辛几何、数学物理与低维拓扑讨论班2023年秋冬学期
Upcoming Seminar
Calendar for the seminar:
Previous Seminar
Time： November 27, Monday, 16:00-18:00
Venue： No. 2 Hainayuan Building, Room 106
Title: Eisenstein series via factorization homology of Hecke categories
Speaker：Penghui Li 李鹏辉 (Tsinghua University）
Abstract: We define the E_2 Hecke category as coherent sheaves on the moduli stack of G-local system on a disk together with P-reduction on the boundary circle, and identify its factorization homology as the (enhenced) spectral Eisenstein series. This is motivated by the gluing partterns in geometric Langlands program where by a theorem of Arinkin-Gaitsgory (in the de-Rham setup), the spectral category is the gluing of (enhenced) spectral Eisenstein series for all parabolic P. It is a joint work with Quoc. P. Ho.
Time： November 20, Monday, 15:00-17:00
Venue： Lecture Hall
Title: Oscillatory integrals in mirror LG models
Speaker： Bohan Fang 方博汉 (Peking University）
Abstract: I will describe, via examples, 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 also describe some applications to Gamma conjectures.
Time： November 13, Monday, 16:00-18:00
Venue： Lecture Hall
Title: Wrapped Floer theory for Lagrangian fillings
Speaker：Yu Pan 潘宇 (Tianjin University）
Abstract: Lagrangian fillings are key objects in symplectic geometry. Wrapped Floer theory can be used to show some rigidity property of embedded Lagrangian fillings. We extend the wrapped Floer theory to immersed Lagrangian fillings and obtain lower bounds of double points of immersed Lagrangian disk fillings.
Time： November 6, Monday, 14:00-16:00
Venue： No. 2 Hainayuan Building, Room 210
Title: Commutative and noncommutative resolutions of singularities
Speaker：Alexey Bondal (Kavli IPMU, Steklov Mathematical Institute Russian Academy of Sciences）
Abstract: Singularity Theory is considered as an important tool in application of geometry to various problems in physics. Singularities of complex and algebraic varieties are recently studied by a combination of geometric and algebraic methods. The structure of a given singularity can be described by means of its classical commutative resolution. Relatively recently, noncommutative resolutions were proposed and intensively studied.
We discuss recent results and conjectures about the derived categories of singularities and their commutative and noncommutative resolutions. These include the null categories of the resolutions, spherical functors, flop-flop functors and Knoerrer periodicity for noncommutative Auslander resolutions.
Time： October 23, Monday, 16:00-18:00
Venue： Lecture Hall
Title: profinite properties of 3-manifold groups
Speaker：刘毅 Yi Liu (Peking University）
Abstract: In this talk, I will survey on progress in studies of profinite properties of 3-manifold groups. Then I will report some recent progress on the profinite invariance or non-invariance regarding Turaev—Viro invariants.
Time： October 19, Thursday, 16:00-18:00
Venue： Lecture Hall
Title: Coulomb branch and enumerative geometry
Speaker：周子浚 Zijun Zhou (Shanghai Jiaotong University）
Abstract: Coulomb branches of 3d N=4 theories and their quantizations, originated appearing in physics and later developed by Braverman-Finkelberg-Nakajima, provide new ways of constructing algebras. In this talk, I will discuss its connection to enumerative geometry, where representations of Coulomb branches are expected to show up from the moduli of quasimaps into Higgs branches. Interesting enumerative invariants, such as I-functions/Okounkov's vertex functions, quantum differential/difference modules, etc. can be described in terms of quantized Coulomb branch. I'll also talk about potential applications, includding variation of GIT and 3d mirror symmetry in the abelian case.
Time： October 13, Friday, 16:00-18:00
Venue： Lecture Hall
Title: Symplectic and smooth excision
Speaker：唐修棣 Xiudi Tang(Beijing Institute of Technology）
Abstract: A symplectic (smooth) excision is a symplectomorphism (diffeomorphism) between a manifold and the complement of a closed subset. We focus on the construction of symplectic excisions by Hamiltonian vector fields and use them to yield symplectomorphisms. Then we explain facts about smooth excisions known and mostly discovered by us. These are not hard but surprising. The talk is based on arXiv:2101.03534 and further developments after.
Time： September 25, Monday, 16:00-18:00
Venue： Lecture Hall
Title: Configuration space integrals and formal smooth structures
Speaker：林剑锋 Jianfeng Lin (Tsinghua University）
Abstract: Watanabe disproved the 4-dimensional Smale conjecture by establishing many disk bundles which are topologically trivial but not smoothly so. Amazingly, Watanabe used Kontsevich's characteristic classes, which are very different from previous invariants that can detect exoticness in dimension 4 (e.g. the Seiberg-Witten invariants and the Donaldson invariants). So one may wonder what's the role played by the smooth structure in this story. In this talk, I will sketch our proof that Kontsevich's characteristic classes only depend on a formal smooth structure (i.e. a vector bundle structure on the topological tangent bundle). This makes the invariant more flexible and allows several new applications. For example, we show that the homeomorphism group of the 4-dimensional sphere or Euclidian space has nontrivial rational homotopy/homology group in infinitely many dimensions. And we show that for any compact orientable 4-manifold, the natural inclusion from the diffeomorphism group to the homeomorphism group is not a homotopy equivalence. Furthermore, we discovered a new MMM (Miller-Morita-Mumford) class, which can obstruct smoothings of 4-dimensional topological bundles. The talk is based on a joint work with Yi Xie.
