# 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

## 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

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## 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]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)