## No Theory* Seminar

### Dept. of Mathematical Sciences, Binghamton University

Organizer: Vaidehee Thatte

### Attendees:

This is a seminar aimed at graduate students working in number theory and related areas (broadly defined). Talks should be accessible to early-career graduate students. Undergraduate students interested in these areas are quite welcome to participate as well.

### Speakers:

Most of our speakers are students or postdocs in the area.

**10 Reasons to sign up for talk**

There will be some special lectures by our faculty members &

"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****!**

### Topics:

Including but not limited to - Number Theory, Arithmetic Geometry, Algebraic Geometry, Commutative Algebra, Valuation Theory, Diophantine Geometry, Tropical Geometry, Arithmetic Dynamics.

## ~ All the upcoming talks will be virtual via Zoom ~

## Fall 2020

### Graduate Assistant: Hari Asokan

### We will meet during one of these four time slots, all times ET.

### Mon 6:30 - 7:30 pm, Tue 3 - 4 pm, Tue 4:30 - 5:30 pm, Tue 5:30 - 6:30 pm

*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.*

September 14, Monday 6:30 - 7:30 pm

**Speaker: Vaidehee Thatte****Informal discussion:**Productivity and self-care during COVID, life as a graduate student/postdoc in mathematics, overview of the job application process , etc.

October 6, Tuesday 3 pm - 4 pm

**Speaker:****Juliette Bruce****Title:**Computing Syzygies**Abstract:**I will discuss recent large-scale computations, which utilize numerical linear algebra and highly distributed, high-performance computing to generate data about the syzygies of various algebraic surfaces. Further, I will discuss how this data has led to several new conjectures.

October 20, Tuesday 3 pm - 4 pm

**Title:**Log Intersection Theory and the Log Product Formula**Abstract:**Kato: Log structures are "magic powder" that makes mildly singular spaces appear smooth. Log schemes literally lie between ordinary schemes and tropical geometry, and are related to Berkovich Spaces. Problems in Gromov-Witten Theory demand intersection theoretic machinery for slightly singular spaces. Log structures have solved similar problems in Hodge Theory, D-modules, Connections and Riemann-Hilbert Correspondences, Abelian Varieties (esp. Elliptic Curves), etc. How can they be used to define a reasonable intersection theory for curve counting on singular spaces? We'll give a product formula as proof-of-concept for a whole toolkit under development to tackle these types of problems. The pre-talk will be several examples of log structures and a bit about moduli spaces of curves and stable maps.

November 3, Tuesday 5:30 - 6:30 pm

**Speaker:****Hari Asokan****Title:**Steiner's conic problem.**Abstract:**Steiner's conic problem asks, given five conics in the (Projective) plane how many conics can one draw that are tangent to all five of them. We will discuss an approach to this problem using Chow rings and related techniques in algebraic geometry.

November 17, Tuesday 3 pm - 4 pm

**Speaker:****Piotr Achinger****(pre-talk)****-****SLIDES****Title:**Regular connections (after Deligne)**Abstract:**I will review Deligne's Riemann--Hilbert correspondence between algebraic integrable connections on a smooth complex algebraic variety which are regular at infinity and complex local systems.

The seminar talk is titled "Regular connections on log schemes and rigid analytic spaces". The speaker will describe an extension of Deligne's results on regular connections to log schemes over C. This is part of a project in progress whose goal is to obtain a Riemann--Hilbert type correspondence for smooth rigid-analytic spaces over C((t)). **SLIDES**

November 23, Monday 6:30 - 7:30 pm

**Speaker:****Andrew Lamoureux****Title:**p-adic manifolds**Abstract:**This talk will introduce the p-adic numbers with an emphasis on their topology and analysis. In particular, we will define the notion of an analytic p-adic manifold and explore both similarities to and differences from real and complex analysis.

December 1, Tuesday 3 pm - 4 pm

**Speaker:****Julia Hartmann****(****pre-talk****)****Title:**Torsors and Patching**Abstract:**The pre-talk will recall nonabelian Galois cohomology and introduce the patching setup used in the main talk.

The seminar talk is titled "Local-global principles for linear algebraic groups over arithmetic function fields". We consider linear algebraic groups over arithmetic function fields, i.e., over one variable function fields over complete discretely valued fields. Such function fields naturally admit several collections of overfields with respect to which one can study local-global principles. We will recall results about local-global principles for rational linear algebraic groups, and then focus on new results concerning certain classes of non-rational groups.

December 7, Monday 6:30 - 7:30 pm

**Speaker:****Meenakshy Jyothis****Title:**History and Construction of Witt Vectors**Abstract:**In the year 1927, Artin and Schreier were studying about cyclic extensions over a field of characteristic p. They proved that if F is a field of characteristic p and E/F a cyclic extension of degree p, then E/F is a a simple extension. Later in the year 1936, Witt generalized this result, stating what happens if the degree of extension is p^n instead of p. In order to prove his result, he defined a new ring called the ring of Witt vectors. The talk will focus on the construction of the ring of Witt vectors and Wittâ€™s generalization of Artin-Schreier theorem.

## If you would like to speak in the seminar next semester, please contact me!

## Fall 2019

### Unless mentioned otherwise, we will meet on Mondays at 4:30 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.*

Monday, September 9

Organizational Meeting

Monday, October 7 (Pre-talk 3:30 - 3:55 pm and regular talk 4 - 4:50 pm)

Speaker: Huy Dang (University of Virginia)

Title: The refined Swan conductor and deformation of Artin-Schreier covers

Abstract: An Artin-Schreier curve is a G:=Z/p-branched cover of the projective line over a field of characteristic p>0. A unique aspect of characteristic p is that there exist flat deformations of a wildly ramified cover so that the number of branch points changes but the genus does not. Using refined Swan conductor, we give the necessary and sufficient conditions for the existence of a deformation between given Artin-Schreier curves. As an application, we show that the moduli space of Artin-Schreier covers of fixed genus g is connected when g is sufficiently large.

Tuesday, October 8

Speaker: Andrew Lamoureux

Title: Integral Extensions - Part 1

Abstract: If A is a subring of B, an element x of B is integral over A when x is a root of a monic polynomial with coefficients in A. We say B itself is integral over A when every element of B is integral over A. In this talk, we explore some basic properties of integral extensions, such as the fact that UFDs are integrally closed, the commutativity of integral closure with localization, and the lying-over and going-up theorems.

Monday, October 14

Speaker: Andrew Lamoureux

Title: Integral Extensions - Part 2

Abstract: We will continue the discussion from October 8.

Notes for both parts (references included).

Monday, October 21

Speaker: Fikreab Solomon Admasu

Title: Zeta functions of groups with examples

Abstract: We will look at a way of encoding various data about the subgroup structure of groups into a Dirichlet series, generally called zeta functions of groups. In important special cases, analogous to classical zeta functions such as Dedekind's, these zeta functions have Euler product decomposition, meromorphic continuation to some right half plane and a kind of local functional equation. Examples and references will be mentioned.

Thursday, November 7

**WH 329 during 2:30 - 4 pm**Speaker: Vaidehee Thatte

Title: Cyclic Extensions and Ramification Breaks

Abstract: The talk will cover parts of Kummer theory, Artin-Schreier theory, and Artin-Schreier-Witt theory. We will also discuss classical ramification groups and jumps in the ramification filtrations (lower and upper) via some examples. Prior knowledge of these topics is not required to understand this talk. This will serve as a pre-pre-talk for Andrew Kobin's talk next week.

References/Notes: Local Fields (J.P.Serre), Algebraic Number Theory (J. Neukirch)

Monday, November 11 (Pre-talk 3:30 - 3:55 pm and main talk 4 - 4:50 pm)

Title: A stacky compactification of the ring of Witt vectors

Abstract: The ring of Witt vectors is an essential tool for understanding relationships between the worlds of characteristic 0 and finite characteristic algebra. In this talk, I will recall how the ring (scheme) of Witt vectors allows one to lift field extensions and covers of curves from characteristic p to characteristic 0. The latter situation motivated Garuti to define a projective scheme which compactifies the ring (scheme) of Witt vectors "equivariantly" (with respect to Witt vector addition). After describing Garuti's construction and its application to the study of covers of curves, I will introduce a new compactification in the category of algebraic stacks that I am currently using to describe the local structure of stacky curves in characteristic p.

Monday, November 18

Speaker: Patrick McGinty

Title: L-functions and Primes in Arithmetic Progression

Abstract: The theory of analytic functions has many applications in number theory. One particular application was discovered by Dirichlet in 1837 where he proved there are infinitely many primes in any arithmetic progression a, a+b, a+2b,... given gcd(a,b)=1. To do this he introduced L-functions. In this talk we will define these functions and discuss other tools needed to prove his theorem.

## Spring 2019

Tuesday, March 12

Organizational Meeting

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.

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."

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.)

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.

:**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.****Speaker:****Renee Bell (UPenn)****Room: WH 329****Time: 3 pm - 4 pm**

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.

April 22/23 -

**No Seminar**(There are two talks in Arithmetic Seminar this week, followed by Ninth Annual Upstate Number Theory Conference at Cornell.)April 29/30 -

**No Seminar**(There are two talks in Arithmetic Seminar this week.)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.