Description
The lecture announcements will be continually updated. The arrangement of the upcoming lectures is as follows:
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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.
To Join Zoom Meeting: https://us02web.zoom.us/j/82573458140?pwd=dE9lTGhVdmljanIvU2djNFg0WFRhdz09
Meeting ID: 825 7345 8140
Password: 626379
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Previous Lectures
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.
Join Zoom Meeting:
https://zoom.us/j/86570181622?pwd=TzIyMnZUL2JZb2RjdjI4amVTWWx1UT09
Meeting ID: 865 7018 1622
Password: 600908
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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.
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.
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.
Download Slides:
/upload/editor/file/20211022/22085533159.pdf
Lecture XXVII - October 7th, 2021
On the Saito-Givental theory of elliptic singularities
Speaker：Dr. Wang Xin (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.
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
Lecture XXV - June 10th，2021
Mirror symmetry for a cusp polynomial
Landau-Ginzburg orbifold
Speaker: Basalaev Alexey Andreevich (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
Lecture XXIV - May 20th，2021
Virasoro conjecture for FJRW theory
Speaker: Dr. He Weiqiang (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.
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://cloud.mail.ru/stock/3XmMgG8dduEKd55qxfS2fCDy
or
https://disk.pku.edu.cn:443/link/69A1A095EFD2E28809739E9664B9B6B9
Valid Until: 2025-01-01
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.
Lecture XXI - March 18th，2021
Virasoro constraints for
Drinfeld-Sokolov hierarchies and equations of Painlevé type
Speaker: Prof. Wu Chaozhong（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.
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.
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
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.
Lecture XVII
Video:https://disk.pku.edu.cn:443/link/79BFC86B8E4BF6B39A0F7342732A65F9
Video Until:2025-01-01
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.
Lecture XVI
Video:https://disk.pku.edu.cn:443/link/3E5C254A4C00FAE8BAB460A757F08C66
Video Until:2025-01-01
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.
Lecture XV
Video:https://disk.pku.edu.cn:443/link/74EB24823ABACB010B4349C229404068
Video Until:2025-01-01
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.
Lecture XIV
Video:https://disk.pku.edu.cn:443/link/DE42FB3F2DD64F94C7A966B6B8AC193B
Video Until: 2025-01-01
Title: Logarithmic GLSM and its
applications
Speaker: Prof.Ruan Yongbin，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.
Lecture XIII
Video:https://disk.pku.edu.cn:443/link/DE42FB3F2DD64F94C7A966B6B8AC193B
Video Until:2025-01-01
Transposed Poisson algebras
Time: 2020-10-15 17:00 Beijing time
(16:00 Novosibirsk time)
Speaker: Prof. Bai Chengming (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.
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. Guo Shuai, 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
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
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: Jiang Qingyuan (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
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.
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: Li
Changzheng （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.
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
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
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
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
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
Lecture II
Stokes phenomenon, reflection equations
and Frobenius manifolds
Time: 2020-05-01 17:00
Speaker: Xu Xiaomeng (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
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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Organization Committee
◆ Huijun FAN
◆ A.E. MIRONOV
◆ I.A. TAIMANOV
◆ Youjin ZHANG
