This is a series seminar organized by MCM on Symplectic Geometry and Mathematical Physics at MCM410 on each Wednesday. If you have any questions, please feel free to contact us (mcmoffice@math.ac.cn).
Yalong Cao and Zhengyi Zhou
Each Wednesday, MCM410
Upcoming talks:
Date: December 30, 2024 (Monday)
Time: 10:30-11:30
Speaker: Xinle Dai (Harvard University)
Title: TBA
Abstract: TBA
Date: January 6, 2024 (Monday)
Time: 10:30-11:30
Speaker: Yuan Yao (Université de Nantes)
Title: TBA
Abstract: TBA
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Date: June 26, 2024
Time: 14:00-15:00
Speaker: Dr. Yu Pan (Tianjin University)
Title: Augmentations and Exact Lagrangian surfaces
Abstract: Exact Lagrangian surfaces are important objects in the derived Fukaya category. Augmentations are objects of the augmentation category, which is the contact analog of the Fukaya category. In this talk, we discuss various relations between augmentations and exact Lagrangian surfaces. In particular, we realize augmentations, which is an algebraic object, fully geometrically via exact Lagrangian surfaces.
Date: April 10, 2024
Time: 10:30-11:30
Speaker: Prof. Si Li (Tsinghua Univ)
Title: Holomorphic Chern-Simons at Large N
Abstract: We describe the coupling of holomorphic Chern-Simons theory at large N with Kodaira- Spencer gravity. The 1st order deformation is realized by the Loday-Quillen-Tsygan Theorem on the Lie algebra cohomology of large N matrices. We show that the dynamics of Kodaira-Spencer gravity is fully recovered from this large N holomorphic Chern-Simons theory.
Date: May 15, 2024
Time: 10:00-11:00
Speaker: Prof. Zhengyu Zong (Tsinghua Univ)
Title: Open WDVV equations for toric Calabi-Yau 3-folds
Abstract: The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations is an important system of equations in the study of genus zero Gromov-Witten invariants. It implies the associativity of the quantum product. The associativity of the quantum product has many important applications including the recursive formula given by Kontsevich and Manin that calculates the Gromov-Witten invariants of the projective plane. The system of open WDVV equations plays an important role in the study of open Gromov-Witten invariants. It can be viewed as an extension of the WDVV equation to the open sector. The natural structure that captures the WDVV equation is that of a Frobenius manifold. Similarly, the system of open WDVV equations determines the structure of an F-manifold, a generalization of a Frobenius manifold.
In this talk, we prove two versions of open WDVV equations for toric Calabi-Yau 3-folds. The first version leads to the construction of a semi-simple (formal) Frobenius manifold and the second version leads to the construction of a (formal) F-manifold. This is a joint work with Song Yu.
Date: May 29, 2024
Time: 10:30-11:30
Speaker: Prof. Bohan Fang (Beijing International Center for Mathematical Research)
Title: Oscillatory integrals in mirror symmetry
Abstract: I will describe the oscillatory and period integrals on the B-side of mirror symmetry. They correspond to Gromov-Witten primary and descendant invariants of Gamma-modified twisted Chern classes of the mirror coherent sheaves. The cycles for integration correspond to these mirror sheaves by homological mirror symmetry, and one may obtain higher genus invariants if using correct higher genus B-model integrands. I will explain some examples in the setting of toric mirrors and Gross-Hacking-Keel mirror LG models, and discuss application to Gamma conjectures in the toric setting.
Date: May 29, 2024 (Wednesday)
Time: 16:30-17:30
Speaker: Prof. Qizheng Yin (Beijing International Center for Mathematical Research)
Title: Cohomological and motivic aspects of compactified Jacobian fibrations
Abstract: Beauville showed using Fourier transforms that the Chow ring/motive of an abelian variety admits a natural, multiplicative decomposition. I will explain how Beauville’s theory can be extended to certain abelian fibrations with singular fibers. One notable consequence of this extension is a proof of the P=W conjecture in nonabelian Hodge theory. In this talk I will focus on aspects beyond P=W, and discuss some related open questions. Joint work in progress with Davesh Maulik and Junliang Shen.
Date: May 31, 2024 (Friday)
Time: 14:00-15:00
Speaker: Dr. Oliver Edtmair (University of California, Berkeley)
Title: Systoles of convex energy hypersurfaces
Abstract: Hofer-Wysocki-Zehnder proved that every strictly convex energy hypersurface in R^4 possesses a disk-like global surface of section. They asked whether a systole, i.e. a periodic orbit of least action, must span such a disk-like global surface of section. In my talk, I will give an affirmative answer to this question, based on joint work in progress with Abbondandolo and Kang. I will explain how this result can be used to obtain a sharp symplectic embedding result for convex domains in R^4. Moreover, I will explain how this relates to the strong Viberbo conjecture on the equivalence of normalized symplectic capacities.
Date: June 12, 2024
Time: 14:00-15:00
Speaker: Dr. Yu-Wei Fan (YMSC)
Title: Finite subgroups of derived automorphisms of general K3 surfaces
Abstract: We will discuss a classification of finite subgroups of the group of autoequivalences of the derived category of coherent sheaves on a general K3 surface. The main tool is the examination of the actions of autoequivalences on the space of Bridgeland stability conditions, which will be explained in the talk, along with related background on mirror symmetry. Joint work with Kuan-Wen Lai.
Date: June 19, 2024
Time: 14:00-15:00
Speaker: Dr. Honghao Gao (YMSC)
Title: Legendrian knots and Lagrangian fillings
Abstract: Legendrian knots and their exact Lagrangian fillings are central objects to study in low dimensional contact and symplectic topology. Therefore, it is an important question to classify exact Lagrangian fillings up to Hamiltonian isotopy. It is conjectured that this classification is controlled by a quiver and some derived algebraic structures. In this talk, I will review the historical developments, and explain the algebraic machinery to distinguish fillings. Then, I will discuss the ideas to obtain a subjectivity result, which involving new ideas such as understanding polygons on surfaces, quiver with potentials, etc. This is based on a joint work with Roger Casals.
Date: September 11, 2024
Time: 10:30-11:30
Speaker: Prof. Wenbin Yan (Tsinghua Univ)
Title: Modules of W-algebras and affine Springer fibers
Abstract: I will discuss results relating representation theory of affine W-algebras and the geometry of affine Springer fibers. If time permits, I will also dicuss generalization of this correspondence to moduli space of Higgs bundles.
Date: September 18, 2024
Time: 10:30-11:30
Speaker: Prof. Yu Zhao (Beijing Institute of Technology)
Title: Hilbert schemes on blowing ups and the free boson
Abstract: The relation between the gauge theory of 4-manifold and 2-dim conformal field theory is a fundamental question in physics and mathematics. In this talk, we will introduce our identification between the cohomology of Hilbert schemes of blow-up of a surface and the tensor product of cohomology of Hilbert schemes on the surface with a Bosonic Fock space for different cohomology theories. The action of the infinite-dimensional Heisenberg algebra, which might be different with the action of Nakajima and Grojnowski, is also given by geometric correspondences.
Date: September 25, 2024
Time: 10:30-11:30
Speaker: Prof. Jie Zhou (Tsinghua Univ)
Title: A-infinity structure constants of the Fukaya category of symplectic torus, and mock modular forms
Abstract: For a symplectic manifold, the generating functions arising from genus zero open Gromov-Witten invariants give the structure constants for the A infinity-structure (i.e., the homotopy version of associative algebra structure) in the Fukaya category. On the one hand, having a clear understanding of these functions is very useful to test ideas and conjectures in homological mirror symmetry; on the other hand, these functions frequently exhibit nice transformation properties, making them interesting on their own in the theory of modular forms.
In this talk, I will focus on the case of the simplest Calabi-Yau manifolds, namely elliptic curves. I will explain how these generating functions are reduced to counting functions of planar polygons, following Polishchuk-Zaslow. I will then explain what mock modular forms are and why these generating functions are mock modular forms. The talk is based on joint works with Kathrin Bringmann and Jonas Kaszian.
Date: October 10, 2024 (Thursday)
Time: 10:30-11:30
Speaker: Prof. Penghui Li (Tsinghua Univ)
Title: Graded sheaves and categorification of q
Abstract: I will present joint work with Quoc P. Ho on our theory of graded sheaves, which offers a uniform construction of "mixed versions" or "graded lifts" in the sense of Beilinson–Ginzburg–Soergel applicable to arbitrary Artin stacks. As a straightforward reinterpretation of classical results, this framework allows for the categorification of many quantum algebras, including Hecke algebras and positive quantum groups, through graded sheaves. In a new application, we introduce a graded version of Lusztig's character sheaves and utilize it to prove a conjecture by Gorsky, Negut, and Rasmussen, which establishes a connection between the HOMFLY homology of knots and links and the cohomology of coherent sheaves on Hilbert schemes.
Date: October 16, 2024
Time: 10:00-11:30
Speaker: Prof. Naichung Conan Leung (The Chinese University of Hong Kong)
Title: 3d mirror symmetry is mirror symmetry
Abstract: 3d mirror symmetry is a mysterious duality for certain pairs of hyperkahler manifolds, or more generally complex symplectic manifolds/stacks. In this talk, I will describe its relationships with 2d mirror symmetry. This could be regarded as a 3d analog of the paper' Mirror symmetry is T-duality' by Strominger, Yau and Zaslow which described 2d mirror symmetry via 1d dualities.
Date: October 30, 2024
Time: 10:30-11:30
Speaker: Dr. Chenglang Yang (AMSS, CAS)
Title: A relation between the topological vertex and multi-component KP hierarchy
Abstract: The topological vertex, developed by Aganagic, Klemm, Marino and Vafa, provides an explicit algorithm to compute the open Gromov-Witten invariants of smooth toric Calabi-Yau threefolds in mathematics, as well as the A-model topological string amplitudes in physics. In this talk, I will introduce our recent work about the connection between the topological vertex and multi-component KP hierarchy. This talk is based on a joint work with Zhiyuan Wang and Jian Zhou.
Date: November 6, 2024
Time: 10:30-11:30
Speaker: Dr. Ce Ji (Tsinghua Univ)
Title: A generalization of the Witten conjecture from spectral curve
Abstract: The celebrated Witten conjecture initiates the study of relation between intersection theories on moduli spaces of curves (axiomatized as cohomological field theories, CohFTs) and integrable hierarchies. In this talk we will briefly review generalizations of the Witten conjecture over decades. Then we introduce our recent result relating CohFTs with the KP integrable hierarchies, via the technique of topological recursion on spectral curves. Our result is expected to recover nearly all generalizations before. It is based on the joint work with Shuai Guo and Qingsheng Zhang.
Date: November 13, 2024
Time: 10:30-11:30
Speaker: Prof. Jianfeng Lin (Tsinghua Univ)
Title: Monodromy of Milnor fibrations detected by Seiberg-Witten theory
Abstract: Milnor fibration is an important object in algebraic geometry. In complex dimension 3, such fibration gives an open book decomposition of S^5. And the monodromy gives a diffeomorphism on the Milnor fiber. When the Milnor fibration is given by a weighted homogeneous polynomial, the algebraic monodromy (i.e. the monodromy induced map on homology) is always of finite order. We will sketch a proof that, except for the ADE singularities, this monodromy is of infinite order in the smooth mapping class group. Our proof makes uses of recent advances in equivariant Seiberg-Witten-Floer theory. I will discuss the motivation of this problem (simultaneous resolution of ADE singularities by Aityah, Brieskorn and Wahl) and further applications in low dimensional topology. (Based on a joint with Hokuto Konno, Anubhav Mukherjee and Juan Munoz Echaniz).
Date: November 20, 2024
Time: 16:30-17:30
Speaker: Prof. Yi Xie (Peking University)
Title: On the mapping class groups of 4-manifolds with 1-handles
Abstract: Budney and Gabai proved that the mapping class group of the product of a circle and a 3-dimensional disk is an abelian group of infinite rank. Their proof relies on an invariant defined using the configuration spaces of ordered points. In this talk, we will generalize Budney-Gabai's invariant and use it to demonstrate that the mapping class groups of certain 4-manifolds with 1-handles contain infinite rank subgroups. This is joint work with Jianfeng Lin and Boyu Zhang.
Date: November 27, 2024
Time: 10:30-11:30
Speaker: Dr. Song Yu (Tsinghua Univ)
Title: Integrality in open and closed Gromov-Witten theory
Abstract: We will discuss an extension of the well-known Gopakumar-Vafa integrality in Gromov-Witten theory to the open Gromov-Witten theory of toric Calabi-Yau 3-folds relative to Aganagic-Vafa Lagrangians. In this setting, integrality and finiteness properties are captured by a resummation formula of Labastida-Mariño-Ooguri-Vafa originally discovered in the context of Chern-Simons knot/link invariants and large-N duality. We will discuss a verification of these properties for the open Gromov-Witten invariants of toric Calabi-Yau 3-folds. As a consequence, we will explain how to apply the open/closed correspondence in Gromov-Witten theory, developed jointly with Chiu-Chu Melissa Liu, to obtain integrality of a class of genus-zero Gromov-Witten invariants of toric Calabi-Yau 4-folds, which provides additional examples to the higher-dimensional integrality conjecture of Klemm-Pandharipande.
Date: December 4, 2024
Time: 10:30-11:30
Speaker: Prof. Yingchun Zhang (Shanghai Jiao Tong University)
Title: Cluster algebra and quantum cohomology ring
Abstract: We propose a relation between the cluster algebra and the quantum cohomology ring of a quiver variety. In this talk, I will explain this relation for A and D type quivers and give a conjectural construction for general quivers. This is a joint work with Weiqiang He.
Date: December 11, 2024
Time: 10:30-11:30
Speaker: Dr. Junxiao Wang (Peking University)
Title: Mirror Symmetric Gamma Conjecture for Toric GIT quotients via Fourier Transform
Abstract: The mirror symmetric Gamma conjecture states the equality between the central charges of a pair of mirror objects under homological mirror symmetry. In this talk, I will show how the mirror symmetric Gamma conjecture for a toric Fano orbifold and its Landau-Ginzburg mirror arises as the Fourier transform of the equivariant version of mirror symmetric Gamma conjecture on C^d. This is a joint work in progress with Konstantin Aleshkin and Bohan Fang.
Date: December 18, 2024
Time: 10:30-11:30
Speaker: Tianqing Zhu (Tsinghua Univ)
Title: Rationality of the capped vertex functions for Nakajima quiver varieties
Abstract: The capped vertex function was originally defined as the generating series for the relative stable quasi-map counting for the GIT quotient. In the case of Nakajima quiver varieties, it has long been conjectured that the generating series is the Taylor expansion of a rational function. In this talk I will give a proof of the conjecture using fusion operator techniques introduced by Smirnov, which will enable us to use stable envelopes to reduce the problem in some limit form of the qKZ equation. If time permits, I will also talk about how to use the capped vertex function to compute the monodromy for the capping operators in Kahler variables in terms of the elliptic stable envelopes.
