### 北京-新西伯利亚论坛 (Beijing-Novosibirsk Seminar on Geometry and Mathematical Physics)

##### Description

Organizing committee of Beijing-Novosibirsk Seminar on Geometry and Mathematical Physics

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

(2)    A.E. MIRONOV (IM SB RAS, dynamical systems, differential geometry, topology, soliton theory)

(3)    I.A. TAIMANOV (IM SB RAS, integrable systems, geometry, mathematical physics, dynamical systems)

(4)    Youjin ZHANG (THU, mathematical physics, theory of integrable systems)

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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 XXXX – June 9, 2022

Valid Until: 2026-07-31 23:59

Speaker: Prof. Yongbin Ruan (Zhejiang University)

Time: 2022-06-09, 16:00-17:00 Beijing time (15:00-16:00 Novosibirsk time)

Abstract: Geometric Langlands can be interpreted as a mirror symmetry between the moduli space of Higgs bundle of group $G$ via the same moduli space of its Langlands dual $G'$. Once we consider the insertion as a form of parabolic structure or more generally coadjoint orbit, it is clear that the conjectural mirror symmetry of moduli space of Higgs bundle requires a version of mirror symmetry between coadjoint orbits. In the talk, we will explore the possible "coadjoint orbit mirror symmetry".

Bio: Yongbin Ruan is currently a professor at IAS of Zhejiang University and is a world well-known mathematician. He was selected as an Academician of the Chinese Academy of Sciences in 2021. He works in the field of symplectic geometry and mathematical physics. He was the founder of many famous theories in mathematics, like (orbifold) Gromov-Witten theory, relative Gromov-Witten theory, Chen-Ruan cohomology and Fan-Jarvis-Ruan-Witten theory. He has made great contributions to the resolution of many famous conjectures, like the Arnold conjecture, the generalized Witten conjecture and LG/CY correspondence conjectures.

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Lecture XXXIX – May 26, 2022

Valid Until: 2026-06-30 23:59

Isoperiodic confocal families of conics and Painleve VI equations

Speaker: Vladimir Dragovic (University of Texas at Dallas)

Time: 2022-05-26, 13:00-14:00 Beijing time (12:00-13:00 Novosibirsk time)

Abstract: We study Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal family, which is a question that naturally arose in the analysis of the numerical range and Blaschke products. We examine the behaviour of the rotation numbers and discover confocal families of conics with the property that each conic from the family is inscribed in k-Poncelet polygons inscribed in the circle, with the same k. Characterization of all such families is given and it is proved that they always correspond to k = 4. Relationship to the solutions to Painleve VI equations is established. This is based on joint work with Milena Radnovic.

Bio: Vladimir Dragovic is a Professor and Head of the Mathematical Sciences Department at the University of Texas at Dallas. Prior to this he was a Full Research Professor at Serbian Academy of Sciences and Arts, the founder and president of the Dynamical Systems group and co-president of The Centre for Dynamical Systems, Geometry and Combinatorics of the Mathematical Institute of the Serbian Academy of Sciences and Arts. Prof. Dragović graduated and received his Doctor of Sciences in Mathematics degree at the Faculty of Mathematics, University of Belgrade. His research interests include integrable dynamical systems with applications to classical and statistical mechanics, extremal polynomials, isomonodromy deformations, Painleve and Schlesinger equations, algebraic geometry and analysis on Riemann surfaces.

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Lecture XXXVIII – May 12, 2022

Valid Until: 2026-06-30 23:59

Non-orientable Lagrangians and ungraded matrix factorizations.

Speaker: Lino Amorim (Kansas State University)

Time: 2022-05-12, 13:00-14:00 Beijing time (12:00-13:00 Novosibirsk time)

Abstract: I will discuss the Fukaya category of unorientable Lagrangians over a field of characteristic 2. I will explain how to construct the mirror to this category using the localized mirror functor. This leads to a new notion of ungraded matrix factorizations. I will then discuss a version of the Homological Mirror Symmetry conjecture in this setting and prove it in one example.

Bio: Lino Amorim received his PhD from the University of Wisconsin-Madison in 2012, with a thesis entitled “A Künneth Theorem in Lagrangian Floer Theory” under the supervision of Prof. Yong-Geun Oh. After postdoc positions at the University of Oxford and Boston University, he has been an Assistant Professor at Kansas State University since 2017. His research centers on the theory of Fukaya categories, their relations to Gromov-Witten invariants and their role in Homological Mirror Symmetry.

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Lecture XXXVII – April 28, 2022

Valid Until: 2026-05-31 23:59

Lagrangian manifolds and Hamiltonian systems, corresponding to asymptotic solutions of PDE's with singularities.

Speaker: Andrey Shafarevich

Time: 2022-04-28, 15:00-16:00 Beijing time (14:00-15:00 Novosibirsk time)

Abstract: Certain class of asymptotic solutions to PDE's with smooth coefficients is deeply connected with geometric objects - Lagrangian surfaces or complex vector bundles over isotropic manifolds. We study modifications of these objects for the case of PDE's with singularities which appear either in coefficients or on manifolds of independent variables.

Bio: Professor 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.

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Lecture XXXVI – April 14, 2022

Valid Until: 2026-05-31 23:59

Hall-Littlewood functions and Virasoro constraints.

Speaker: Xiaobo Liu

Time: 2022-04-14, 17:00-18:00 Beijing time (16:00-17:00 Novosibirsk time)

Abstract: Hall-Littlewood functions are generalizations of Schur Q-functions which have been used to study Kontsevich-Witten and Brezin-Gross-Witten tau functions. Recently Mironov and Morozov proposed to use Hall-Littlewood functions specialized at roots of unity to study generalized Kontsevich matrix models. Virasoro constraints are powerful tools in the study of matrix models and Gromov-Witten invariants. In this talk, I will describe how Virasoro operators act on Hall-Littlewood functions and applications of such formulas. This is based on joint works with Chenglang Yang.

Bio: Xiaobo Liu obtained his Ph.D. degree from University of Pennsylvania. He was a professor at University of Notre Dame, and obtained Sloan Research Fellowship. He was also an invited speaker of ICM in 2006. Currently, he is a professor at Peking University and is a deputy director of Beijing International Center for Mathematical Research.

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Previous Lectures

Lecture XXXV – March 31, 2022

An explicit formula for the full Gromov-Witten potential of an elliptic curve.

Speaker: Alexander Buryak

Time: 2022-03-31, 17:00-18:00 Beijing time (16:00-17:00 Novosibirsk time)

Abstract: A algorithm to determine all the Gromov-Witten (GW) invariants of any smooth projective curve was obtained by Okounkov and Pandharipande in 2006. They identified stationary invariants with certain Hurwitz numbers and then presented Virasoro type constraints that allow to determine all the other GW invariants (containing insertions from the unit and odd cohomology classes of the target curve) in terms of the stationary ones. In the case of an elliptic curve, I will show that these Virasoro constraints can be explicitly solved leading to a very explicit formula for the full GW potential in terms of the stationary invariants. In particular, this implies that the Dubrovin-Zhang hierarchy for the elliptic curve is Miura equivalent to its dispersionless limit.

Bio: Alexander Buryak is an associate professor at HSE University. His research interests are mathematical physics, algebraic geometry, topology and combinatorics.

Valid Until：2026-05-31 23:59

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Lecture XXXIV – March 10, 2022

Open space index theorems in physics

Speaker: Prof. Guo Chuan Thiang, BICMR, Peking University

Time: 2022-03-10, 17:00-18:00 Beijing time (16:00-17:00 Novosibirsk time)

Abstract: The incredible stability of topological phases such as quantum Hall systems indicates an underlying index theorem protecting the spectral phenomenon. In contrast to the Atiyah-Singer theory used for compactified problems, what is required here is an index theory on noncompact manifolds, with interplay between discrete and continuous spectra. This is the subject of coarse geometry and index theory, and I will explain how they are manifested physically in actual experiments.

Bio: Guo Chuan Thiang is an assistant professor at BICMR. He was a DECRA Research Fellow at the University of Adelaide, and also a Postdoc at the School of Mathematical Sciences, University of Adelaide.

Prior to this, Prof. Thiang completed a DPhil in Mathematics at the University of Oxford, a Master's degree at the University of Cambridge, and studied physics and mathematics at the National University of Singapore.

Guo Chuan Thiang’s research interest revolves around K-theory, algebraic topology, noncommutative geometry, index theory, operator algebras, and functional analysis, usually in the physical contexts of topological phases of matter, quantum theory, and strings. Currently, he is investigating the role of coarse geometry, gerbes, spectral flows, and dualities in contemporary physics problems.

Valid Until：2026-04-30 23:59

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Lecture XXXIII – February 17, 2022

Asymptotic data and Stokes data for the tt*-Toda equations, and some relations with physics

Speaker: Prof. Martin Guest, Waseda University, Japan

Time: 2022-02-17, 17:00 Beijing time, 16:00 Novosibirsk time

Abstract: We give a concrete example of an "integrable" nonlinear p.d.e. related to the 2D Toda equations, whose solutions can be parametrized by asymptotic data and also by Stokes data. The p.d.e. was first studied by the physicists Cecotti and Vafa; certain special solutions are related to Frobenius manifolds (such as quantum cohomology or unfoldings of singularities).  The explicit nature of the data leads to relations with geometry and physics; we describe some of these briefly.

Valid Until：2026-04-30 23:59

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Lecture XXXII - December 16th, 2021

Dispersive shock waves: theory and observations

Speaker: Anatoly M. Kamchatnov (Institute of Spectroscopy, Russian Academy of Sciences)

Time: 2021-12-16 17:00 Beijing time, 16:00 Novosibirsk time

Abstract: In this talk, I give a brief introduction to physics of dispersive shock waves (DSWs) and to basic principles of Gurevich-Pitaevskii theory of such waves. I show that many important characteristics of DSW, such as speeds of its edges and the amplitude of the leading soliton, can be calculated by an elementary method based on the asymptotic theory of propagation of high-frequency wave packets along a smooth background evolved from an intensive nonlinear pulse. In particular, this method allows one to find the number of solitons produced from an initial pulse for a wide class of evolution equations and initial conditions.

Valid Until：2026-04-30 23:59

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Lecture XXXI - December 2nd, 2021
A new construction of the moduli space of pointed stable curves of genus 0
Speaker: Young-Hoon Kiem (Seoul National University)
Time: 2021-12-02 17:00 Beijing time, 16:00 Novosibirsk time
Abstract: The moduli space of n points on a projective line up to projective equivalence has been a topic of research since the 19th century. A natural moduli theoretic compactification was constructed by Deligne and Mumford as an algebraic stack. Later, Knudsen, Keel, Kapranov and others provided explicit constructions by sequences of blowups. The known inductive constructions however are rather inconvenient when one wants to compute the cohomology of the compactified moduli space as a representation space of its automorphism group because the blowup sequences are not equivariant. I will talk about a new inductive construction of the much studied moduli space from the perspective of the Landau-Ginzburg/Calabi-Yau correspondence. In fact, we consider the moduli space of quasimaps of degree 1 to a point over the moduli stack of n pointed prestable curves of genus 0. By studing the wall crossing, we obtain an equivariant sequence of blowups which ends up with the moduli space of n+1 pointed stable curves of genus 0. As an application, we provide a closed formula of the character of the cohomology of the moduli space. We also provide a partial answer to a question which asks whether the cohomology of the moduli space is a permutation representation or not. Based on a joint work with J. Choi and D.-K. Lee.

Valid Until：2026-04-30 23:59

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Lecture XXX - November 18th, 2021
Towards a mirror theorem for GLSMs
Speaker:  Mark Shoemaker (Colorado State University)
Time: 2021-11-18 10:00 Beijing time, 09:00 Novosibirsk time
Abstract: A gauged linear sigma model (GLSM) consists roughly of a complex vector space V, a group G acting on V, a character \theta of G, and a G-invariant function w on V.  This data defines a GIT quotient Y = [V //_\theta G] and a function on that quotient.  GLSMs arise naturally in a number of contexts, for instance as the mirrors to Fano manifolds and as examples of noncommutative crepant resolutions. GLSMs provide a broad setting in which it is possible to define an enumerative curve counting theory, simultaneously generalizing FJRW theory and the Gromov-Witten theory of hypersurfaces. Despite a significant effort to rigorously define the enumerative invariants of a GLSM, very few computations of these invariants have been carried out.  In this talk I will describe a new method for computing generating functions of GLSM invariants.  I will explain how these generating functions arise as derivatives of generating functions of Gromov-Witten invariants of Y.

Valid Until：2026-04-30 23:59

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Lecture XXIX - November 4th, 2021
Poisson manifolds with semi-simple modular symmetry
Speaker: Prof. Xiaojun Chen (Sichuan University)
Time: 2021-11-04 17:00 Beijing time, 16:00 Novosibirsk time
Abstract: In this talk, we study the “twisted” Poincare duality of smooth Poisson manifolds, and show that, if the modular symmetry is semisimple, that is, the modular vector is diagonalizable, there is a mixed complex associated to the Poisson complex which, combining with the twisted Poincare duality, gives a Batalin-Vilkovisky algebra structure on the Poisson cohomology, and a gravity algebra structure on the negative cyclic Poisson homology. This generalizes the previous results obtained by Xu et al for unimodular Poisson algebras. We also show that these two algebraic structures are preserved under Kontsevich's deformation quantization, and in the case of polynomial algebras they are also preserved by Koszul duality. This talk is based on a joint work with Liu, Yu and Zeng.

Valid Until：2026-04-30 23:59

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Lecture XXVIII - October 21st, 2021
Non-diagonalisable Hydrodynamic Type Systems, Integrable by Tsarev's Generalised Hodograph Method
Speaker: Maxim Pavlov
Time：2021-10-21 17:00 Beijing time, 16:00 Novosibirsk time
Abstract: We present a wide class of non-diagonalisable hydrodynamic type systems, which can be integrated by Tsarev' s generalised hodograph method. This class of hydrodynamic type systems contains Jordan blocks 2x2 only. The Haantjes tensor has vanished. This means such 2N component hydrodynamic type systems possess N Riemann invariants and N double eigenvalues only.
First multi-component example was extracted from El's nonlocal kinetic equation, describing dense soliton gas. All conservation laws and commuting flows were found. A general solution is constructed.

Valid Until：2026-04-30 23:59

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Lecture XXVII - October 7th, 2021

On the Saito-Givental theory of elliptic singularities
Speaker: Dr. Xin
Wang (Shandong University)
Time：2021-10-07 17:00
Abstract: In this talk, we will first discuss genus zero Givental I function for Saito’s singularity theory of any invertible singularities. Then we show how to use Givental formalism to do explicit computation about higher genus invariants of Saito-Givental theory. As an example, we compute the genus-1 and genus-2 G function for the associated semisimple Frobenius manifold of elliptic singularities. At the end, we discuss the higher genus structures about the generating function of  Saito-Givental  invariants for Fermat elliptic singularities.

Valid Until：2026-04-30 23:59

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Lecture XXVI - September 16th, 2021

Integration of algebraic functions, polynomial approximation, nonclassical boundary problems and Poncelet-type theorems.
Speaker: Prof. Sergey Tsarev (Siberian Federal University, Krasnoyarsk)
Time: 2021-09-16 17:00-18:00 Beijing time, 16:00-17:00 Novosibirsk time, 12:00-13:00 Moscow time

Abstract: In this review talk we expose remarkably tight relations between the four topics mentioned in the title. Starting from the paper by H.Abel published in 1826 and subsequent results of Chebyshev and Zolotarev we finish at the recent results by Burskii, Zhedanov, Malyshev (et al.)  devoted to algorithmic decidability of some identities for the values of the Weierstrass P-function, unexpected elementary geometric applications and many, many more hidden equivalences in seemingly unrelated areas of analysis, modern computer algebra and geometry.

Valid Until: 2026-10-01 23:59

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Lecture XXV - June 10th，2021

Mirror symmetry for a cusp polynomial Landau-Ginzburg orbifold
Speaker: Alexey Andreevich
Basalaev (HSE)
Time: 2021-06-10 17:00 Beijing time, 16:00 Novosibirsk time, 12:00 Moscow time

Abstract:  We will establish mirror symmetry between  the cusp polynomials considered with a nontrivial symmetry group and Geigle-Lenzing orbifold projective lines. In particular, we will introduce Dubrovin-Frobenius manifold of equivariant Saito theory of any cusp polynomial and show that it is isomorphic to Dubrovin-Frobenius manifold of the respective Geigle-Lenzing orbifold.
We will also show that in the case of simple-elliptic singularities this mirror isomorphism is equivalent the certain relations in the ring of modular forms.
This is a joint work with A.Takahashi (Osaka).

Expiration Time：2026-06-01 23:59

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Lecture XXIV - May 20th，2021

Virasoro conjecture for FJRW theory
Speaker: Dr. Weiqiang
He (Sun Yat-sen University)
Time: 2021-05-20 17:00 Beijing time, 16:00 Novosibirsk time, 12:00 Moscow time

Abstract:  Virasoro conjecture is one of the most fascinating conjecture in Gromov-Witten theory, which is introduced by Eguchi-Hori-Xiong. It state that the Gromov-Witten potential Z is a solution of a sequence of nonlinear differential equation: L_k(Z)=0, k>=-1. And L_k satisfies the following Virasoro relation [L_m, L_n]=(m-n)L_{m+n}

In this talk, I will give a survey on Virasoro conjecture. I will also talk about the explicit  form of Virasoro constraints on FJRW theory and prove it in some simple case, base on the joint work with Yefeng Shen.

Valid Until：2026-04-30 23:59

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Lecture XXIII - April 29th，2021

Spinorial description of G_2 and SU(3)-manifolds
Speaker: I. Agricola (Marburg, Germany)
Time: 2021-04-29 17:00 Beijing time, 16:00 Novosibirsk time, 12:00 Moscow time

Valid Until: 2025-01-01

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Lecture XXII - April 15th，2021

Homological mirror symmetry for chain type polynomials
Speaker: Umut Varolgunes
Time: 2021-04-15 12:00 Beijing time, 11:00 Novosibirsk time

Abstract: I will start by explaining Takahashi's homological mirror symmetry (HMS) conjecture regarding invertible polynomials, which is an open string interpretation of Berglund-Hubsch-Henningson mirror symmetry. In joint work with A. Polishchuk, we resolve this HMS conjecture in the chain type case up to rigorous proofs of general statements about Fukaya-Seidel categories. Our proof goes by showing that the categories in both sides are obtained from the category Vect(k) by applying a recursion. I will explain this recursion categorically and sketch the argument for why it is satisfied on the A-side assuming the aforementioned foundational results. If time permits, I will also mention what goes into the proof in the B-side.

Valid Until：2026-04-30 23:59

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Lecture XXI - March 18th，2021

Virasoro constraints for Drinfeld-Sokolov hierarchies and equations of Painlevé type

Speaker: Prof. Chaozhong Wu (Sun Yat-Sen University)

Time: 2021-03-18 17:00 Beijing time, 16:00 Novosibirsk time

Abstract: By imposing Virasoro constraints to Drinfeld-Sokolov hierarchies, we obtain their solutions of Witten-Kontsevich and of Brezin-Gross-Witten types, and those characterized by certain ordinary differential equations of Painlevé type. We also show the existence of affine Weyl group actions on solutions of such Painlevé-type equations, which generalizes the theory of Noumi and Yamada on affine Weyl group symmetries of the Painlevé-type equations. This work is joint with Si-Qi Liu and Youjin Zhang.

Valid Until：2026-04-30 23:59

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Lecture XX - February 18th，2021

Fukaya category for Landau-Ginzburg orbifolds and Berglund-H\"ubsch homological mirror symmetry for curve singularities.

Speaker: Prof. Cheol-Hyun Cho (Seoul National University)

Time: 2021-02-18 17:00 Beijing time, 16:00 Novosibirsk time

Abstract: For a weighted homogeneous polynomial and a choice of a diagonal symmetry group, we define a new Fukaya category based on wrapped Fukaya category of its Milnor fiber together with monodromy information. It is analogous to the variation operator in singularity theory.  As an application, we formulate a complete version of Berglund-H\"ubsch homological mirror symmetry and prove it for two variable cases.  This is a joint work with Dongwook Choa and Wonbo Jung.

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Lecture XIX

Real-valued semiclassical approximation for the asymptotics with complex-valued phases of the  Hermitian type orthogonal polynomials

S. Yu. Dobrokhotov,  A.V. Tsvetkova (Ishlinsky Institute for Problems in Mechanics RAS)

based on joint work with A.I. Aptekarev, D. N. Tulyakov (Keldysh Institute of Applied Mathematics RAS)

Time: 2021-02-04 17:00 Beijing time, 16:00 Novosibirsk time

Valid Until：2026-04-30 23:59

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Lecture XVIII

Multiple Orthogonal Polynomials with respect to Hermite weights: applications and asymptotics
Speaker: A.I. Aptekarev, (Keldysh Institute of Applied Mathematics RAS),
Joint work with S. Yu. Dobrokhotov, A.V. Tsvetkova (Ishlinsky Institute for Problems in Mechanics RAS) and D. N. Tulyakov (Keldysh Institute of Applied Mathematics RAS)
Time: 2021-01-21 17:00

Abstract: We start with the definition of the Hermite multiple orthogonal polynomials by means of orthogonality relations. Then we present several recent applications, such as eigenvalues distribution of random matrices ensembles with external field and Brownian bridges. The main goal of the talk will be the asymptotics of this polynomial sequence when the degree of the polynomial is growing in the scale corresponding to its variable (so called Plancherel – Rotach type asymptotics). The starting point for our asymptotical analysis is the recurrence relations for multiple orthogonal polynomials. We will present an approach based on the construction of decompositions of bases of homogeneous difference equations. Another approach, based on the  semiclassical  asymptotics in the case of complex-valued phases will be presented in S. Yu. Dobrokhotov’s talk.

Valid Until：2026-04-30 23:59

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Lecture XVII

A discretization of complex analysis for triangulated surfaces.
Speaker: Ivan Dynnikov (Steklov Mathematical Institute, Russia)

Time：2020-12-10 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)

Abstract: I will overview results on a particularly simple discrete version of the notion of a holomorphic function. It was suggested by S.P.Novikov and myself in 2002 and stemmed from the idea that the discrete analogue of the Cauchy--Riemann operator must be a first order difference operator. This is most naturally defined on a triangular lattice or a triangulated surface admitting a checkerboard coloring.

Video Until: 2025-01-01

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Lecture XVI

Exponential Networks and enumerative invariants of local CY

Speaker: Mauricio Romo (Tsinghua University)
Time：2020-11-26 17:00
Abstract: Exponential networks (EN) are a variant of the spectral networks of Gaiotto-Moore-Neitzke, for the case of logarithmic differentials and they naturally lead to Donaldson-Thomas type invariants of local CY 3-folds. I will define ENs and subsequently describe how to get the invariants, illustrated by some examples. If time permits I will show some recent development for cases with compact 4-cycles.

Video Until:2025-01-01

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Lecture XV

Higher, Super, and Quantum
Speaker: Vincent Bouchard (University of Alberta, Саnada)
Time：2020-11-12 17:00
Abstract: Kontsevich and Soibelman recently introduced the concept of quantum Airy structures, which may be understood as generalizations of Virasoro constraints in enumerative geometry. In this talk I will present two broad generalizations, namely higher and super quantum Airy structures. I will explain how many examples of these structures can be constructed as modules of vertex operator algebras, in particular W-algebras. I will comment (and speculate) on the enumerative interpretation of these new constructions in terms of intersection numbers on various moduli spaces. If time permits, I may also briefly explain how these higher and super quantum Airy structures further expand the definition of the Eynard-Orantin topological recursion.

I will overview results on a particularly simple discrete version of the notion of a holomorphic function. It was suggested by S.P.Novikov and myself in 2002 and stemmed from the idea that the discrete analogue of the Cauchy--Riemann operator must be a first order difference operator. This is most naturally defined on a triangular lattice or a triangulated surface admitting a checkerboard coloring.

Video Until: 2025-01-01

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Lecture XIV

Title: Logarithmic GLSM and its applications
Speaker: Prof. Yongbin
Ruan, IAS, Zhejiang University,Hangzhou.
Time: 2020-10-29 17:00-18:00
Abstract: In early 2010, a mathematical theory of GLSM was proposed by Fan-Jarvis-Ruan to generalize both Gromov-Witten theory and FJRW-theory. The mathematical GLSM theory produced an open moduli space, in contrast to the traditional moduli theory where the compactness is required. Then, a cosection (constructed out of superpotential) localized the theory to the critical locus. The above theory is theoretically beautiful, but not so useful in computation. Recently, a delicate compactification of GLSM (logarithmic GLSM) was constructed to remedy the above defect. Its localization formula is proved to be extremely effective to solve many outstanding problems in the subject of Gromov-Witten theory, including BCOV axioms of higher genus Gromov-Witten theory of quintic 3-fold, r-spin conjecture relating r-spin virtual cycle and locus of holomorphic differential, modularity of Gromov-Witten theory of elliptic fibration and so on. In the talk, we will survey the above developments.
These are joint works with Shuai Guo, Felix Janda and Qile Chen.

Valid Until: 2025-01-01

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Lecture XIII

Transposed Poisson algebras

Time: 2020-10-15 17:00 Beijing time (16:00 Novosibirsk time)
Speaker: Prof. Chengming
Bai (Nankai Institute)
Abstract：
We introduce a notion of transposed Poisson algebra which is a dual notion of the Poisson algebra by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra. We interpret the close relationships between it and some structures such as Novikov-Poisson and pre-Lie Poisson algebras including the example given by a commutative associative algebra with a derivation, and 3-Lie algebras.

Valid Until: 2025-01-01

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Lecture XII
The Landau-Ginzburg/Calabi-Yau correspondence for the quintic threefold
Time: 2020-07-10 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)
Speaker: Prof. Shuai
Guo, Department of Mathematics, School of Mathematical Sciences, Peking University
Abstract: In this talk, we will first introduce the physical and mathematical versions of the Landau-Ginzburg/Calabi-Yau correspondence conjecture for the Calabi-Yau threefolds. Then we will explain our approach to prove this conjecture for the most simple Calabi-Yau threefold - the quintic threefold. This is a work in progress joint with Felix Janda and Yongbin Ruan.

Valid Until: 2025-08-31 23:59

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Lecture XI

Kostant, Steinberg, and the Stokes matrices of thett*-Toda equations

Time: 2020-07-03 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)

Speaker: Ho Nan-Kuo (Department of Mathematics, NTHU)

Abstract:

We propose a Lie-theoretic definition of the tt*-Toda equations for anycomplex simple Lie algebra, based on the concept of topological-antitopological fusion which was introduced by Cecotti and Vafa. Our main result concerns the Stokes dataof a certain meromorphic connection, whose isomonodromic deformations are controlled by these equations. First, by exploiting a framework introduced by Boalch,we show that this data has a remarkable structure. It can be described using Kostant’stheory of Cartan subalgebras in apposition and Steinberg’s theory of conjugacy classesof regular elements, and it can be visualized on the Coxeter Plane. Second, we compute canonical Stokes data for a certain family of solutions of the tt*-Toda equationsin terms of their asymptotics.This is joint work with Martin Guest.

Valid Until: 2025-08-31 23:59

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Lecture X

Derived categories and Chow theory of Quot-schemes of Grassmannian type.

Time: 2020-06-26 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)

Speaker: Qingyuan Jiang  (University of Edinburgh)

Abstract:

Quot-schemes of Grassmannian type naturally arise as resolutions of degeneracy loci of maps between vector bundles over a scheme. In this talk we will discuss the relationships of the derived categories and Chow groups among these Quot-Schemes. This provides a unified way to understand many known formulae such as blowup formula, Cayley's trick, projectivization formula, Grassmannian bundles formula and formula for Grassmannain type flops and flips, as well as provide new phenomena such as virtual flips. We will also discuss applications to the study of moduli of linear series on curves, blowup of determinantal ideals, generalized nested Hilbert schemes of points on surfaces, and Brill--Noether problem for moduli of stable objects in K3 categories.

Valid Until: 2025-07-31 23:59

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Lecture IX

Mirror Symmetry for quasi-smooth Calabi-Yau hypersurfaces in weighted projective spaces

Time: 2020-06-19 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)

Speaker: Victor Batyrev (University of Tubingen)

Abstract:

In the talk based on my joint work with K.Schaller I will explain a general combinatorial framework for constructing mirrors of d-dimensional Calabi-Yau orbifolds defined by arbitrary non-degenerate weighted homogeneous polynomials W. Our mirror construction generalizes the one of Berglund-Huebsch-Krawitz in the case of invertible polynomials W.

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Lecture VIII
Gamma conjecture I for del Pezzo surfaces
Time: 2020-06-12 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)
Speaker: Changzheng
Li （Sun Yat-Sen University）
Abstract:
Gamma conjectures were proposed to relate the quantum cohomology of a Fano manifold and the Gamma class interms of differential equations. Gamma conjectures consist of the underlying conjecture O and Gamma conjecture I and II. In this talk, I will first introduce the conjecture O for del Pezzo surfaces, then I will talk about the Gamma conjecture I for del Pezzo surfaces. This talk is based on a joint work with Jianxun Hu, Hua-Zhong Ke and Tuo Yang.

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Lecture VII

Open r-spin intersection theory and the open analog of Witten’s r-spin conjecture.

Speaker: Ran Tessler (Weizmann Institute of Science)

Time：2020-06-05 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)

Abstract：

We will describe the moduli of r-spin disks and its associated vector bundles. We will then define intersection theory on the moduli of r-spin disks, and relate its potential to the r-KdV hierarchy. We will also make a high genus conjecture, generalizing Witten’s r-spin conjecture to the open setting. Based on joint works with A. Buryak and E. Clader.

Valid Until: 2026-10-01 23:59

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Lecture VI

Quantum integrable systems and Symplectic Field Theory

Speaker: Paolo Rossi (University of Padua, Italy)

Time：2020-05-29 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)

Abstract：

Eliashberg, Givental and Hofer's Symplectic Field Theory is a large project aiming to subsume under a unified topological field theoretical approach several techniques from symplectic topology (Floer homology, contact homology and more). Similarly to what happens in Gromov-Witten theory, at its core we find holomorphic curve counting. The general target manifold considered in SFT is a symplectic cobordism between contact manifolds (or more generally between stable Hamiltonian structures). When the cobordism is just a cylinder from a contact manifold to itself, the corresponding operator in SFT is, in particular, a collection of mutually commuting quantum Hamiltonians in a Weyl algebra.

These ideas were behind the introduction, by Buryak and myself, of the quantum double ramification hierarchy, which can be seen as a transposition of the SFT  approach to the algebraic category together with several enhancements. I will introduce the double ramification hierarchy with an eye to its origins in Symplectic FIeld Theory and showcase some examples that we were able to fully compute.

Valid Until: 2026-10-01 23:59

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Lecture V

Geometrization, integrability and knots.

Speaker: A.P. Veselov  (Loughborough, UK and Moscow, Russia)

Time：2020-05-22 17:30 Beijing time (16:30 Novosibirsk time)

Abstract：

I will discuss the coexistence of the chaos and Liouville integrability in relation with Thurston’s geometrization programme, using as the main example the geodesic flows on the 3-folds with SL(2,R)-geometry.

A particular case of such manifold SL(2,R)/SL(2,Z) is known after Milnor and Quillen to be topologically equivalent to the complement of the trefoil knot in 3-sphere. I will explain that the remarkable results of Ghys about modular and Lorenz knots can be naturally extended to the integrable region, where these knots are replaced by the cable knots of trefoil.

The talk is based on a joint work with Alexey Bolsinov and Yiru Ye.

Valid Until: 2026-10-01 23:59

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Lecture IV

Spaces with indefinite metrics and the spectral theory of singular Schrodinger operators

Time: 2020-05-15 17:30

Speaker: P.G. Grinevich (Steklov Mathematical Institute)

Abstract: The famous Korteweg- de Vries (KdV) equation admits important singular solutions, but only very special singularities are compatible  with the KdV dynamics. We show, that for the Schrodinger operators from the KdV Lax pair with such special singularities the spectral theory can be naturally formulated in terms of pseudo-Hilbert spaces with indefinite metrics. IN particular, the number of negative squares in this metric  provides a new conservation law for such solutions. The talk is based on joint works with S.P. Novikov.

Valid Until2026-10-01 23:59

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Lecture III

Topological recursion and KP tau-functions

Time: 2020-05-08 17:00

Speaker:  Sergey Shadrin (University of Amsterdam, Netherlands)

Abstract：

We would like to recall some basic definitions of the so-called Chekhov-Eynard-Orantin theory of topological recursion. Originally it was developed to compute the cumulants for a class of matrix model, but since then it has evolved to one of the key tools on the edge between combinatorics and algebraic geometry that helped to resolve some famous open conjectures. In particular, it has appeared that the topological recursion can be proved for a large class of KP tau-functions from the Orlov-Scherbin family. We'll explain what extra properties of these tau-functions can be derived this way.An example of a direct application of this circle of ideas is a recent proof (our joint work with Dunin-Barkowski, Kramer, and Popolitov) of the so-called r-ELSV formula conjectured by Zvonkine in mid 2000's. We'll try to explain that formula, and, if time permits, sketch the main steps of the proof.

Valid Until: 2026-10-01 23:59

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Lecture II

Stokes phenomenon, reflection equations and Frobenius manifolds

Time: 2020-05-01 17:00

Speaker:  Xiaomeng Xu (Peking university)

Abstract: Reflection equations, arsing from quantum integrable systems with boundary conditions, are the analog of Yang-Baxter equations on a half line. Geometrically, they encode the cylinder braid groups. Algebraically they are closely related to quantum homogenous spaces. In this talk, we first give an introduction to the Stokes phenomenon of an ODE with irregular singularities. We then prove that the Stokes matrices of cyclotomic Knizhnik–Zamolodchikov (KZ) equations give universal solutions to reflection equations. As an application, we show that the isomonodromy deformation of the KZ equations is a quantization of the Dubrovin connections of Frobenius manifolds from various aspects.

Valid Until: 2026-10-01 23:59

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Lecture I

Flat F-manifolds in higher genus and integrable hierarchies

Time: 2020-04-24 17:00

Speaker: Alexandr Buryak (National Research University Higher School of Economics)

Abstract: By Dubrovin--Zhang theory, there is a deep relation between dispersive deformations of the hierarchies of hydrodynamic type corresponding to Frobenius manifolds and the geometry of the moduli spaces of stable algebraic curves. I will talk about a generalization of some of the results of the Dubrovin--Zhang theory for flat F-manifolds, which we obtained in joint works with A. Arsie, P. Lorenzoni and P. Rossi.

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