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 (email@example.com) and Rongxiao Mi (firstname.lastname@example.org).
|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
|10/27/2020||Franco Rota (Rutgers University)||Title: Kuznetsov components of Fano threefolds of index 2 and moduli spaces.|
Abstract: The derived category of a Fano threefold Y of Picard rank 1 and index 2 admits a semiorthogonal decomposition. This defines a non-trivial subcategory Ku(Y) called the Kuznetsov component, which encodes most of the geometry of Y. I will present joint work with M. Altavilla and M. Petkovic, in which we describe certain moduli spaces of Bridgeland-stable objects in Ku(Y), via the stability conditions constructed by Bayer, Macrì, Lahoz and Stellari. Furthermore, in our work we study the behavior of the Abel-Jacobi map on these moduli space. As an application in the case of degree d = 2, we prove a strengthening of a categorical Torelli Theorem by Bernardara and Tabuada.
|11/10/2020||Matej Penciak (Northeastern University)||Title: A new perspective on the 2D Toda-RS correspondence|
Abstract: The 2D Toda system consists of a complicated set of infinitely many coupled PDEs in infinitely many variables that is known to assemble into an infinite-dimensional integrable system. Krichever and Zabrodin made the remarkable observation that the poles of some special meromorphic solutions to the 2D Toda system are known to evolve in time according to the Ruijsenaars-Schneider many particle integrable system. In this talk I will describe work in progress to establish this 2D Toda-RS correspondence via a Fourier-Mukai equivalence of derived categories: a category of “RS spectral sheaves” on one side, and a category of “Toda micro-difference operators” on another. This description of the 2D Toda-RS correspondence mirrors that of the KP-CM correspondence previously established by two of the authors and suggests the existence of a conjectural elliptic integrable
|11/17/2020||Valentino Tosatti (McGill University)||Title: Smooth asymptotics for collapsing Ricci-flat metrics|
Abstract: I will discuss the problem of understanding the collapsing behavior of Ricci-flat Kahler metrics on a Calabi-Yau manifold that admits a holomorphic fibration structure, when the Kahler class degenerates to the pullback of a Kahler class from the base. I will present recent work with Hans-Joachim Hein where we obtain a priori estimates of all orders for the Ricci-flat metrics away from the singular fibers, as a corollary of a complete asymptotic expansion.
|11/24/2020||Yang Li (MIT)||Title: Metric SYZ conjecture|
Abstract: One possible interpretation of the SYZ conjecture is that for a polarized family of CY manifolds near the large complex structure limit, there is a special Lagrangian fibration on the generic region of the CY manifold. Generic here means a set with a large percentage of the CY measure, and the percentage tends to 100% in the limit. I will discuss my recent progress on this version of the SYZ conjecture, with emphasis on how differential geometers think about this problem, and give some hint about where nonarchimedean geometry comes in.