### Student Algebra-Number Theory Archives for Academic Year 2021

#### Introduction to Deformation Theory

When: Mon, October 4, 2021 - 5:00pm

Where: Kirwan Hall 3206

Speaker: Shin Eui Song (UMD) - https://www.math.umd.edu/~sesong/

#### Moduli Space of Elliptic Curves

When: Mon, October 11, 2021 - 5:00pm

Where: Kirwan Hall 3206

Speaker: Samuel Bachhuber (UMD) -

#### Global moduli problems and representability

When: Mon, October 18, 2021 - 5:00pm

Where: Kirwan Hall 3206

Speaker: Jackson Hopper (UMD) -

#### Local deformation and Algebraization

When: Mon, October 25, 2021 - 5:00pm

Where: Kirwan Hall 3206

Speaker: Shin Eui Song (UMD) - https://www.math.umd.edu/~sesong

Abstract: We first review local deformation functors. Studying local deformations is useful as there may not be any global moduli space. We will review the definition of versal deformations and effective formal deformation. One of the fascinating results is that having an effective formal versal deformation implies that an algebraization exists. This follows from a deep approximation theorem of Artin.

#### The cotangent complex and applications to deformation theory

When: Mon, November 1, 2021 - 5:00pm

Where: Kirwan Hall 3206

Speaker: Steven Jin (UMD) -

Abstract: Suppose $X$ is a scheme (or Artin stack) over a base scheme $S$. The cotangent complex of $X/S$ is a certain complex $L_{X/S}$ of locally free $\mathcal{O}_X$-modules in the nonpositive degrees that in some sense generalizes the sheaf of relative Kahler differentials. In this talk, we will motivate why such an object might be desirable, and we will sketch some of its most important properties. We will conclude with a discussion of how the cotangent complex controls the deformation theory of $X/S$.