During the Fall 2020 semester, the Differential Geometry Seminar will take place on Tuesday’s at 8:00am ET. This seminar is a joint event between Harvard CMSA and Tsinghua University’s Yau Mathematical Science Center. To learn how to attend, please contact Yun Shi (firstname.lastname@example.org) and Rongxiao Mi (email@example.com).
|9/29/2020||Tristan Collins (MIT)||Title: SYZ mirror symmetry for del Pezzo surfaces and rational elliptic surfaces|
Abstract: I will discuss some aspects of SYZ mirror symmetry for pairs (X,D) where X is a del Pezzo surface or a rational elliptic surface and D is an anti-canonical divisor which is either smooth or a wheel of rational curves. In particular I will explain the existence of special Lagrangian fibrations and mirror symmetry for (suitably interpreted) Hodge numbers. If time permits, I will describe a proof of SYZ mirror symmetry for del Pezzo surfaces. This is joint work with A. Jacob and Y.-S. Lin.
|8/6/2020||Lutian Zhao (UIUC)||Title: The Gopakumar-Vafa invariants for local P2.|
Abstract: In this talk, I will introduce the Gopakumar-Vafa(GV) invariant and show one calculation on the nonreduced cycle. The GV invariant is an integral invariant predicted by physicists that counts the number of curves inside a given Calabi-Yau threefold. The definition has been conjectured by Maulik-Toda in 2016 in terms of perverse sheaf. I’ll use this definition on the total space of the canonical bundle of P2 and compute the associated invariants. This verifies a physical formula based on the work of Katz-Klemm-Vafa in 1997.
|Siu-Cheong Lau (Boston University)||Title: Kaehler quiver geometry in application to machine learning|
Abstract: Quiver theory and machine learning share a common ground, namely, they both concern about linear representations of directed graphs. The main difference arises from the crucial use of non-linearity in machine learning to approximate arbitrary functions; on the other hand, quiver theory has been focused on fiberwise-linear operations on universal bundles over the quiver moduli.
Compared to flat spaces that have been widely used in machine learning, a quiver moduli has the advantages that it is compact, has interesting topology, and enjoys extra symmetry coming from framing. In this talk, I will explain how fiberwise non-linearity can be naturally introduced by using Kaehler geometry of the quiver moduli.
|10/20/2020||Henry Liu (Columbia University)||Title: Self-duality in quantum K-theory|
Abstract: When we upgrade from equivariant cohomology to equivariant K-theory, many important algebraic/geometric tools such as dimensional vanishing become inapplicable in general. I will explain some nice conditions we can impose on K-theory classes to restore some of these tools. These conditions hold for many types of curve-counting theories (e.g. quasimaps) and are crucial for the development of those flavors of quantum K-theory, but they notably are not present in Gromov-Witten theory. I will describe an attempt to twist GW theory to fulfill these