During the 2020-2021 academic year, the Differential Geometry Seminar will take place on **Tuesday’s at 8:00am or 8:00pm 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 (yshi@cmsa.fas.harvard.edu) and Rongxiao Mi (rongxiao@cmsa.fas.harvard.edu).

Date | Speaker | Title/Abstract |
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2/16/20218:00am ET | Sara Tukachinsky (IAS) | Title: A potential function for open and closed Gromov-Witten theoryAbstract: Classical, or closed, Gromov-Witten invariants count pseudo-holomorphic curves in a symplectic manifold. In particular, genus zero Gromov-Witten invariants count pseudo-holomorphic spheres. Analogously, genus zero open Gromov-Witten invariants count pseudo-holomorphic disks with natural boundary conditions. It turns out that in exploring open Gromov-Witten theory, the closed theory pops up and can be integrated naturally into the conversation, giving rise to a combined, or relative, theory. In this talk we will discuss how this happens and introduce the tensor potential, a function that encodes the heart of this relative theory. This is joint work with Jake Solomon. |

3/2/20218:00am ET | Yoosik Kim (Pusan National University) | Title: Disc potential functions of QuadricsAbstract: A disc potential function plays an important role in studying a symplectic manifold and its Lagrangian submanifolds. In this talk, I will explain how to compute the disc potential function of quadrics. The potential function provides the Landau—Ginzburg mirror, which agrees with Przyjalkowski’s mirror and a cluster chart of Pech—Rietsch—Williams’ mirror. |

3/9/20218:00pm ET | Nawaz Sultani (University of Michigan) | Title: Gromov–Witten invariants of some non-convex complete intersectionsAbstract: For convex complete intersections, the Gromov-Witten (GW) invariants are often computed using the Quantum Lefshetz Hyperplane theorem, which relates the invariants to those of the ambient space. However, even in the genus 0 theory, the convexity condition often fails when the target is an orbifold, and so Quantum Lefshetz is no longer guaranteed. In this talk, I will showcase a method to compute these invariants, despite the failure of Quantum Lefshetz, for a class of orbifold complete intersections. This talk will be based on joint work with Felix Janda (Notre Dame) and Yang Zhou (Harvard) and upcoming work with Rachel Webb (Berkeley). |

3/16/2021 | Junliang Shen (MIT) | Title: Cohomology of the moduli of Higgs bundles and the Hausel-Thaddeus conjectureAbstract: In this talk, I will discuss some structural results for the cohomology of the moduli of semi-stable SL_n Higgs bundles on a curve. One consequence is a new proof of the Hausel-Thaddeus conjecture proven previously by Groechenig-Wyss-Ziegler via p-adic integration. If time permits, we will also discuss the case where the rank of the Higgs bundle is not coprime to the degree, so that the moduli spaces are singular due to the presence of the strictly semi-stable loci. We will explain how intersection cohomology comes into play naturally. Based on joint work with Davesh Maulik. |

3/23/20219:00pm ET | Shiyu Shen (University of Toronto) | Title: Topological mirror symmetry for parabolic Higgs bundles Abstract: I will present work on establishing the correspondence between the (appropriately defined) Hodge numbers of the moduli spaces of parabolic Higgs bundles for the structure groups SL_n and PGL_n, building on previous results of Groechenig-Wyss-Ziegler on the non-parabolic case. I will first describe the strategy used by Groechenig-Wyss-Ziegler, which combines p-adic integration with the generic duality between the Hitchin systems. Then I will talk about the new ingredients that come into play in the parabolic setting. |

3/30/20218:00am ET | Noah Arbesfeld (Imperial College London) | Title: K-theoretic invariants of Hilbert schemes of points and Quot schemes on surfacesAbstract: In the first part of the talk, I will establish the rationality of generating series formed from Euler characteristics of tautological bundles over Hilbert schemes of points on surfaces. In the second part, I will present results on virtual invariants of Quot schemes parameterizing rank zero quotients of trivial bundles on surfaces. The second part of the talk is based on work with Y. Kononov and work with D. Johnson, W. Lim, D. Oprea and R. Pandharipande. |

4/6/2021 8:00am ET | Pierrick Bousseau (ETH) | Title: Quasimodular forms from Betti numbersAbstract: This talk will be about refined curve counting on local P^2, the noncompact Calabi-Yau 3-fold total space of the canonical line bundle of the projective plane. I will explain how to construct quasimodular forms starting from Betti numbers of moduli spaces of dimension 1 coherent sheaves on P^2. This gives a proof of some stringy predictions about the refined topological string theory of local P^2 in the Nekrasov-Shatashvili limit. Partly based on work with Honglu Fan, Shuai Guo, and Longting Wu. |

4/20/2021 9:00pm ET | Jason Lo (CSUN) | Title: Group actions and stability on elliptic surfacesAbstract: There are two natural group actions on the Bridgeland stability manifold of a triangulated category: a left action by the group of autoequivalences, and a right action by the universal covering space of $\mathrm{GL}^+(2,\mathbb{R})$.The left action is much harder to compute than the right action in general. In this talk, we will discuss a method for recognising when a left action is equivalent to that of a right action, and apply it to a non-standard autoequivalence on elliptic surfaces. This work is partly motivated by an attempt to understand equivalences of triangulated categories in representation theory and algebraic geometry at the same time. |

4/27/20218:00am ET | Zhiwei Zheng (Max Planck Institute for Mathematics) | Title: Ball Quotients from Deligne-Mostow Theory and Periods of K3 SurfacesAbstract: In this talk I will first briefly review the Deligne-Mostow theory of moduli spaces of weighted points on the projective line, and a construction of ball quotients from periods of (possibly singular) K3 surfaces with non-symplectic group action. Then I will discuss how these two constructions can be unified for some examples. I will focus on a new case about a 6-dimensional family of K3 surfaces with D4-singularity. This is a joint work with Yiming Zhong. |

5/4/20219:00pm ET | Kwokwai Chan (Chinese University of Hong Kong) | Title: An algebraic model for smoothing Calabi-Yau varietiesAbstract: We are interested in smoothing of a degenerate Calabi-Yau variety or a pair (degenerate CY, sheaf). I will explain an algebraic framework for solving such smoothability problems. The idea is to glue local dg Lie algebras (or dg Batalin-Vilkovisky algebras), coming from suitable local models, to get a global object. The key observation is that while this object is only an almost dg Lie algebra (or pre-dg Lie algebra), it is sufficient to prove unobstructedness of the associated Maurer-Cartan equation (a kind of Bogomolov-Tian-Todorov theorem) under suitable assumptions, so the former can be regarded as a singular version of the Kodaira-Spencer DGLA. Our framework applies to degenerate CY varieties previously studied by Kawamata-Namikawa and Gross-Siebert, as well as a more general class of varieties called toroidal crossing spaces (by the recent work of Felten-Filip-Ruddat). This talk is based on joint works with Conan Leung, Ziming Ma and Y.-H. Suen. |

Date | Speaker | Title/Abstract |
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9/29/2020 | Tristan Collins (MIT) | Title: SYZ mirror symmetry for del Pezzo surfaces and rational elliptic surfacesAbstract: 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. |

10/13/2020 8:00pm ET | Siu-Cheong Lau (Boston University) | Title: Kaehler quiver geometry in application to machine learningAbstract: 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-theoryAbstract: 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 theseconditions. |

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 correspondenceAbstract: 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 metricsAbstract: 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 conjectureAbstract: 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. |