## No Theory* Seminar

### Dept. of Mathematical Sciences, Binghamton University

Organizer: Vaidehee Thatte

## Speakers:

### Most of our speakers are students or postdocs in the Department of Mathematical Sciences.

10 Reasons to sign up for talk

### "Pre-Talks" by Arithmetic Seminar speakers, whenever possible.

If you have any questions/comments about the seminar OR would like to give a talk OR would like to request a talk on a specific topic, please contact me!

## Spring 2019

### Unless mentioned otherwise, we will meet on Mondays at 3:30 p.m. or on Tuesdays at 4:15 p.m. in WH 100E.

References and supplementary notes will be made available for these topics, whenever possible.

The talks will be self-sufficient, however, and it is not necessary to study the suggested material before the seminar meetings.

1. Tuesday, March 12
• Organizational Meeting
2. Monday, March 25
• Speaker: Andrew Lamoureux ("Pre-Talk")
• Title: Introduction to the p-adic numbers
• Abstract: This talk will discuss the construction and basic properties of the p-adic numbers and p-adic integers, such as their arithmetic, their topology, and Hensel's Lemma.
• Slides from the corresponding Arithmetic Seminar talk.
3. Monday, April 1
• Speaker: Daniel Rossi
• Title: Linear Algebraic Groups
• Abstract: A linear algebraic group is an affine variety which is also a group, in such a way that the group operation is ``compatible'' with the geometric structure. In this talk, I will provide the necessary background to make this notion precise. Then, I will try to accomplish two goals. The first is to describe some of the key ideas involved in understanding the structure of these objects. Linear algebraic groups arise in a variety of contexts; my own interest in them is mainly in how they relate to the finite simple groups of Lie type. This relationship underlies much of the modern understanding of these groups. So, my second goal is to describe the method by which the simple groups of Lie type arise from these linear algebraic groups.
• References: Contents of the talk - Malle, Testermann, "Linear algebraic groups and finite groups of Lie type." Background (Alg. Geom.) - Geck, "An introduction to algebraic geometry and algebraic groups."
4. Monday, April 8
• Speaker: Vaidehee Thatte
• Title: Complete Discrete Valuation Rings (CDVRs) - Properties and Examples
• Abstract: We will discuss some basic properties of CDVRs and look at a few examples of Artin-Schreier extensions of such rings. We will explicitly compute some classical invariants of ramification theory in these cases, the definitions of these invariants will be reviewed during the talk. Essentially, we will look at power series rings A= k[[X]] with X-adic valuation, where k is a "nice enough" field. Let K be the field of fractions of A. We will look at Artin-Schreier extensions of degree p of such valuation fields K, when the characteristic of k is p>0. These are the (non-trivial) extensions L of K obtained by attaching to K roots of an Artin-Schreier polynomial T^p-T=f; where f is an element of K. (This may also serve as a "pre-pre-talk" for Dr. Bell's talk on April 9.)
• Rough notes.
• References: Local Fields (J.P.Serre), Algebraic Number Theory (J. Neukirch), Commutative Algebra II (O. Zariski, P. Samuel), A survey paper by L. Xiao, I. Zhukov.
5. Tuesday, April 9 : This will be a Pre-Talk for the Arithmetic Seminar Talk titled "Local-to-Global Extensions for Wildly Ramified Covers of Curves ". Please note the room and time for this special talk.
6. Tuesday, April 16
• Speaker: Michael Gottstein
• Title: Introduction to Kummer Theory - I
• Abstract: A Kummer extension of exponent n is a Galois field extension L|K such that K contains n distinct roots of unity and the Galois group G is abelian of exponent n. In particular, every element of G has a finite order and the lcm of all these orders is n. When G is a finite group, the exponent of G divides |G|. Let K be a field that contains n distinct roots of unity. Adjoining to K the nth root of any element h of K creates a (finite) Kummer extension of degree d, where d divides n. Kummer theory is concerned with classifying all Kummer extensions L of K that have exponent n. To do this as stated, the machinery needed is more sophisticated than for the finite case. In this talk we will discuss the problem and the techniques involved so that we can see some of the first results in the theory.
7. April 22/23 - No Seminar (There are two talks in Arithmetic Seminar this week, followed by Ninth Annual Upstate Number Theory Conference at Cornell.)
8. April 29/30 - No Seminar (There are two talks in Arithmetic Seminar this week.)
9. Tuesday, May 7
• Speaker: Michael Gottstein
• Title: Introduction to Kummer Theory - II
• Abstract (tentative): We will continue the discussion about classifying all Kummer extensions L of K that have exponent n.

## ~~~~~~~

*: Originally the name of a student number theory seminar at the University of Chicago. "No Theory" was the result of a typo for number (no.) theory.