Patrick Lei

Seminar on K-theoretic computations in enumerative geometry (Spring 2025)
About this seminar

Enumerative geometry is a field where in many cases, the answer to a question is an actual formula. In this informal seminar, we will teach each other our tricks for actually computing these formulae. In the process, we will hopefully gain a better understanding of the underlying theory in the field.

Rules
Schedule
1/22
Organizational meeting
1/29
Konstantin Aleshkin (IPMU)
Quasimap central charges for GLSM and Fourier transforms
Mathematically, a Gauge Linear Sigma Model is a curve counting theory associated to a critical locus of a holomorphic function on a GIT quotient of a vector space. Quasimap central charges are generating functions of certain quasimap invariants related to I-functions or vertex functions or hemisphere partition functions in different contexts. I plan to explain how to compute these objects and perform variations of GIT quotient generalizing earlier results for abelian Calabi-Yau GLSM. The talk is based on a joint work in progress with Melissa Liu.
2/05
Patrick Lei
Some aspects of genus-zero mirror symmetry
I will describe some interesting properties of the I-function of the (formal) quintic threefold which are useful in understanding genus-zero mirror symmetry.
Reference: Zagier-Zinger
2/12
No seminar
2/19
Hechen Hu
Hurwitz numbers and the ELSV formula
Hurwitz numbers count the ramified coverings of \(\mathbb{P}^1\) with a given ramification type. Classically, they are studied using the representation theory of symmetric groups. After sketching the classical proof, I'll talk about the formula proposed by Ekedahl, Lando, Shapiro, and Vainshtein (ELSV) that relates the Hurwitz numbers to Hodge integrals and how it can be proven by virtual localization on the moduli space of relative stable maps. If time permits, I'll also talk about the implication of the ELSV formula such as a proof of Witten's conjecture by Okounkov and Pandharipande.
2/26
Davis Lazowski
Heine's \({}_2\phi_1\) and K-theoretic vertex functions
I will explain how Heine's classical \(q\)-hypergeometric function arises as the K-theoretic vertex function for \(\mathbb{P}^1\), and how monodromy for the K-theoretic vertex accounts for many of the classical identities known in the theory of \(q\)-hypergeometric functions.
3/5
Hechen Hu
The Mariño-Vafa formula for one-partition Hodge integrals
In this talk I'll describe a recipe for computing one-partition Hodge integrals in terms of the representation theory of symmetric groups. I'll also describe some applications such as the proof of the ELSV formula.
3/12
No seminar
3/19
No seminar (spring break)
3/26
Patrick Lei
GW theory of Calabi-Yau threefolds I
I will explain an approach to proving structural results about the all-genus Gromov-Witten theory of Calabi-Yau threefolds. We will see how to implement this approach in the one-parameter case and give a proof of the finite generation conjecture of Yamaguchi-Yau in this setting. Along the way, we will learn how to package the contributions of graphs arising from virtual localization using ideas of Givental.
4/2
Patrick Lei
GW theory of Calabi-Yau threefolds II
In this talk, I will explain a proof of the fact that the genus g Gromov-Witten invariants of some Calabi-Yau threefold can be determined from lower-genus invariants up to 3g-3 ambiguities.
4/9
Melissa Liu
Remodeling Conjecture with descendants
The Remodeling Conjecture proposed by Bouchard-Klemm-Mariño-Pasquetti relates open and closed Gromov-Witten invariants of a semi-projective toric Calabi-Yau 3-manifold/3-orbifold to invariants of its mirror curve defined by the Chekhov-Eynard-Orantin topological recursion. It can be viewed as a version all-genus open-closed mirror symmetry, and has been proved in full generality by Bohan Fang, Zhengyu Zong, and the speaker. In this talk, I will describe a mirror theorem relating all-genus equivariant descendant Gromov-Witten invariants of a general semi-projective toric Calabi-Yau 3-orbifold X to certain integrals on its equivariant mirror curve, based on joint work with Bohan Fang, Song Yu, and Zhengyu Zong. Our genus-zero descendant mirror theorem can be viewed as an equivariant Hodge theoretic mirror symmetry with integral structures. We also establish a correspondence between quantum cohomology central charges of coherent sheaves with compact support on X and period integrals of a holomorphic 3-form on the Hori-Vafa mirror of X. This resolves a conjecture proposed by Hosono in 2004.
4/16
Hechen Hu
TBA
4/23
Hechen Hu
TBA
4/30
Tommaso Botta
TBA