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 9:30pm – 10:30pm (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.

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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