Mathematics

Moduli space of compact Riemann surfaces

Jin Hyung To (UIUC)

Abstract: We will overview the moduli space of compact Riemann surfaces.

Riemann-Roch Formula and The Dimension of Our Universe

Lutian Zhao   [email] (UIUC)

Abstract: In this talk we'll introduce the classical Riemann-Roch formula, which appears as a vast generalization of the Euler-Maclaurin formula for the integrals. As an interesting application, the critical dimension for the bosonic string theory can be calculated by these formula to be d=26, which matches with the physical prediction using light-cone quantization. No basic knowledge on string theory and Riemann-Roch will be assumed.

Cancelled

Abstract: Cancelled

Introduction to Cohomological Field Theory

Sungwoo Nam (UIUC)

Abstract: Cohomological field theory(CohFT) was first introduced by Kontsevich and Manin to organize the data of Gromov-Witten theory and quantum cohomology into a list of axioms. Although its main model is Gromov-Witten theory, it has been also successful dealing with problems outside of Gromov-Witten theory. In this talk, I will introduce the notion of CohFT, Givental-Teleman’s classification of semisimple CohFTs and some concrete examples. Basic knowledge of Gromov-Witten theory will be helpful, but it is not assumed in this talk.

A Bird's-Eye View of Seiberg Witten Integrable Systems

Matej Penciak (UIUC)

Abstract: In this talk I will give a rudimentary description of supersymmetric gauge theories, and focus on the particular case of $N=2$ supersymmetry in dimension $4$ with gauge group $SU(2)$. In this setting, originally noticed and explained by Seiberg and Witten in 1994, the moduli of vacua exhibits the structure of an algebraic integrable system. I will explain how this structure manifests itself, and the give a sketch of the calculation that Seiberg and Witten made in their original paper. If time permits, I will explain the generalization of this story to more general gauge groups, and with possible additional matter fields included in the theory.

Introduction to GIT

Itziar Ochoa de Alaiza Gracia (UIUC)

Abstract: The aim of this talk is to give the motivation for the GIT quotient. We will do so by introducing different notions of quotients, illustrated by examples. Finally we will define the Affine and projective GIT quotients.

Introduction to GIT, II

Itziar Ochoa de Alaiza Gracia

Abstract: The aim of this talk is to give the motivation for the GIT quotient. We will do so by introducing different notions of quotients, illustrated by examples. Finally we will define the Affine and projective GIT quotients.

Algebraic Morse theory from GIT

Jesse Huang (UIUC)

Abstract: Birational geometry is closely tied to GIT quotients and variations. In this episode of GIT series, I will apply the machinery to a countable set of basic examples, through which we shall see how the change of linearization produces elementary birational transformations.

GIT quotients of flag varieties

Joshua Wen (UIUC)

Abstract: As we’ve seen, GIT quotients depend on a choice of line equivariant line bundle, and varying this choice can lead to drastic or subtle changes between quotients. After introducing a framework for ‘variation of GIT’ by Dolgachev and Hu, I want to consider a case of the flag variety and its action either by a torus or semisimple group. Here, one already knows many equivariant line bundles, and studying dimensions of invariant sections leads to results of representation-theoretic significance.

Moduli of Twisted Curve

Hao Sun (UIUC)

Abstract: I'll give an introductory talk about the twisted curves. Twisted curves are related to the study of r-spin Witten classes and r-spin geometry of the moduli space of curves.

Organizational Meeting

(Crystalline) differential operators in positive characteristic

Shiyu Shen (UIUC Math)

Abstract: I will talk about several features of (Crystalline) differential operators in characteristic $p$, including the Azumaya property and two theorems by Cartier.

Differential Equations from Hodge Theory

Lutian Zhao   [email] (UIUC Math)

Abstract: The classical theory of elliptic integrals is the milestone in the history of various fields in math: algebraic geometry, differential equations, number theory,.. etc. In this talk, I'll use this as the motivating example for the theory of periods. I'll talk about how we get some equations for the periods and interpret these equations in terms of Hodge theory. As an interesting application, I'll calculate the number of rational curves on quintic threefold by these differential equations. Only complex analysis is assumed.

Window equivalences and spherical functors

Jesse Huang   [email] (UIUC Math)

Abstract: We will appreciate some recent results on the derived categories of GIT quotients through the basic example of a flip/flop.

Line bundles on abelian varieties

Matej Penciak   [email] (UIUC Math)

Abstract: In this talk I want to give classical results on line bundles on abelian varieties. I’ll begin with the Appel-Humbert theorem. Then I will give an introduction to theta functions and show that they can be identified with sections of line bundles. This will connect the older approach with the more well-known description in terms of divisors. Finally, I’ll consider more general problem of classifying vector bundles on elliptic curves, and describe Atiyah’s solution to the classification problem.

Flops and derived categories of threefolds, Part 1

Sungwoo Nam (UIUC Math)

Abstract: In his paper, Bridgeland showed that derived categories of threefolds, which are related by flopping operations, are equivalent. Besides its own interest, this result can be used to study behavior of invariants of threefolds under birational morphisms. In this talk, we will present Bridgeland's work for two weeks. As the main idea involves constructing flop as a moduli space of perverse point sheaves, I'll introduce some notions such as derived categories and t-structures and their properties relevant to the proof. After that, I will give application of the theorem on birational Calabi-Yau threefolds and curve counting invariants.

Flops and derived categories of threefolds, part 2

Ciaran O'Neill (UIUC Math)

Abstract: We will give the proof of Bridgelands theorem, stated last time. We will introduce Fourier-Mukai transforms, an important part of the proof.

Derived categories of abelian categories and applications

Jin Hyung To

Abstract: TBD

Derived categories of abelian categories, II

Jin Hyung To

Abstract: TBA

Congruences of modular forms from an algebro-geometric perspective

Ningchuan Zhang (UIUC Math)

Abstract: In this talk, I’ll give an algebro-geometric explanation of congruences of normalized Eisenstein series following chapter 4 of Nicholas Katz’s paper “$p$-adic properties of modular schemes and modular forms”. The key idea in Katz’s paper is to establish a $p$-adic Riemann-Hilbert correspondence that can translate congruences of normalized Eisenstein series to that of continuous $\mathbb{Z}_p^\times$-representations in rank $1$ free $\mathbb{Z}_p$-modules. The latter is very easy to compute given that $\mathbb{Z}_p^\times$ is topologically cyclic when $p\neq 2$.

Model theory and ideas from algebraic geometry

Chieu Minh Tran (UIUC Math)

Abstract: In this (very soft) talk, I will give a model theory crash course and explain how ideas from classical (or ancient?) algebraic geometry have played an important role in shaping the field as we know it today.

Tiered UFD

Justin Kelm (UIUC Math)

Abstract: We proved the proporty that a strongly tiered UFD will satisfy.