Spring 2017 Schedule

April 7

Speaker: David Wen

Title: Introduction to the Minimal Model Program: Surfaces III

Abstract: This talk will continue the discussion of the minimal model program of surfaces from last quarter with a proof of Castelnuovo's Contraction Theorem and it's uses in the birational classification of algebraic surfaces. If there is time, I will discuss the interlude towards higher dimensions specifically the obstructions in running a minimal model program for threefolds.

April 14

Speaker: Nadir Hajouji

Title: Classification of fibers in elliptic fibrations

Abstract: In this talk, I will explain Kodaira's classification of singular fibers in elliptic fibrations, and discuss some applications in number theory and algebraic geometry.

April 21

Speaker: Zach Blumenstein

Title: Stacks I

Abstract: The first talk in a three-part introduction to stacks. We introduce three foundational concepts: Grothendieck topologies, topoi, and fibered categories.

April 28

Speaker: Nadir Hajouji

Title: Introduction to Modular Form

Abstract: In this talk, I will show how modular forms arise naturally in the study of moduli spaces of elliptic curves. I will discuss applications to the theory of elliptic curves and number theory more generally.

May 5

Speaker: Nathan Schley

Title: Cure for a Splitting Headache

Abstract: The Brauer group Br(F) for a field F is the equivalence classes of finite dimensional central simple algebras over F with the equivalence relation generated by taking F-tensor products with algebras of the form M_n(F).

A useful way to study the Brauer group is to study subgroups that are kernels of Brauer group homomorphisms Br(F) into Br(K), each induced by taking an extension of scalars over K. In particular, extending a finite dimensional central simple algebra A by a splitting field K will send [A] to the identity element in Br(K).

In this talk, I will give a brief presentation of the Braur group of a field, Group cohomology for finite groups and a brief survey that (hopefully) ends with motivation for the isomorphism H^2(Gal(F^{sep}/F), F^*) \cong Br(F), using only cohomology to express the full Brauer group of any field.

May 12

Speaker: David Wen

Title: Zariski Decomposition, and So Can You!

Abstract: The Zariski decomposition is originally a theorem about effective divisors on algebraic surfaces, mainly saying that effective divisors on surfaces can be decomposed into a "positive" and a "negative" part. While the statement of the theorem will seem somewhat technical with algebraic geometric jargon, the proof at it's core can be reduced to pure linear algebra. This talk is based off of a paper that can be found on the arXiv called "Zariski Decomposition: a new (old) chapter of linear algebra" by Bauer, Caibar and Kennedy, and will attempt to realize and prove the Zariski decomposition as a statement in the framework of linear algebra.

May 19

No Seminar

May 26

Speaker: Zach Blumenstein

Title: Stacks II

Abstract: Stacks are a fundamental tool in the modern theory of invariants and moduli in algebraic geometry, needed to address the pathologies of quotients of varieties by group actions. We introduce the definition of a stack and try to make sense of it. We also explain the stacks-based approach to moduli theory and relate it to the more classical theory.

June 2

Speaker: Zach Blumenstein

Title: Stacks III

Abstract: We finish our series on stacks by discussing root stacks and the construction of the moduli space of genus-g curves.

June 9

No Seminar