Seminars on Integrable Systems, USTC, 2019
March, 4, 2019, Room 1418, 14:30-16:00, Jie Zhou (YMSC, Tsinghua University)
Title: On a theory of generalized Jacobi forms and its applications in enumerative theories
Abstract: Motivated by the studies on the Gromov-Witten theory of elliptic curves,
I will explain a differential-geometric approach in contructing a theory of quasi-Jacobi forms.
This theory is then applied to study the Landau-Ginzburg/Calabi-Yau correspondence, which relates
the Fan-Jarvis-Ruan-Witten theory of certain singularities to the Gromov-Witten theroy, for the case of elliptic curves.
Moreover, it explains the holomorphic anomaly phenomenon in the generating series of the Gromov-Witten theory. If time permits,
I will also mention some applications of this theory in the studies of Gromov-Witten/Hurwitz correspondence for elliptic curves,
and geometric quantization.
March, 21, 2019, Room 1418, 10:00-11:30, Hanxiong Zhang （中国矿业大学）(GAP seminar)
Title: Hurwitz numbers and the cut-and-join equation
Abstract:
Hurwitz numbers are classical objects in enumerative geometry. In this introductory talk,
I will review the geometric and algebraic definitions of Hurwitz numbers, and explain why these two definitions are equivalent.
Then I will introduce the generating function of Hurwitz numbers using symmetric functions,
which turns out to be a solution of the cut-and-join equation. Its link with integrable hierarchies via Boson-Fermion correspondence will also be gently addressed.
March, 26, 2019, Room 1218, 16:00-17:30, Ying Zhang (Suzhou University)
Title: The minimal equal length of a pair of simple closed curves in a once punctured torus as the torus runs over its relative Teichmuller space
Abstract: We consider the problem of minimizing the equal length of a pair of simple closed geodesics of given topological type in a once punctured
hyperbolic torus with fixed geometric boundary data as the torus runs over its relative Teichmuller space. For specific pairs with symmetry,
we are able to determine the minimizing torus and hence the minimal length. It is natural to compare the minimal lengths for inequivalent pair
of the same intersection number. As computer experiments show, there is a conjecture that the specific pair of slopes (1/0, 1/n) has its minimal
length smaller than any other pair of slopes (1/0, m/n),
regardless of the geometric boundary data. In joint work with Da Lei, we are able to establish a stronger result as the geometric boundary
of the torus is a conic point and the cone angle is approaching 2π.
April, 23, 2019, 5205, 13:50-14:50, Guofu Yu (Shanghai Jiao Tong University)
Title: On the Biconfluent Heun and its connection with Painleve IV equation
Abstract:
In this talk, we first apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions of certain Hill equation are
Liouvillian solutions. That are solutions obtainable by suitable differential field extensions construction. We establish a full correspondence between
solutions of non-oscillatory type equations and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained for this
Hill equation whose potential owns four exponential functions in the Bank-Laine theory. The differential equation is a periodic form of biconfluent Heun equation.
We further show that these Liouvillian solutions exhibit novel single and double orthogonality and a Fredholm integral equation over suitable integration regions
in complex plane that mimic single/double orthogonality for the corresponding Liouvillian solutions of the Lame and Whittaker-Hill equations, discovered
by Whittaker and Ince almost a century ago. In the second part, we discuss special solutions of Painleve IV equation, that comes from the pioneering works of
Okamoto and Noumi. We report that the linear equation that gives raise to $P_{IV}$ via isomonodromy deformation in the classical works of Garnier and
Jimbo-Miwa also possesses special properties with the same parameter space as the $P_{IV}$. This is a joint work with Yik-Man Chiang and Chun-Kong Law.
April, 23, 2019, 5205, 14:50-15:50, Dali Shen (Tsinghua University)
Title: Frobenius manifolds and root systems
Abstract: Frobenius manifolds are devised to give a coordinate-free way to describe equations of associativity (or WDVV equations)
arising from 2D topological field theory. One such manifold can be viewed as a space parametrizing a family of Frobenius algebras,
endowed with some additional conditions. In this talk I will construct Frobenius structures on the \mathbb{C}^{\times}-bundle
of the complement of a toric arrangement associated with a root system, by making use of a family of torsion free and flat connections on it.
This gives rise to a trigonometric version of Frobenius algebras in terms of root systems.
April, 24, 2019, 5507, 14:30-16:00, Si Li (YMSC, Tsinghua University)
Title: Holomorphic Chern-Simons theory at large N
Abstract: We describe the coupling of holomorphic Chern-Simons theory at large N with Kodaira- Spencer gravity.
We explain a new anomaly cancellation mechanism at all loops in perturbation theory for open-closed topological B-model.
At one loop this anomaly cancellation is analogous to the Green-Schwarz mechanism. As an application, we introduce a type
I version of Kodaira-Spencer theory in complex
dimension 5 and show that it can only be coupled to holomorphic Chern-Simons theory with gauge group SO(32) at quantum level.
This coupled system is conjectured to be a supersymmetric localization of type I string theory.
April, 26, 2019, Room 1418, 16:00-17:30, Si Li (YMSC, Tsinghua University) (GAP seminar)
Title: Singularities: from L^2 Hodge theory to Seiberg-Witten geometry
Abstract: Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on X with compact critical locus, satisfying a general asymptotic condition.
We establish a version of twisted L^2 Hodge theory for the pair (X,f) and prove the corresponding Hodge-to-de Rham degeneration property.
It can be viewed as a generalization of Kyoji Saito's higher residue theory and primitive forms for isolated singularities.
In the second part of the talk, I will explain a connection between primitive period maps and 4d N=2 Seiberg-Witten geometry.
April, 30, 2019, 14:30-16:00, Alexander Cruz Morales (National University of Colombia)
Title: Geometric and algebraic approach to the notion of F-algebroids
Abstract: Motivated by Dubrovin's duality in Frobenius manifolds we introduce the notion of F-algebroids
as a link between the notion of F-manifolds (defined by Hertling-Manin) and the notion of Lie algebroids.
In this talk we will describe the geometric structure of F-algebroids
and discuss an algebraic approach of these objects as a generalization of Lie-Rinehart pairs of algebras.
This is based on joint works with A. Torres-Gomez and J. Gutierrez.
May, 10, 2019, Room 1418, 14:30-16:00, Alexander Cruz Morales (National University of Colombia) (GAP seminar)
Title: Quantum cohomology for isotropic Grassmannians
Abstract: We will discuss the big quantum cohomology ring of isotropic Grassmannians IG(2,2n). After introducing the basic notions we will show that these rings are regular.
In particular, by “generic smoothness”, we will give a conceptual proof of generic semisimplicity of the big quantum cohomology for these Grassmannians.
We will also relate certain decomposition of the ring with a exceptional collection of the derived category of IG(2,2n).
This is based on joint work with A. Mellit, A. Kuznetsov, N. Perrin and M. Smirnov.
May 30, 2019, 14:30-15:30, 周子翔 (复旦大学)
Title: 非局部 Davey-Stewartson I 方程的 Darboux 变换与整体显式解
Abstract:
对非局部 (PT对称的) 可积非线性偏微分方程, Darboux 变换的构造同传统方程并无太大差别, 但由于非局部方程的 Lax 对没有逐点的Lie代数对称性,
用 Darboux 变换或其他任何方法得到的显式解往往不能直接保证是整体解. 我们以非局部 Davey-Stewartson I 方程为例, 通过 Darboux 变换构造了它的显式解,
证明了当参数满足一定条件时, 所得到的显式解一定是整体的, 并且得到了时间趋于无穷时解的渐近性质.
May 30, 2019, 15:30-16:30, 林机 (浙江师范大学)
Title: 周期量级超短激光脉冲的解析解
Abstract:
介绍超短激光的产生和在非线性介质中传播的特点，运用孤立子可积方法研究相关几类非线性偏微分方程的解析解和数值解，得到了新颖的解结构。
June 6, 2019, 10:30-12:00, Room 1308, Shuai Guo (Peking University)
Title: Mathematical approaches to the higher genus Gromov-Witten invariants of compact Calabi-Yau threefolds
Abstract:
During the last two decades, it has been one of the central problem to compute the
Gromov-Witten invariants of Calabi–Yau 3-folds in geometry and physics. In this talk,
we will discuss the recent mathematical approach to the all genera Gromov-Witten
potential functions of the quintic threefolds. We will also discuss about the
possible generalizations of these methods.
June 6, 2019, 16:00-17:30, Room 1418, Shuai Guo (Peking University) (GAP seminar)
Title: An Introduction to Gromov-Witten theory and Mirror Symmetry
Abstract:
In this talk, I will first briefly introduce the motivation of defining and studying Gromov-Witten invariants.
Then I will explain the physics mirror symmetry conjectures related to the Gromov-Witten invariants of the quintic threefolds.
Finally I will list the recent progresses related to these conjectures.
June 12, 2019, 10:00-11:00, 五教5305, Shi-Hao Li (University of Melbourne)
Title: Orthogonal polynomials and integrable systems
Abstract：
Toda lattice, as one of the most famous integrable systems, is the first example connected with orthogonal polynomials.
Later on, a variety of orthogonality are shown to have connections with integrable systems. In this talk, I will present that
several different orthogonal polynomials, originated from matrix models, are linked to different integrable lattices.
June 21, 2019, 16:00-17:30, Room 1418, Duo Li (YMSC, Tsinghua University) (GAP seminar)
Title: Categorical characterization of quadrics
Abstract：
We give a characterization of smooth quadrics in terms of the existence of full exceptional collections
of certain type, which generalizes a result of C.Vial for projective spaces.
July 9, 2019, 14:30-16:00, 五教 5107, Yang Liu (MPIM, SISSA)
Title: Hypergeometric functions in heat coefficients
Abstract：
In this talk, I would like to report some recent progress which relates (pseudo) differential calculus with respect to noncommutative variables to hypergeometric functions.
Such calculus serves as the analytic backbone of the conformal geometry on noncommutative two tori initiated in Connes and Moscovici's 2014 JAMS paper.
The essential difficulty behind the computation is the noncommutativity between the coordinates and their derivatives.
Hypergeometric functions arise in some rearrangement process in the computation of the heat kernel asymptotic.
The main result is a full set of recurrence relations among the hypergeometric family which allows us to obtain their explicit expressions.
More geometric applications will be disused if time permits.
July 12, 2019, 16:00-17:30, Room 1418, Yang Liu (MPIM, SISSA) (GAP seminar)
Title: Modular geometry on noncommutative tori
Abstract：
A general question behind the project is to explore the notion of
intrinsic curvature for noncommutative spaces given in terms of Connes's spectral triple paradigm. It has only recent began (2014) to be
understood on noncommutative two tori in the paper of Connes and Moscovici. The notion of curvature (or general local invariants) is modeled on
spectral geometry of Riemannian manifolds. The new input, purely due to the quantum feature, is the modular theory of weights for Von Neumann algebras. In this talk,
I will focus on some intriguing spectral functions arising from the interplay between the two components. I will present a bending phenomenon in contrast to the
commutative world. My recent progress relates the calculus for those spectral functions to hypergeometric functions and cyclic homology/cohomology.
August, 27, 2019, 10:00-11:00, Room 1318, Youjin Zhang (Tsinghua University) (吴文俊讲座)
Title: Virasoro symmetries of the Drinfeld-Sokolov hierarchies and Painlevé type equations
Abstract:
For the Drinfeld-Sokolov hierarchy associated to an affine Kac-Moody algebra,
we construct its tau-cover and derive its Virasoro symmetries. By imposing the Virasoro constraints,
we obtain solutions of Witten-Kontsevich and of Brezin-Gross-Witten types of the integrable hierarchy,
and those characterized by certain ODEs of Painlevé type. We also establish certain Weyl group actions
on such Painlevé type equations.
October, 8, 2019, 14:30-16:00, Giulio Ruzza (SISSA, UCLouvain)
Title: Matrix models and isomonodromic tau functions
Abstract：
After briefly recalling the theory of isomonodromic
deformations and tau functions, I will explain how to interpret certain
matrix integrals as tau functions. Interestingly this is applied in one
direction (from matrix integrals to tau functions) in the examples of
the Kontsevich-Witten and Brezin-Gross-Witten tau function, but can also
be applied in the opposite direction (from tau functions to matrix
integrals) in other cases (e.g. the Gromov-Witten theory of P^1).
October, 15, 2019, 16:00-17:30, 5506, Christian Blohmann (MPIM, Bonn)(GAP seminar)
Title: Hamiltonian actions: from integrable systems to hamiltonian Lie algebroids
Abstract：
I will give a tour of the notion of hamiltonian actions from basic concepts to recent developments.
In the first hour I will give a self contained review of the following topics: conserved quantities in classical mechanics;
integrability; hamiltonian group actions; toric manifolds, Guillemin-Sternberg convexity theorem, Delzant reconstrucion; symplectic reduction;
Bochner-Weinstein linearizaton at critical points. In the last half hour I will describe a recent generalization of hamiltonian actions to Lie algebroids and Lie groupoids,
which is motivated by a study of the symmetries of General Relativity. This is joint work with Alan Weinstein.
October, 25, 2019, 15:00-16:30, 5107, Giulio Ruzza (SISSA, UCLouvain) (GAP seminar)
Title: Random matrices, integrable hierarchies and isomonodromic deformations
Abstract：
I will give a basic introduction to random matrices, focusing in particular on the Gaussian Unitary Ensemble (GUE),
Wigner Semicircle Law, and combinatorics of the correlators. Depending on time I will move towards more recent
results about computation of correlators of the GUE and of related models.
October, 25, 2019, 16:30-18:00, 5107, Xingjun Lin (Wuhan University) (GAP seminar)
Title: On constructions and classification of rational vertex operator algebras
Abstract：
In this talk, we first talk about some basic results about vertex operator algebras.
We then talk about the classical classification of even unimodular lattices of rank 24. Finally,
we talk about the classification of holomorphic vertex operator algebras of central charge 24.
December, 9, 2019, 10:00-11:00, Room 1418, Alexander Zheglov (Moscow State University)
Title: Commuting partial differential operators and higher-dimensional
algebraic varieties in connection with higher-dimensional analogues of the KP theory
Abstract：
The well-known KP hierarchy is an infinite system of nonlinear partial differential equations,
which describes, among other things, the isospectral deformations of the rings of commuting ordinary differential operators.
In geometric terms, these deformations are described as flows on the Jacobians of the spectral curves of such rings,
which can also be regarded as restriction of the flows defined by the hierarchy on the Sato Grassmannian.
I will talk about the analogue of this theory in the two-dimensional case. In this case, rings of commuting
differential operators of two variables, or more general rings of differential-difference or pseudodifferential operators,
and their isospectral deformations are considered. Deformations are described by analogues of the KP hierarchy — the modified Parshin hierarchies,
which define flows on the moduli space of torsion-free sheaves of a spectral surface.
December, 25, 2019, 9:40-10:40, Room 1418, Liu, Yue (University of Texas Arlington TX)
Title: Stability of peakons of the shallow water modeling with cubic nonlinearity
Abstract：
In this talk, I will start by demonstrating the underlying complexity of the physical system, and then
I will discuss possible simplifications in the shallow water regime along with the relevant physical phenomena.
In particular, I will derive some simplified nonlocal shallow-water models with cubic nonlinearity,
such as integrable Novikov and Modified Camassa-Holm-type equations.
It is shown these approximating model equations possess a single peaked soliton
and multi-peakon solutions. Finally I will prove the single peaked soliton is orbitally stable in the energy space.
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