NTSGrad Fall 2015/Abstracts

Sep 08

Vladimir Sotirov

Untitled

This is a prep talk for Sean Rostami's talk on September 10.

Sep 15

David Bruce

The Important Questions

Did the Universe come from nothing? Why are we moral? Where did we come from? According to some signs on Bascom Hill these are the important questions in life. Sadly the poor person who made these signs does not know what the really important questions are: What is David Zureick-Brown going to saying in his NTS talk? How many rational points are on the projective curve given (in affine coordinates) by: $$y^2 = x^6 + 8x^5 + 22x^4 + 22x^3 + 5x^2 + 6x + 1?$$ If you would like to be enlightened by the answers to these truly important questions come to my talk where everything will be illuminated… Or at least some of the background for Coleman and Chabauty’s method for finding rational points on curves will be discussed PS: The number of references to Elijah Wood will be bounded — just like the number of rational points on our curves.

Sep 29

Eric Ramos

Generalized Representation Stability and FI_d-modules.

Let FI denote the category of finite sets and injections. Representations of this category, known as FI-modules, have been shown to have incredible applications to topology and arithmetic statistics. More recently, Sam and Snowden have begun looking at a more general category, FI_d, whose objects are finite sets, and whose morphisms are pairs (f,g) of an injection f with a d-coloring of the compliment of the image of f. These authors discovered that while this category is very nearly FI, its representations are considerably more complicated. One way to simplify the theory is to use the combinatorics of FI_d and the symmetric groups to our advantage. In this talk we will approach the representation theory of FI_d using mostly combinatorial methods. As a result, we will be about to prove theorems which restrict the growth of these representations in terms of certain combinatorial criterion. The talk will be as self contained as possible. It should be of interest to anyone studying representation theory or algebraic combinatorics.

Oct 20

Wanlin Li

ABSTRACT

Oct 27

Megan Maguire

How I accidentally became a topologist: a cautionary tale

The Grothendieck Ring of Varieties is super cool (that's a technical term) and can be used to predict things about the complex topology of complex varieties (like what their Betti numbers should be). However, you can't prove these topological things just using the Grothendieck ring. You have to get down and dirty with topology. It's still pretty cool.

Nov 3

Solly Parenti

Golod-Shafarevich or: How I learned to Stop Worrying and Love Cohomology

Start with a number field K. Let K_1 be the Hilbert class field of K. Let K_2 be the Hilbert class field of K_1. If we keep defining K_n like this, does it eventually stabilize? In 1964, Golod and Shafarevich proved that this tower of fields can be infinite. The proof of this fact comes down to some facts about group theory and more specifically group cohomology. This talk will be an introduction to group cohomology and we'll even try to prove Golod and Shafarevich's result if we have time.

Nov 04

Vlad Matei

Modular forms for definite quaternion algebras

The Jacquet-Langlands theorem states that given two quaternion algebras, then certain automorphic forms for one of them are in canonical bijection with certain automorphic forms for the other. This seems far too general and also a bit vague.So if one translates the statement of the JL theorem down a bit, we should have that certain classical modular forms should be related to certain "modular forms" on other quaternion algebras. We will define modular forms for quaternion algebras, and we will see that for definite quaternion algebras they are very concrete algebraic objects.

Nov 11

Ryan Julian

What is a K3 surface, and why are K1 and K2 surfaces only studied by mountain climbers?

In preparation for Thursday's talk on the Shafarevich conjecture for K3 surfaces, I will attempt to build up enough of the definitions and background theory of differential geometry to define what a K3 surface is. In particular, I hope to explain how K3 surfaces fit into a larger classification of algebraic surfaces, allowing us to prove theorems in a more restricted setting before tackling surfaces of general type. Time permitting, I might even give a couple examples of K3 surfaces. If we're really lucky, I might even have time to explain the hilarious semi-joke in the title above.

Nov 18

SPEAKER

TITLE

ABSTRACT

Nov 25

SPEAKER

TITLE

ABSTRACT

Dec 01

Daniel Ross

Number theory and modern cryptography

This will be a survey-level talk. We'll start with the state of practical RSA and then discuss some cryptosystems that address security-related questions for which there isn't a known answer in the case of RSA. Time permitting, we'll also discuss applications of class field theory to one promising class of such systems.

Dec 09

Jiuya Wang

Parametrization of Cubic Field

The discriminant parametrizes quadratic number fields well, but it will not work for cubic number fields. In order to develop a parametrization of cubic number fields, we will introduce the correspondence between a cubic ring with basis and a binary cubic form. The fact that there is a nice correspondence between orbits under [math]\displaystyle{ GL_2(\mathbb{Z}) }[/math]-action will give the parametrization of cubic fields.

Organizer contact information

Megan Maguire (mmaguire2@math.wisc.edu)

Ryan Julian (mrjulian@math.wisc.edu)

Sean Rostami (srostami@math.wisc.edu)

---

Return to the Number Theory Graduate Student Seminar Page

Return to the Number Theory Seminar Page

Return to the Algebra Group Page