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 53 —— April 4，2024（17:00-18:00 Beijing time, 16:00-17:00 Novosibirsk time）
To Join Tencent Meeting：https://meeting.tencent.com/dm/Ab8dyTLadpbu
Meeting ID：367-2633-1572
Password：202403
On the logarithmic Landau-Ginzburg mirrors of projective toric manifolds
Speaker: Hao Wen (Nankai University)
Time: 17:00-18:00 Beijing time (16:00-17:00 Novosibirsk time)
Abstract: The notion of a logarithmic Landau-Ginzburg (log LG) model for projective toric manifolds is introduced, which is essentially given by equipping the central degenerate fiber of the Landau-Ginzburg mirror family with a log structure. I will show how to recover the cohomological data of the original toric manifold from such a log mirror. I will also describe a perturbative construction of primitive forms by studying the deformation theory of such a log LG model along the line of the work of Si Li, Changzheng Li and Kyoji Saito. This yields a logarithmic Frobenius manifold structure on the base space of the universal unfolding. This talk is based on recent joint work with Kwokwai Chan and Ziming Nikolas Ma.
Bio: Hao Wen got his PhD at Peking University in 2016 under the supervision of Prof. Huijun Fan and then he worked as postdoc at YMSC for three years. He is now an assistant professor at Nankai University.
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Lecture 52 —— March 21，2024（17:00-18:00 Beijing time, 16:00-17:00 Novosibirsk time）
Recording: https://disk.pku.edu.cn/link/AA63FAECFCC23A4D2582C6160D0D055FB9
Valid Until: 2028-04-30 16:33
Mirror maps for Landau-Ginzburg models
Speaker: Alexey Basalaev (NRU Higher School of Economics)
Time：17:00-18:00 Beijing time（16:00-17:00 Novosibirsk time）
Abstract: This talk will discuss mirror maps of the LG-LG and CY-LG type for the quasihomogeneous singularities. We will comment on the old results regarding the so-called invertible polynomials with diagonal symmetries and also present new mirror maps for nondiagonal symmetry groups. In particular, we will propose a mirror partner for the Klein quartic.
Bio: Alexey Basalaev is an assistant professor in the HSE University, researcher in the Skolkovo Institute of Science and Technology
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Lecture 53 —— April 4，2024（17:00-18:00 Beijing time, 16:00-17:00 Novosibirsk time）
To Join Tencent Meeting：https://meeting.tencent.com/dm/Ab8dyTLadpbu
Meeting ID：367-2633-1572
Password：202403
On the logarithmic Landau-Ginzburg mirrors of projective toric manifolds
Speaker: Hao Wen (Nankai University)
Time: 17:00-18:00 Beijing time (16:00-17:00 Novosibirsk time)
Abstract: The notion of a logarithmic Landau-Ginzburg (log LG) model for projective toric manifolds is introduced, which is essentially given by equipping the central degenerate fiber of the Landau-Ginzburg mirror family with a log structure. I will show how to recover the cohomological data of the original toric manifold from such a log mirror. I will also describe a perturbative construction of primitive forms by studying the deformation theory of such a log LG model along the line of the work of Si Li, Changzheng Li and Kyoji Saito. This yields a logarithmic Frobenius manifold structure on the base space of the universal unfolding. This talk is based on recent joint work with Kwokwai Chan and Ziming Nikolas Ma.
Bio: Hao Wen got his PhD at Peking University in 2016 under the supervision of Prof. Huijun Fan and then he worked as postdoc at YMSC for three years. He is now an assistant professor at Nankai University.
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Lecture 52 —— March 21，2024（17:00-18:00 Beijing time, 16:00-17:00 Novosibirsk time）
Recording: https://disk.pku.edu.cn/link/AA63FAECFCC23A4D2582C6160D0D055FB9
Valid Until: 2028-04-30 16:33
Mirror maps for Landau-Ginzburg models
Speaker: Alexey Basalaev (NRU Higher School of Economics)
Time：17:00-18:00 Beijing time（16:00-17:00 Novosibirsk time）
Abstract: This talk will discuss mirror maps of the LG-LG and CY-LG type for the quasihomogeneous singularities. We will comment on the old results regarding the so-called invertible polynomials with diagonal symmetries and also present new mirror maps for nondiagonal symmetry groups. In particular, we will propose a mirror partner for the Klein quartic.
Bio: Alexey Basalaev is an assistant professor in the HSE University, researcher in the Skolkovo Institute of Science and Technology
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Lecture 48 —— May 11, 2023 (17:00-18:00 Beijing time, 16:00-17:00 Novosibirsk time)
Recording: https://disk.pku.edu.cn:443/link/81DAD6E74DECF04671D52DC6DE859B3F
Valid Until: 2027-06-30 23:59
Rotations and integrability in Euclidean space
Speaker: Andrey Tsiganov (St. Petersburg State University)
Time: 17:00-18:00 Beijing time (16:00-17:00 Novosibirsk time)
Abstract: Integrals of motion for finite-dimensional systems can be obtained using the brute force method, symmetries of configuration space, relations with nonlinear evolution equations, using Lax matrices, methods of bi-Hamiltonian geometry, etc. In this talk, we discuss a new combination of Nether's symmetry method with the brute force method that allows us to get three new families of integrable systems in n-dimensional Euclidean space. A few of such systems are related to the Hermitian symmetric space theory and to the multi component nonlinear Schrodinger equations hierarchy.
Bio: Andrey Tsiganov is a Professor at the Department of Computational Physics, St. Petersburg State University. He graduated from St. Petersburg State University in 1980 and obtained Doctor of Sciences degree in 2003. Prof. Tsiganov’s research focuses on mathematical physics, integrable systems of classical and quantum physics, etc.
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Lecture 47 —— April 20, 2023 (17:00-18:00 Beijing time, 16:00-17:00 Novosibirsk time)
Recording: https://disk.pku.edu.cn:443/link/59108A9C0E7650C20621084CA1F64AD5
Valid Until: 2027-05-31 23:59
A higher-dimensional Chevalley restriction theorem for orthogonal groups
Speaker: Jinxing Xu (University of Science and Technology of China)
Time: 17:00-18:00 Beijing time (16:00-17:00 Novosibirsk time)
Abstract: The classical Chevalley restriction theorem asserts that for a semisimple complex Lie group G, the ring of G-invariant polynomials on the Lie algebra g is isomorphic through restriction to the ring of Weyl group invariant polynomials on the Cartan subalgebra. In studying the Hitchin morphism of principal G-Higgs bundles over higher dimensional varieties, Chen and Ngo conjectured a multi-variable generalization of the Chevalley restriction theorem, and they proved the GL and Sp cases. We then prove the orthogonal group case and our treatment can apply to the GL and Sp cases in a uniform way. This result has some interesting corollaries on the determinants and Pfaffians for commutative skew symmetric matrices over an arbitrary char. 0 commutative ring. This is a joint work with Lei Song and Xiaopeng Xia.
Bio: Jinxing Xu is an associate professor at the School of Mathematical Sciences, University of Science and Technology of China. He received PhD at Peking University in 2011 (supervisor: Acad. Gang TIAN). Xu specializes in Algebraic Geometry and has obtained a number of original results on the moduli spaces of Calabi-Yau varieties and the cohomology group of p-adic algebraic varieties. He has published several papers on Adv. Math., Int. Math. Res. Not. etc.
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Lecture 46 —— March 23, 2023 (17:00-18:00 Beijing time, 16:00-17:00 Novosibirsk time)
Recording: https://disk.pku.edu.cn:443/link/36551E85507B5A06010718181535ABA9
Valid Until: 2027-04-30 23:59
Finite-gap approach to anomalous waves in 2 spatial dimensions and the Davey-Stewartson equation
Speaker: Petr Grinevich (Steklov Mathematical Institute of RAS)
Time: 17:00-18:00 Beijing time (16:00-17:00 Novosibirsk time)
Abstract: P. G. Grinevich, P.M. Santini.
One of the mathematical models used in the theory of anomalous (rogue) waves in the Nonlinear Schrodinger equation with special Cauchy data: at the initial time we have a small perturbation of the constant solution, corresponding to the unstable background. In the spatially-periodic setting, the finite-gap method is used. In the previous papers was demonstrated that due to the presence of a small parameter in this problem, namely the amplitude of initial perturbation, the finite-gap formulas can be essentially simplified in the leading order. These leading-order solutions are in close agreement with the results of numerical simulations.
In the present work, we show that this approach can be applied to the focusing Davey-Stewardson 2 equation, which is a 2+1 dimensional integrable analog of the Nonlinear Schrodinger equation, admitting rogue waves type solutions. Formulas expressing the leading order approximation in terms of elementary functions of Cauchy data are obtained in the doubly-periodic setting.
Bio: Petr Grinevich is a leading scientific researcher at the Steklov Mathematical Institute, RAS; he also has part-time positions at the Landau Institute of Theoretical Physics, RAS, and Lomonosov Moscow State University. Scientific interests: Solition equations, including algebro-geometric method of integration, scattering theory.
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Lecture 45 —— March 9, 2023 (17:00-18:00 Beijing time, 16:00-17:00 Novosibirsk time)
Recording: https://disk.pku.edu.cn:443/link/BB50F921DEA78EC68051BF234DC635E1
Valid Until: 2027-04-30 23:59
From the B-toda to the BKP hierarchy
Speaker: Yuancheng Xie (BICMR)
Time: 17:00-18:00 Beijing time (16:00-17:00 Novosibirsk time)
Abstract: ../../docs/20230301111439686441.pdf
Bio: Yuancheng Xie is currently a postdoc at Beijing International Center for Mathematical Research, and he obtained his Ph.D. degree from The Ohio State University in 2021 under the guidance of Yuji Kodama. His current interests lie in integrable systems and related algebras and geometries.
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Lecture 44 —— December 15, 2022 (17:00-18:00 Beijing time, 16:00-17:00 Novosibirsk time)
Recording: https://disk.pku.edu.cn:443/link/421E9A256FF207AEF1CD23364820C76B
Valid Until: 2027-01-31 23:59
A local index formula on some noncommutative orbifolds
Speaker: Anton Savin (Peoples' Friendship University of Russia)
Time: 17:00-18:00 Beijing time (16:00-17:00 Novosibirsk time)
Abstract: The Fredholm index of an elliptic differential operator on a smooth manifold is a global homotopy invariant and it is equal to the topological index defined by Atiyah and Singer. Later Atiyah, Bott and Patodi obtained a local index theorem for twisted Dirac operator which refines the Atiyah--Singer index formula to a pointwise equality, which after integration over the manifold recovers the Atiyah--Singer index formula. Noncommutative geometry is a natural framework for index theories. Connes and Moscovici obtained a general local index formula in noncommutative geometry in terms of coefficients of heat trace expansions. We compute these coefficients and hence obtain an explicit local index formula in the following situation.
On Rn, we consider the Euler operator (an elliptic differential operator of order one and index one, written in terms of creation and annihilation operators) acting on differential forms and show that this operator is local with respect to the action of the affine unitary group by metaplectic operators. Our main result then is an explicit algebraic expression for the Connes-Moscovici coefficients. As a corollary we obtain local index formulae for noncommutative tori and toric orbifolds. The results were obtained in a joint work with Elmar Schrohe (Hannover) and published in: Savin A., Schrohe E. Local index formulae on noncommutative orbifolds and equivariant zeta functions for the affine metaplectic group. Adv. Math. 409 (2022), part A, Paper No. 108624, 37 pp.
Bio: Anton Savin is a Professor at the Peoples' Friendship University of Russia (RUDN University). He obtained his PhD from Moscow State University in 2000 and a Doctor of Sciences degree from RUDN University in 2012. He worked at Moscow State University, the University of Potsdam, the Independent University of Moscow and the University of Hannover before joining RUDN University. He was a winner of the Pierre Deligne Contest for young Russian mathematicians. His research interests include the index theory of elliptic operators and noncommutative geometry.
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Lecture XXXXIII – December 1, 2022
Recording: https://disk.pku.edu.cn:443/link/68F472C86C06B88997B7E000164B2C45
Valid Until: 2026-12-31 23:59
Symplectic and birational geometry
Speaker: Zhiyu Tian (BICMR)
Time: 2022-12-01, 17:00-18:00 Beijing time (16:00-17:00 Novosibirsk time)
Abstract: Algebraic structures are generally considered to be quite rigid, while symplectic structures are more flexible. However, since the 1990s, there have been some striking discoveries that suggest the distinction between the two different worlds is not as strict as one naively thinks. Thanks to the work of many people, we have a better understanding of the symplectic nature of certain algebraic invariants, ranging from basic building blocks of algebraic varieties in birational geometry to invariants of singularities. In this talk, I plan to discuss these connections between symplectic and algebraic geometry, with a particular emphasis on the so-called symplectic birational geometry, some intriguing questions and partial answers.
Bio: Zhiyu Tian is an associate professor at BICMR, Peking University. He obtained his PhD from Stony Brook University in 2011 and worked at Caltech (2011-2014), Bonn (fall 2014) and Institut Fourier (2015-2018) before joining Peking University. He received Qiushi Award for young scientists. His research interest is algebraic geometry.
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Lecture XXXXII – November 17, 2022
Recording: https://disk.pku.edu.cn:443/link/B891C175BE71CD1B4FAEC5E6E4930AB5
Valid Until: 2026-12-31 23:59
Stark-Wannier resonances and cubic exponential sums
Speaker: Alexander Fedotov (St. Petersburg State University)
Time: 2022-11-17, 16:00-17:00 Beijing time (15:00-16:00 Novosibirsk time)
Abstract: /pub/zesxzx2021/docs/20221115101000139036.pdf
Bio: Alexander Fedotov is a Professor at the Department of Higher Mathematics and Mathematical Physics, Faculty of Physics, Saint Petersburg State University. Doctor of Physical and Mathematical Sciences. Prof. Fedotov was an invited speaker at several prestigious conferences including XII International Congress on Mathematical Physics (Brisbane, Australia) and the 20th European Math. Congress. Main scientific interests:
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Lecture XXXXI – November 3, 2022
Recording: https://disk.pku.edu.cn:443/link/1D9D2E557171189F284B44629FC2ED6F
Valid Until: 2026-12-31 23:59
3-Manifold groups and finite quotients
Speaker: Prof. Yi Liu (Beijing International Center for Mathematical Research)
Time: 2022-11-03, 16:00-17:00 Beijing time (15:00-16:00 Novosibirsk time)
Abstract: In this talk, I will discuss profinite completions of 3-manifold groups. I will explain a result of profinite almost rigidity, namely, that finite-volume hyperbolic 3-manifold groups are determined up to finitely many possibilities by their profinite completions.
Bio: Yi Liu is a professor at Beijing International Center for Mathematical Research (BICMR) at Peking University. His research interest lies primarily in 3-manifold topology and hyperbolic geometry. He received his Ph.D. degree in 2012 from the 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. He is an invited speaker of ICM 2022.
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Lecture XXXX – June 9, 2022
Recording: https://disk.pku.edu.cn:443/link/305E993ABF9949487C876F9D881BDE6C
Valid Until: 2026-07-31 23:59
Geometric Langlands and co-adjoint orbits.
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
Recording: https://disk.pku.edu.cn:443/link/E5B7644DC430D723D286320314989157
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
Recording: https://disk.pku.edu.cn:443/link/5ECE85D9E9A340DA1938AF9DE67CECCD
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
Video: https://disk.pku.edu.cn:443/link/79E436355FC4F8183D5BB31C132F51C0
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
Video: https://disk.pku.edu.cn:443/link/BCD8AE63BCE6528E7201AD3A19D81AE0
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.
Video: https://disk.pku.edu.cn:443/link/00D78290DBF63E607B728CC1D34FCC31
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.
Video: https://disk.pku.edu.cn:443/link/C9DFF6AE358405DA91FA713A3A0BD6B0
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.
Video: https://disk.pku.edu.cn:443/link/5E7C0013F32289CA4B2AD57253C0C1C0
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.
Video: https://disk.pku.edu.cn:443/link/0B0A0A0655BA41698F62581B9875820B
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.
Video: https://disk.pku.edu.cn:443/link/8C94582898EB435A172C3CA7731DF4F4
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.
Video: https://disk.pku.edu.cn:443/link/C6D3DE67BA36629464B4CC8D622DC0AA
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.
Video: https://disk.pku.edu.cn:443/link/EB6A80DEAF5CE5C2E40543666B0687BE
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.
Video: https://disk.pku.edu.cn:443/link/2C0F8F5F401539DA1174FC5310B72517
Valid Until：2026-04-30 23:59
Download Slides:
/upload/editor/file/20211022/22085533159.pdf
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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.
Video: https://disk.pku.edu.cn:443/link/E811B836AA43500B1B9BAD6373A54084
Valid Until：2026-04-30 23:59
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.
Slides: https://disk.pku.edu.cn:443/link/F1CA39F7D183F668A242DEFBDEC43AE4
Video: https://disk.pku.edu.cn:443/link/EC62753E3F712DC7EA925DA9CB7BB11B
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).
Video: https://disk.pku.edu.cn:443/link/038A4056A5DFA2E111A615470EFAA6B5
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.
Video: https://disk.pku.edu.cn:443/link/CBBAD79B354C943633CFB00E8896F2F6
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
Video: https://disk.pku.edu.cn:443/link/69A1A095EFD2E28809739E9664B9B6B9
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.
Video: https://disk.pku.edu.cn:443/link/9D0A742DC37AE552CF71BDE948703413
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.
Video: https://disk.pku.edu.cn:443/link/D3FEEF949EB263940D3D4B0DFFD6F356
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
Video: https://disk.pku.edu.cn:443/link/F2CF6D4AC32064CAA1FA056CC9BF1ACE
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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.
Video: https://disk.pku.edu.cn:443/link/AAFA7C7F3BF6CAB45E1062EC4179FA7D
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: https://disk.pku.edu.cn:443/link/79BFC86B8E4BF6B39A0F7342732A65F9
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: https://disk.pku.edu.cn:443/link/3E5C254A4C00FAE8BAB460A757F08C66
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: https://disk.pku.edu.cn:443/link/74EB24823ABACB010B4349C229404068
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.
Video: https://disk.pku.edu.cn:443/link/DE42FB3F2DD64F94C7A966B6B8AC193B
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.
Video: https://disk.pku.edu.cn:443/link/DE42FB3F2DD64F94C7A966B6B8AC193B
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.
Video: https://disk.pku.edu.cn:443/link/F54316E400E92E93AEA5D31C6DE880B2
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.
Video: https://disk.pku.edu.cn:443/link/F3047C92A5299C446000061029BB1915
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.
Download Slides: https://disk.pku.edu.cn:443/link/E89CA08FEABF9D84D2569EE6B5529037
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.
Video: https://disk.pku.edu.cn:443/link/28299E6CBADB355CE99D8E869E298CA7
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.
Video: https://disk.pku.edu.cn:443/link/A2FC001A09694D73B18E709AD0CB2BA5
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.
Video: https://cloud.mail.ru/stock/kv7Re1gF8pM3Jk2us6F9CNS8
or https://disk.pku.edu.cn:443/link/9E7818092043A5BE2BBC442433970C73
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.
Video: https://disk.pku.edu.cn:443/link/349ADE069333C744171CA06241D030A1
Valid Until: 2026-10-01 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.
Video: https://disk.pku.edu.cn:443/link/AA7F132E5B69726DED4743D0B57368C2
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.
Slides Download: https://disk.pku.edu.cn:443/link/4AF416B93C5C1656C693CAA784A33B0C
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.
