
Speaker: Yalong Cao (RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), Japan)Title: GopakumarVafa type invariants of holomorphic symplectic foldsVenue: virtualAbstract: GromovWitten invariants of holomorphic symplectic 4folds vanish and one can consider the corresponding reduced theory. In this talk, we will explain a definition of GopakumarVafa type invariants for such a reduced theory. These invariants are conjectured to be integers and have alternative interpretations using sheaf theoretic moduli spaces. Our conjecture is proved for the product of two K3 surfaces, which naturally leads to a closed formula of Fujiki constants of Chern classes of tangent bundles of Hilbert schemes of points on K3 surfaces. On a very general holomorphic symplectic 4folds of K3^[2] type, our conjecture provides a YauZaslow type formula for the number of isolated genus 2 curves of minimal degree. Based on joint works with Georg Oberdieck and Yukinobu Toda. 

Speaker: Si Li (Yau Mathematics Science Center, Tsinghua University)Title: Elliptic chiral homology and chiral indexVenue: VirtualAbstract: We present an effective quantization theory for chiral deformation of twodimensional conformal field theories. We explain a connection between the quantum master equation and the chiral homology for vertex operator algebras. As an application, we construct correlation functions of the curved betagamma/bc system and establish a coupled equation relating to chiral homology groups of chiral differential operators. This can be viewed as the vertex algebra analogue of the trace map in algebraic index theory. The talk is based on the recent work arXiv:2112.14572 [math.QA]. 

Speaker: Steve Zelditch (Northwestern)Title: Birkhoff’s conjecture on integrable billiards and Kac’s problem “hearing the shape of a drum”Venue: VirtualAbstract: Billiards on an elliptical billiard table are completely integrable: phase space is foliated by invariant submanifolds for the billiard flow. Birkhoff conjectured that ellipses are the only plane domains with integrable billiards. AviladeSimoi Kaloshin proved the conjecture for ellipses of sufficiently small eccentricity. KaloshinSorrentino proved local results for all eccentricities. On the quantum level, the analogous conjecture is that ellipses are uniquely determined by their Dirichlet (or, Neumann) eigenvalues. Using the results on the Birkhoff conjecture, Hamid Hezari and I proved that for ellipses of small eccentricity are indeed uniquely determined by their eigenvalues. Except for disks, which Kac proved to be uniquely determined, these are the only domains for which it is known that one can hear their… 

Speaker: YatHin Suen (IBSCenter for Geometry and Physics, Korea)Title: Tropical Lagrangian multisections and locally free sheavesVenue: VirtualAbstract: The SYZ proposal suggests that mirror symmetry is Tduality. It is a folklore that locally free sheaves are mirror to a Lagrangian multisection of the SYZ fibration. In this talk, I will introduce the notion of tropical Lagrangian multisections and discuss how to obtain from such object to a class of locally free sheaves on the log CalabiYau spaces that GrossSiebert have considered. I will also discuss a joint work with Kwokwai Chan and Ziming Ma, where we proved the smoothability of a class of locally free sheaves on some log CalabiYau surfaces by using combinatorial data obtained from tropical Lagrangian multisections. 

Speaker: Andre Neves (University of Chicago)Title: Geodesics and minimal surfacesVenue: VirtualAbstract: There are several properties of closed geodesics which are proven using its Hamiltonian formulation, which has no analogue for minimal surfaces. I will talk about some recent progress in proving some of these properties for minimal surfaces. 

Speaker: Tom Bridgeland (University of Sheffield)Title: DonaldsonThomas invariants and hyperkahler manifolds: the example of theories of class S[A]Venue: VirtualAbstract: I will report on a project which aims to encode the DT invariants of a CY3 triangulated category in a geometric structure on its stability space. I will focus on a class of categories whose stability spaces were studied in previous joint work with Ivan Smith, and which correspond in physics to theories of class S[A1]. I will describe the resulting geometric structures using a kind of complexified Hitchin system parameterising bundles on curves equipped with pencils of flat connections. 

Speaker: Davesh Maulik (Massachusetts Institute of Technology)Title: Cohomology of the moduli of Higgs bundles via positive characteristicVenue: VirtualAbstract: In this talk, I will survey the P=W conjecture, which describes certain structures of the cohomology of the moduli space of Higgs bundles on a curve in terms of the character variety of the curve. I will then explain how certain symmetries of this cohomology, which are predictions of this conjecture, can be constructed using techniques from nonabelian Hodge theory in positive characteristic. Based on joint work with Mark de Cataldo, Junliang Shen, and Siqing Zhang. 

Speaker: Nigel Hitchin (University of Oxford)Title: Lagrangians and mirror symmetry in the Higgs bundle moduli spaceVenue: VirtualAbstract: The talk concerns recent work with Tamas Hausel in asking how SYZ mirror symmetry works for the moduli space of Higgs bundles. Focusing on C^*invariant Lagrangian submanifolds, we use the notion of virtual multiplicity as a tool firstly to examine if the Lagrangian is closed, but also to open up new features involving finitedimensional algebras which are deformations of cohomology algebras. Answering some of the questions raised requires revisiting basic constructions of stable bundles on curves. 

Speaker: Nick Sheridan (School of Mathematics, University of Edinburgh)Title: Quantum cohomology as a deformation of symplectic cohomologyVenue: VirtualAbstract: Let X be a compact symplectic manifold, and D a normal crossings symplectic divisor in X. We give a criterion under which the quantum cohomology of X is the cohomology of a natural deformation of the symplectic cochain complex of X \ D. The criterion can be thought of in terms of the Kodaira dimension of X (which should be nonpositive), and the log Kodaira dimension of X \ D (which should be nonnegative). We will discuss applications to mirror symmetry. This is joint work with Strom Borman and Umut Varolgunes. 

Speaker: Richard Thomas (Department of Mathematics, Imperial College London)Title: Higher rank DT theory from rankVenue: VirtualAbstract: Fix a CalabiYau 3fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of JoyceSong count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants. By the MNOP conjecture the latter are determined by the GW invariants of X. Along the way we also show they are determined by rank 0 invariants counting sheaves supported on surfaces in X. These invariants are predicted by Sduality to be governed by (vectorvalued, mock) modular forms. 

Speaker: Jim Bryan (Department of Mathematics, University of British Columbia)Title: Counting invariant curves on a CalabiYau threefold with an involutionVenue: VirtualAbstract: GopakumarVafa invariants are integers n_beta(g) which give a virtual count of genus g curves in the class beta on a CalabiYau threefold. In this talk, I will give a general overview of two of the sheaftheoretic approaches to defining these invariants: via stable pairs a la PandharipandeThomas (PT) and via perverse sheaves a la MaulikToda (MT). I will then outline a parallel theory of GopakumarVafa invariants for a CalabiYau threefold X with an involution. They are integers n_beta(g,h) which give a virtual count of curves of genus g in the class beta which are invariant under the involution and whose quotient by the involution has genus h. I will give two definitions of n_beta(g,h) which are conjectured to… 

Speaker: Christopher Woodward (Rutgers University)Title: Tropical disk countsVenue: VirtualAbstract: (joint with S. Venugopalan) I will describe version of the Fukaya algebra that appears in a tropical degeneration with the Lagrangian being one of the “tropical fibers”. An example is the count of “twentyone disks in the cubic surface” (suggested by Sheridan) which is an open analog of the twentyseven lines. As an application, I will explain why the Floer cohomology of such tropical fibers is welldefined; this is a generalization fo a result of FukayaOhOhtaOno for toric varieties. 

Speaker: Jason D. Lotay (University of Oxford)Title: Some remarks on contact CalabiYau manifoldsVenue: VirtualAbstract: In geometry and physics it has proved useful to relate G2 and CalabiYau geometry via circle bundles. Contact CalabiYau 7manifolds are, in the simplest cases, such circle bundles over CalabiYau 3orbifolds. These 7manifolds provide testing grounds for the study of geometric flows which seek to find torsionfree G2structures (and thus Ricci flat metrics with exceptional holonomy). They also give useful backgrounds to examine the heterotic G2 system (also known as the G2HullStrominger system), which is a coupled set of PDEs arising from physics that involves the G2structure and gauge theory on the 7manifold. I will report on recent progress on both of these directions in the study of contact CalabiYau 7manifolds, which is joint work with H. Sá Earp and J…. 

Speaker: MuTao Wang (Columbia)Title: Angular momentum in general relativityVenue: VirtualAbstract: The definition of angular momentum in general relativity has been a subtle issue since the 1960′, due to the discovery of “supertranslation ambiguity”: the angular momentums recorded by two distant observers of the same system may not be the same. In this talk, I shall show how the mathematical theory of optimal isometric embedding and quasilocal angular momentum identifies a correction term, and leads to a new definition of angular momentum that is free of any supertranslation ambiguity. This is based on joint work with PoNing Chen, Jordan Keller, YeKai Wang, and ShingTung Yau. 