Joint Harvard-CUHK-YMSC Differential Geometry Seminar

During the 2021-22 academic year, the CMSA will be co-hosting the Joint Harvard-CUHK-YMSC Differential Geometry Seminar, organized by Prof. Conan Leung (Chinese University of Hong Kong) and Yun Shi. This seminar is joint between the CMSA, the Harvard Math Department, the Chinese University of Hong Kong, and Tsinghua University’s Yau Mathematical Science Center.

This seminar will take place weekly on Tuesdays at 8:30pm – 9:30pm (Boston time). The meetings will take place virtually on Zoom. To learn how to attend, please contact Yun Shi (yshi@cmsa.fas.harvard.edu)

The schedule below will be updated as talks are confirmed.

Spring 2022

DateSpeakerTitle/Abstract
1/26/2022**
**(9:30 am Hong Kong; 8:30 pm 1/25/2022 ET)
Davesh Maulik (Massachusetts Institute of Technology)Title: Cohomology of the moduli of Higgs bundles via positive characteristic

Abstract: 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 non-abelian Hodge theory in positive characteristic.  Based on joint work with Mark de Cataldo, Junliang Shen, and Siqing Zhang.
2/8/2022 8:30 –9:30 pm ET

2/9/2022 9:30–10:30 HK time
Andre Neves (University of Chicago)TitleGeodesics and minimal surfaces

Abstract: 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. 
2/23/2022 3:00–4:00 am ET


4:00–5:00 pm HK time
Tom Bridgeland (University of Sheffield)Title: Donaldson-Thomas invariants and hyperkahler manifolds: the example of theories of class S[A1]

Abstract: 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.
3/1/2022
8:30–9:30 pm ET

3/2/2022
9:30–10:30 am HK time
Yat-Hin Suen (IBS-Center for Geometry and Physics, Korea)Title: Tropical Lagrangian multi-sections and locally free sheaves

Abstract: The SYZ proposal suggests that mirror symmetry is T-duality. It is a folklore that locally free sheaves are mirror to a Lagrangian multi-section of the SYZ fibration. In this talk, I will introduce the notion of tropical Lagrangian multi-sections and discuss how to obtain from such object to a class of locally free sheaves on the log Calabi-Yau spaces that Gross-Siebert 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 Calabi-Yau surfaces by using combinatorial data obtained from tropical Lagrangian multi-sections.
3/15/2022
9:30 – 10:30 pm ET

3/16/2022
9:30 – 10:30 am HK time
Steve Zelditch 
(Northwestern)
Title: Birkhoff’s conjecture on integrable billiards and Kac’s problem “hearing the shape of a drum”

Abstract: 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. Avila-deSimoi- Kaloshin proved the conjecture for ellipses of sufficiently small eccentricity. Kaloshin-Sorrentino 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 shape.


3/29/2022
9:30 – 10:30 pm ET

3/30/2022
9:30 – 10:30 am HK time
Si Li (Yau Mathematics Science Center, Tsinghua University)Title: Elliptic chiral homology and chiral index

Abstract: We present an effective quantization theory for chiral deformation of two-dimensional 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 beta-gamma/b-c 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].


4/5/2022
9:30 – 10:30 pm ET

4/6/2022
9:30 – 10:30 am HK time
Yalong Cao (RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), Japan)Title: Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds

Abstract: Gromov-Witten invariants of holomorphic symplectic 4-folds vanish and one can consider the corresponding reduced theory. In this talk, we will explain a definition of Gopakumar-Vafa 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 4-folds of K3^[2] type, our conjecture provides a Yau-Zaslow type formula for the number of isolated genus 2 curves of minimal degree. Based on joint works with Georg Oberdieck and Yukinobu Toda.

Fall 2021

DateSpeakerTitle/Abstract
10/05/2021Mu-Tao Wang (Columbia)Title: Angular momentum in general relativity

Abstract: 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 Po-Ning Chen, Jordan Keller, Ye-Kai Wang, and Shing-Tung Yau. 
10/13/2021

*special time: 4am-5am (Boston time)*
Jason D. Lotay (University of Oxford)Title: Some remarks on contact Calabi-Yau 7-manifolds

Abstract: In geometry and physics it has proved useful to relate G2 and Calabi-Yau geometry via circle bundles. Contact Calabi-Yau 7-manifolds are, in the simplest cases, such circle bundles over Calabi-Yau 3-orbifolds. These 7-manifolds provide testing grounds for the study of geometric flows which seek to find torsion-free G2-structures (and thus Ricci flat metrics with exceptional holonomy). They also give useful backgrounds to examine the heterotic G2 system (also known as the G2-Hull-Strominger system), which is a coupled set of PDEs arising from physics that involves the G2-structure and gauge theory on the 7-manifold. I will report on recent progress on both of these directions in the study of contact Calabi-Yau 7-manifolds, which is joint work with H. Sá Earp and J. Saavedra.
10/19/2021Christopher Woodward (Rutgers University)Title: Tropical disk counts

Abstract: (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 “twenty-one disks in the cubic surface” (suggested by Sheridan)  which is an open analog of the twenty-seven lines.  As an application, I will explain why the Floer cohomology of such tropical fibers is well-defined; this is a generalization fo a result of Fukaya-Oh-Ohta-Ono for toric varieties.
11/02/2021Jim Bryan (Department of Mathematics, University of British Columbia)Title: Counting invariant curves on a Calabi-Yau threefold with an involution

Abstract: Gopakumar-Vafa invariants are integers n_beta(g) which give a virtual count of genus g curves in the class beta on a Calabi-Yau threefold. In this talk, I will give a general overview of two of the sheaf-theoretic approaches to defining these invariants: via stable pairs a la Pandharipande-Thomas (PT) and via perverse sheaves a la Maulik-Toda (MT). I will then outline a parallel theory of Gopakumar-Vafa invariants for a Calabi-Yau 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 be equivalent, one in terms of a version of PT theory, and one in terms of a version of MT theory. These invariants can be computed and the conjecture proved in the case where X=SxC where S is an Abelian or K3 surface with a symplectic involution. In these cases, the invariants are given by formulas expressed with Jacobi modular forms. In the case where S is an Abelian surface, the specialization of n_beta(g,h) to h=0 recovers the count of hyperelliptic curves on Abelian surfaces first computed by B-Oberdieck-Pandharipande-Yin. This is joint work with Stephen Pietromonaco.
11/10/21

* note special date/time 3am-4am ET*
Richard Thomas (Department of Mathematics, Imperial College London)Title: Higher rank DT theory from rank 1

Abstract: Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of Joyce-Song 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 S-duality to be governed by (vector-valued, mock) modular forms.
11/24/21
*note special date/time 3am–4am ET*
Nick Sheridan (School of Mathematics, University of Edinburgh)Title: Quantum cohomology as a deformation of symplectic cohomology

Abstract: 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 non-positive), and the log Kodaira dimension of X \ D (which should be non-negative). We will discuss applications to mirror symmetry. This is joint work with Strom Borman and Umut Varolgunes.
12/1/21
*note special date/time 3am–4am ET*
Nigel Hitchin (University of Oxford)Title: Lagrangians and mirror symmetry in the Higgs bundle moduli space

Abstract: 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 finite-dimensional algebras which are deformations of cohomology algebras. Answering some of the questions raised  requires revisiting basic constructions of stable bundles on curves.

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