by the "Excellent Young Scientists Fund"(2015) and "National Science Fund for Distinguished Young Scholars" (2018) of the National Natural Science Foundation of China.

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**Lecture Series XIV —— January 29, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/0930E258DA1D1B986E4238DB4E21D826

Valid Until：2026-04-30 23:59

**Lecture 1 ——The Aerothermal Performance of Tip Leakage Flow in High Pressure Turbines.**

**Speaker: **Prof. Chao Zhou, College of Engineering, Peking University, Beijing, China

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **Future aero engines are expected to achieve higher efficiencies and lower emissions, which brings new challenges for turbine designers. In a high-pressure turbine, tip clearances exist between the turbine rotor blade tip and the casing to prevent rubbing. The pressure difference between the blade pressure side and the suction side drives the gas across this tip clearance gap. The high-temperature gas results in excessively high metal temperatures on the blade tip, which lead to thermal erosion and oxidation. Obtaining good aerothermal performance is the key to maintain the performance of the high pressure turbines. The current talk will present a combined experimental and numerical study, which aims to understand the performance of the tip leakage flow and to develop new tip configurations for higher engine efficiency. First, the aerodynamic and heat transfer performance of squealer tips will be discussed. The effects of the squealer height and thickness will be investigated. Then, the tips with

coolant injection are investigated to understand the effects of the cooling air on the loss mechanism and tip heat transfer. Finally, winglet configurations are used on blade tips to reduce the tip leakage loss. The results showed that by using the winglet tip developed in the current study, the turbine stage efficiency increases.

**Bio:** Chao Zhou is a tenured associate professor and the director of turbomachinery laboratory at the college of Engineering in Peking University, China. Before join Peking University, he obtained his doctorate degree at the Whittle Laboratory of Cambridge University in 2010. He is also educated in Nanjing University of Aeronautics and Astronautics, China, and received his Master degree and Bachelor in 2006 and 2003 respectively. His main research area is the aerodynamic and heat transfer of turbomachinery, including aerothermal performance of high pressure turbines, high-lift low pressure turbines; inter-turbine duct flows, unsteady flows and loss mechanism of turbomachinery, advanced cooling methods and highly loaded compressors. He has published 9 papers on the Journal of Turbomachinery, which is the top Journal in the research area. Dr. Zhou is a member of ASME IGTI Tubomachinery committee. He serves as the review co-chair of 2020 GPPS (Global Power and Propulsion Society) conference.

**Lecture 2 —— Mathematical Models of Mediums in Continuum Mechanics.**

**Speaker:** Dmitrii Georgievskii

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: **Elements of the theory of constitutive relations. The tangent modulus and the tangent compliance. Physical nonlinearity, tensor linearity (quasilinearity), and nonlinearity. The material functions. Rheonomic and scleronomic media. Homogeneous and inhomogeneous media. Composites. Elastic bodies. Viscous liquids. Media with memory. Non-local media. Tensor functions and their invariants in the theory of constitutive relations. Potential media and conditions of potentiality. Incompressible materials (liquids).Nonlinear elastic-viscoplastic models. Classification of incompressible media (quasilinear models, Bingham bodies, perfectly plastic media, Newtonian viscous fluids). Statement of the linearized boundary value problem of flow stability with respect to small perturbations of the initial data.

**Bio: **Dmitrii Georgievskii received his PhD degree at MSU, 1989, and DSc degree at MSU, 1996. He is a Professor of Russian Academy of Sciences since 2015and is a Chair of Lab. of Elasticity and Plasticity in Institute of Mechanics (MSU) since 2020. His research interests include the Theory of constitutive relations in continuum mechanics, Phenomenological description of stress-strain state by multiscale simulation, Asymptotic methods in theory of thin solids, see also web-page http://mech.math.msu.su/~georgiev/first_e.htm for more details.

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**Lecture Series XIII —— January 15, 2021 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/9ED10161227A7A8F325E9F07F7388A80

Valid Until：2026-04-30 23:59

**Lecture 1 ——Linear stability of pipe Poiseuille flow at high Reynolds number regime**

**Speaker:** Zhifei Zhang, School of Mathematical Sciences, Peking University

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **The linear stability of pipe Poiseuille flow is a long standing problem since Reynolds experiment in 1883. Joint with Qi Chen and Dongyi Wei, we solve this problem at high Reynolds regime. We first introduce a new formulation for the linearized 3-D Navier-Stokes equations around this flow. Then we establish the resolvent estimates of this new system under favorable artificial boundary conditions. Finally, we solve the original system by constructing a boundary layer corrector.

**Bio:** Zhifei Zhang received his PhD from Zhejiang university in 2003. Then he spent 2 years in Mathematics Institute of AMSS as a Postdoc. He joined Peking university in 2005. His research interest is in the mathematical problems in the fluid mechanics such as the well-posedness of the Navier-Stokes equations, free boundary problem, hydrodynamic stability.

**Lecture 2 —— Partial spectral flow and the Aharonov–Bohm effect in graphene.**

**Speaker:** Vladimir E. Nazaikinskii Ishlinsky Institute for Problems in Mechanics RAS

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** We study the Aharonov–Bohm effect in an open-ended tube made of a graphene sheet whose dimensions are much larger than the interatomic distance in graphene. An external magnetic field vanishes on and in the vicinity of the graphene sheet, and its flux through the tube is adiabatically switched on. It is shown that, in the process, the energy levels of the tight-binding Hamiltonian of π-electrons unavoidably cross the Fermi level, which results in the creation of electron–hole pairs. The number of pairs is proven to be equal to the number of magnetic flux quanta of the external field. The proof is based on the new notion of partial spectral flow, which generalizes the ordinary spectral flow introduced by Atiyah, Patodi, and Singer and already having well-known applications (such as the Kopnin forces in superconductors and superfluids) in condensed matter physics.

**Bio:** Vladimir Nazaikinskii received PhD degree from Moscow Institute of Electronic Engineering in 1981 and DSc degree from Steklov Mathematical Institute of RAS in 2014 and was elected Corresponding Member of RAS in 2016. He works at Ishlinsky Institue for Problems in Mechanics of RAS as a principal researcher. His research interests include asymptotic methods in the theory of differential equations and mathematical physics; asymptotic methods in the statistics of many-particle systems and relations to number theory; C*-algebras and noncommutative geometry; elliptic theory and index theory.

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Recording: https://disk.pku.edu.cn:443/link/E9B034DE79B98247C575EAFA19567D82

Valid Until：2026-04-30 23:59

First, we will explain the result on the existence of smooth compactifications of derived categories of coherent sheaves on separated schemes of finite type over a field of characteristic zero. Namely, such a derived category can be represented as a quotient of the derived category of a smooth projective variety, by a triangulated subcategory generated by a single object. Then we will give an example of a homotopically finite DG category which does not have a smooth compactification: a counterexample to one of the Kontsevich's conjectures on the generalized Hodge to de Rham degeneration.

Finally, we will formulate a K-theoretic criterion for existence of a smooth categorical compactification, using DG categorical analogue of Wall's finiteness obstruction from topology.

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Recording: https://disk.pku.edu.cn:443/link/40515B49A258749CAB6F1CC4CEE0D61D

Valid Until：2026-04-30 23:59

Valid Until：2026-04-30 23:59

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**Lecture Series X —— November 20th, 2020 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/D258CE3FD7B6994E558251263993EFA6

Valid Until：2026-04-30 23:59

**Lecture 1 —— Right-angled polytopes, hyperbolic manifolds and torus actions**

**Speaker：**Taras Panov, Moscow State University, Russia

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

This is a joint work with V. Buchstaber, N. Erokhovets, M. Masuda and S.Park.

**Lecture 2 —— Finite covers of 3-manifolds**

**Speaker：**Yi Liu, Beijing International Center for Mathematical Research.

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: **In this talk, I will discuss some developments in 3-manifold topology of this century regarding finite covering spaces. These developments led to the resolution of Thurston’s virtual Haken conjecture and other related conjectures around 2012. Since then, people have been seeking for new applications of those techniques and their combination with other branches of mathematics.

**Bio：**Yi Liu is a professor at Beijing International Center for Mathematical Research (BICMR) in Peking University. His research interest lies primarily in 3-manifold topology and hyperbolic geometry. He received his Ph.D. degree in 2012 in University of California at Berkeley. In 2017, he received the Qiushi Outstanding Young Scholar Award. He has been a principal investigator of the NSFC Outstanding Young Scholar since 2019. Below are some selected research works of Yi Liu: (1) proving J. Simon’s conjecture about knot groups (joint with I. Agol, 2012); (2) resolving fundamental properties of the L2 Alexander torsion for 3-manifolds, (2017); (3) proving C. T. McMullen’s conjecture about virtual homological spectral radii of surface automorphisms (2020).

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**Lecture Series Ⅸ ——November 6th, 2020 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/18091C862640B23DC5A05E771BACB4B9

Valid Until：2026-04-30 23:59

**Lecture 1 —— Spectrum rigidity and integrability for Anosov diffeomorphisms.**

**Speaker：**Assistant Prof. Yi Shi, School of Mathematical Sciences, Peking University

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:**

**Bio: **Yi Shi obtained PhD from Peking University and Universite de Bourgogne in 2014, and then did postdoc in IMPA. He is now an assistant professor in School of Mathematical Sciences at Peking University. His research field is differentiable dynamical systems, including partially hyperbolic dynamics and singular star vector fields.

**Lecture 2 —Аn application of algebraic topology and graph theory in microeconomics**

**Speaker：**Lev Lokutsievskiy (Steklov Mathematical Institute of RAS)

Obviously, some allocation rules are not implementable. Rochet’s theorem states that an allocation rule is implementable iff it is cyclically monotone. During the talk, I’ll present a new convenient topological condition that guarantees that cyclic monotonicity is equivalent to ordinary monotonicity. The last one is easy to check (in contrary to cyclic one). Graph theory and algebraic topology appear to be very useful here.

**Bio: **Lokutsievskiy L.V. is a specialist in geometric optimal control theory. He proved his habilitation thesis in 2015. Starting from 2016 he works at Steklov Mathematical Institute as a leading researcher.

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**Lecture Series Ⅷ ——October 23th, 2020 ****(18:15-20:15 Beijing time, 13:15-15:15 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/37E312BF51E4B5ACEEA83926427C8670

Valid Until：2026-04-30 23:59

**Lecture 1 - Topological cyclic homology for p-adic local fields.**

**Speaker:** Prof. Ruochuan Liu, School of Mathematical Sciences, Peking University

**Time:** 18:15-19:15 Beijing time (13:15-14:15 Moscow time)

**Abstract:** We introduce a new approach to compute topological cyclic homology using the descent spectral sequence and the algebraic Tate spectral sequence. We carry out computations in the case of a p-adic local field with coefficient Fp. Joint work with Guozhen Wang.

**Bio: **Ruochuan Liu is working on p-adic aspects of arithmetic geometry and number theory, especially p-adic Hodge theory, p-adic automophic forms and p-adic Langlands program. He got his PhD from MIT at 2008. After several postdoc experience at Paris 7, McGill, IAS and Michigan, he joined the Beijing International Center for Mathematical Research at 2012. Starting from this year, he holds professorship at the School of Mathematical Sciences of Peking University.

**Lecture 2 — Additive divisor problem and Applications**

**Speaker: **Dimitry Frolenkov, Steklov Mathematical Institute (Moscow)

**Time:** 19:15-20:15 Beijing time (14:15-15:15 Moscow time)

**Abstract:** Additive Divisor Problem (ADP) is concerned with finding an asymptotic formula for the sum $\sum_{n<X}d(n)d(n+a)$, where $d(n)=\sum_{d|n}1$ is the divisor function. Surprisingly, the ADP arises naturally in quite different problems of number theory. For example, it is related to the investigation of the 4th moment of the Riemann zeta-function, the second moment of automorphic $L$-functions and the mean values of the length of continued fractions. In the talk, I will describe the ADP and its applications.

**Bio: **Dmitry Frolenkov received his PhD degree from Steklov Mathematical Institute in 2013. Starting from 2014 he works at Steklov Mathematical Institute as a senior researcher. Besides he got the RAS award for young scientists of Russia. His research interests are centered around an analytic number theory with a special emphasis on the theory of L-functions associated to automorphic forms.

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The conference is on the occassion of the 70-th anniversary of our friend and colleague Vyacheslav Valentinovich Nikulin, to celebrate his huge contributions to the theory of K3 surfaces and other areas of geometry and arithmetic including reflections groups, automorphic forms and infinite-dimensional Lie algebras. The topics covered at the conference reflect the mathematical interests of V.V. Nikulin.

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**Lecture Series Ⅶ - October 16th, 2020 ****(17:30-19:30 Beijing time, 12:30-14:30 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/CD2C01E30B99227AAF28668DE073F517

Valid Until：2026-04-30 23:59

**Lecture 1**** - Restriction of unitary representations of Spin(N,1) to parabolic subgroups**

**Speaker**: Prof. Yu Jun, Beijing International Center for Mathematical Research

**Time**：17：30-18：30 Beijing time （12：30-13：30 Moscow time）

**Abstract:** The orbit method predicts a relation between restrictions of irreducible unitary representations and projections of corresponding coadjoint orbits. In this talk we will discuss branching laws for unitary representations of Spin(N,1) restricted to parabolic subgroups and the corresponding orbit geometry. In particular, we confirm Duflo's conjecture in this setting. This is a joint work with Gang Liu (Lorraine) and Yoshiki Oshima (Osaka).

**Bio:** Jun Yu obtained PhD from ETH Zurich in 2012, and then did postdoc in IAS Princeton and MIT. He is now an assistant professor in Beijing international center for mathematical research at Peking University. His research field is representation theory and Langlands program, including the branching rule problem, the orbit method philosophy, and the beyond endoscopy program.

**Lecture 2: Characterizing homogeneous rational projective varieties with Picard number 1 by their varieties of minimal rational tangents.**

**Speaker:** Prof. Dmitry Timashev, Moscow State University

**Time**：18：30-19：30 Beijing time （13：30-14:30 Moscow Time)

**Abstract:** It is well known that rational algebraic curves play a key role in the geometry of complex projective varieties, especially of Fano manifolds. In particular, on Fano manifolds of Picard number (= the 2nd Betti number) one, which are sometimes called "unipolar", one may consider rational curves of minimal degree passing through general points. Tangent directions of minimal rational curves through a general point $x$ in a unipolar Fano manifold $X$ form a projective subvariety $\mathcal{C}_{x,X}$ in the projectivized tangent space $\mathbb{P}(T_xX)$, called the variety of minimal rational tangents (VMRT).

In 90-s J.-M. Hwang and N. Mok developed a philosophy declaring that the geometry of a unipolar Fano manifold is governed by the geometry of its VMRT at a general point, as an embedded projective variety. In support of this thesis, they proposed a program of characterizing unipolar flag manifolds in the class of all unipolar Fano manifolds by their VMRT. In the following decades a number of partial results were obtained by Mok, Hwang, and their collaborators.

Recently the program was successfully completed (J.-M. Hwang, Q. Li, and the speaker). The main result states that a unipolar Fano manifold $X$ whose VMRT at a general point is isomorphic to the one of a unipolar flag manifold $Y$ is itself isomorphic to $Y$. Interestingly, the proof of the main result involves a bunch of ideas and techniques from "pure" algebraic geometry, differential geometry, structure and representation theory of simple Lie groups and algebras, and theory of spherical varieties (which extends the theory of toric varieties).

**Bio**: Dmitry Timashev recieved PhD degree from Moscow State University in 1997. From 1997, he is working at the Department of Higher Algebra in the Faculty of Mathematics and Mechanics, Moscow State University, currently at the position of associate professor. His research interests include Lie groups and Lie algebras, algebraic transformation groups and equivariant algebraic geometry, representation theory and invariant theory.

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**Lecture Series VI - September 25th, 2020 ****(16:00-18:00 Beijing time, 11:00-13:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/9FA7C0224D4364D456419199A4373EF7

Valid Until：2026-04-30 23:59

**Lecture 1** - ** Geometric description of the Hochschild cohomology of Group Algebras**

**Speaker**: A. S. Mishchenko (Lomonosov Moscow State University)

**Time**: 2020-09-25 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract**: /upload/editor/file/20200921/21143204927.pdf

**Bio**:

Professor A.S. Mischenko graduated from Moscow State University in 1965. He became a Professor of the Department of Higher Geometry and Topology, Faculty of Mechanics and Mathematics of this University in 1979. He also holds a position of Leading researcher at the Mathematical Steklov Institute. He is a Honored Professor of Moscow University since 2006.

His research interests include geometry and topology and their applications. The main direction of his work is related to the study and application of algebraic and functional methods in the theory of smooth manifolds.

**Lecture 2 - Unipotent representations and quantization of classical nilpotent varieties**

**Speaker**: Prof. Daniel Wong (黄家裕）, Chinese University of Hongkong at Shenzhen.

**Time**: 2020-09-25 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Bio**: Graduated at Cornell University in 2013. Research area is on Representation theory of real reductive Lie groups.

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__Lecture Series V - July 10th, 2020__ **(20:00-22:00 Beijing time, 15:00-17:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/167BD5B7868DBD2425132A8F522E4B7C

Valid Until：2026-04-30 23:59

Speaker:

Time: 2020-07-10 20:00-21:00 Beijing time (15:00-16:00 Moscow time)

Abstract: In this talk, I will present recent works on the hydrodynamic limits to the generic Riemann solutions to the compressible Euler system from the Boltzmann equation.

Bio: Prof. Huang, Feimin got Ph. D in Chinese Academy Sinica in 1997，and then did postdoc in ICTP, Italy and Osaka University. His research field is hyperbolic equations and conservative laws, including fluid dynamical systems, Navier-Stokes equations, and other various Partial Differential Equations. He was awarded the SIAM Outstanding Paper Prize by Society of American Industrial and Applied Mathematics in 2004. He won the Second Prize of National Natural Science Award in 2013.

Speaker:

Time: 2020-07-10 21:00-22:00 Beijing time (16:00-17:00 Moscow time)

Abstract: We consider the Riemann problem for a system of conservation laws. For non-strictly hyperbolic in the sense of Petrovskii step-like systems, a new method of constructing a solution is described. The proposed method allows us to construct a unique solution to the Riemann problem, which for each $t$ is a picewise smooth function of $x$ with discontinuities of the first kind. Moreover, for the scalar conservation law, the solution constructed by the proposed method coincides with the known admissible solution.

Bio: Vladimir Palin recieved higher education degree from Moscow State University in 2005, PhD degree from Moscow State University in 2009. He is now a senior lecturer in the Faculty of Mathematics and Mechanics, Moscow State University. His research interests include hyperbolic equations and systems, conservation laws and matrix equations.

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__Lecture Series IV - June 26th, 2020__ **(20:00-22:00 Beijing time, 15:00-17:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/A5C7E39DAC87BAEEAEBA5CD1EBED76E0

Valid Until：2026-04-30 23:59

**Lecture 1 - Tying Knots in Fluids**

Speaker: **Prof. Yue Yang**, Department of Mechanics and Engineering Science, College of Engineering, Peking University,Beijing

Time: 2020-06-26 20:00-21:00 Beijing time (15:00-16:00 Moscow time)

Abstract:

We develop a general method for constructing knotted vortex/magnetic tubes with the finite thickness, arbitrary shape, and tunable twist. The central axis of the knotted tubes is determined by a given smooth and non-degenerated parametric equation. The helicity of the knotted tubes can be explicitly decomposed into the writhe, localized torsion, and intrinsic twist. We construct several knotted vortex/magnetic tubes with various geometry and topology, and investigate the effect of twist on their evolution in hydrodynamic or magnetohydrodynamic flows using direct numerical simulation. In addition, we illustrate a knot cascade of magnetic field lines through the stepwise reconnection of a pair of orthogonal helical flux tubes with opposite chirality.

Bio:

Yue Yang received BE degree from Zhejiang University in 2004, MS degree from the Institute of Mechanics, Chinese Academy of Sciences in 2007, and PhD degree from California Institute of Technology in 2011, then he was sponsored by the CEFRC Fellowship for postdoc research at Princeton University and Cornell University. Yang joined the Department of Mechanics and Engineering Science in College of Engineering, Peking University in 2013, and was promoted to full professor in 2020. He received the “National Distinguished Young Researcher” award and “Qiu Shi Outstanding Young Scholar Award”. His research interests include turbulence, transition, and combustion.

**Lecture 2 - Supercomputer simulations of aerodynamics and aeroacoustics problems using high-accuracy schemes on unstructured meshes.**

Speaker: **Prof. Andrey Gorobets**, Keldysh Institute of Applied Mathematics of RAS, Moscow

Time: 2020-06-26 21:00-22:00 Beijing time (16:00-17:00 Moscow time)

Abstract:

This talk is devoted to scale-resolving simulations of compressible turbulent flows using edge-based high-accuracy methods on unstructured mixed-element meshes. The focus is on parallel computing. Firstly, the family of edge-based schemes that we are developing will be outlined. Then our simulation code NOISEtte will be presented. It has multilevel MPI+OpenMP+OpenCL parallelization for a wide range of hybrid supercomputer architectures. A description of the parallel algorithm will be provided. Finally, our supercomputer simulations of aerodynamics and aeroacoustics problems will be demonstrated.

Bio:

Andrey Gorobets graduated from Moscow State University in 2003. He then outlived three thesis defenses: 2007, Candidate of Sciences (к. ф.-м. н., equivalent to Ph.D.) at IMM RAS; 2008, European Ph.D. degree at UPC, Barcelona, Spain; 2015, Doktor nauk (д. ф.-м. н., higher doctoral degree) at the Keldysh Institute of Applied Mathematics of RAS (KIAM), Moscow, Russia. He is now a leading researcher at KIAM. His work is focused on algorithms and software for large-scale supercomputer simulations of turbulent flows.

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__Lecture Series III - June 12th, 2020__ **(20:00-22:00 Beijing time, 15:00-17:00 Moscow time)**

**Lecture 1 - Slopes of modular forms and ghost conjecture of Bergdall and Pollack**

Speaker：**Prof. Xiao Liang **（Beijing International Center for Math. Research ）

Time: 2020-06-12 20:00-21:00 Beijing time (15:00-16:00 Moscow time)

Abstract: In classical theory, slopes of modular forms are p-adic valuations of the eigenvalues of the Up-operator. On the Galois side, they correspond to the p-adic valuations of eigenvalues of the crystalline Frobenius on the Kisin's crystabelian deformations space. I will report on a joint work in progress in which we seems to have proved a version of the ghost conjecture of Bergdall and Pollack. This has many consequences in the classical theory, such as some cases of Gouvea-Mazur conjecture, and some hope towards understanding irreducible components of eigencurves. On the Galois side, our theorem can be used to prove certain integrality statement on slopes of crystalline Frobenius on Kisin's deformation space, as conjectured by Breuil-Buzzard-Emerton. This is a joint work with Ruochuan Liu, Nha Truong, and Bin Zhao.

**Lecture 2 -** **Higher-dimensional Contou-Carrere symbols**

Speaker：**Prof. RAS Denis V. Osipov** (Steklov Mathematical Institute of Russian Academy of Sciences)

Time: 2020-06-12 21:00-22:00 Beijing time (16:00-17:00 Moscow time)

Abstract: The classical Contou-Carrere symbol is the deformation of the tame symbol, so that residues and higher Witt symbols naturally appear from the Contou-Carrere symbol. This symbol was introduced by C. Contou-Carrere itself and by P. Deligne. It satisfies the reciprocity laws. In my talk I will survey on the higher-dimensional generalization of the Contou-Carrere symbol. The n-dimensional Contou-Carrere symbol naturally appears from the deformation of a full flag of subvarieties on an n-dimensional algebraic variety and it is also related with the Milnor K-theory of iterated Laurent series over a ring. The talk is based on joint papers with Xinwen Zhu (when n=2) and with Sergey Gorchinskiy (when n>2).

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__Lecture Series II - May 29th 2020__ **(20:00-22:00 Beijing time, 15:00-17:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/D369C31F18FD26B1F910B4648F4F6C67

Valid Until：2026-04-30 23:59

**Lecture 1 - Geometry of Landau--Ginzburg models.**

Speaker：Prof. **Victor V. Przyjalkowski** (Steklov Mathematical Institute of Russian Academy of Sciences)

Time：2020-05-29 20:00-21:00 Beijing time (15:00-16:00 Moscow time)

Abstract: We discuss geometric and numerical properties of Landau--Ginzburg models of Fano varieties that reflect geometric and numerical properties of the initial Fano varieties. The main example is the threefold case.

**Lecture 2 - ****Deformation theory of Schroedinger equation arising from singularity theory**

Speaker: Prof. **Huijun Fan** (Peking University)

Time：2020-05-29 21:00-22:00 Beijing time (16:00-17:00 Moscow time)

Abstract: Mirror symmetry phenomenon relates many mathematical branches in a mysterious way. For example, it is conjectured that the quantum geometry of a Calabi-Yau hypersurface is equivalent to the quantum singularity theory of the corresponding defining function. When we consider the complex structure deformation of the two sides, we get the B model mirror conjecture, where the exciting structures of deformation moduli space, Gauss Manin connection, period mapping and etc.will appear. In this lecture, I will report another way to study the deformation theory of singularity via Schroedinger equation. By study the spectral theory of Schroedinger equation, we can build the variation of Hodge theory, Gauss-Manin connection by wave function, Frobenius manifold for some cases and even BCOV type torsion invariants for singularity.

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__Lecture Series I - May 15th 2020__ **(20:00-22:00 Beijing time, 15:00-17:00 Moscow time)**

Recording: https://disk.pku.edu.cn:443/link/AC9C06289D33D936B486150C6AD3876F

Valid Until：2026-04-30 23:59

**Lecture 1 - Representation volumes and dominations of 3-manifolds**

Speaker：**Shicheng Wang** (Peking University)

Time：2020-05-15 20:00-21:00 Beijing time (15:00-16:00 Moscow time)

Abstract: We will discuss recent results on virtual representation volumes and finiteness of the mapping degree set on 3-manifolds.

**Lecture 2 - Topology of integrable systems on 4-manifolds**

Speaker：**Elena Kudryavtseva** (Moscow State University)

Time：2020-05-15 21:00-22:00 Beijing time (16:00-17:00 Moscow time)

Abstract: We will give a survey on the topology of integrable Hamiltonian systems on 4-manifolds. Open questions and problems will be also discussed. Recall that, from a topological point of view, an integrable Hamiltonian system can be treated as a singular Lagrangian fibration on a smooth symplectic 2n-manifold whose generic fibres are n-dimensional tori. By a singularity, we mean either a singular point or a singular fibre of the fibration. The topological structure of such singularities is very important for understanding the dynamics of integrable systems both globally and locally. Our goal is to describe topological invariants of such singularities and obtain their classification up to fibrewise homeomorphism (for time being we forget about symplectic structure). The next step is to combine these singularities together to study the global structure of the fibration. For many integrable systems, this structure is completely determined by topological properties of singularities.

**Slides：**https://disk.pku.edu.cn:443/link/3FFA6A7EF0B27A568A9C20B99FA076F6

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**Organized by:**

◆ Sino-Russian Mathematics Center

◆ Mathematics Department, School of Mathematical Sciences (SMS), Peking University

◆ Beijing International Center for Mathematical Research (BICMR), Peking University

◆ Department of Mechanics and Engineering Science, College of Engineering (EC), Peking University

◆ Mathematics Institute of Academy of Mathematics and Systems Science of Chinese Academy of Sciences (AMSS)

◆ Moscow State University (MSU)

◆ Steklov Mathematical Institute (MIAN)

◆ Steklov International Mathematical Center

◆ Moscow Center of Fundamental and Applied Mathematics

Logo and website of Moscow Center of Fundamental and Applied Mathematics