NTSGrad Fall 2015/Abstracts
This is a prep talk for Sean Rostami's talk on September 10.
|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.
|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.
|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.|
|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.
|Introduction to Singular Moduli|
The values of j-function at imaginary quadratic argument are called singular moduli as they correspond to the j-invariants of singular elliptic curves. These singular moduli turn out to be highly divisible as predicted by a remarkable theorem of Gross and Zagier. I will give a brief introduction of singular moduli and some interesting results about them.
|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.
|Generating random factored numbers and ideals, easily|
|Say we want to generate a number, up to some bound N, uniformly at random, but we also want to know its factorization. We could generate a number and then factor it, but factoring isn't known to be polynomial time. In his dissertation, Eric Bach gave a polynomial time way to do this. We will present an alternative polynomial time algorithm for generating a number and its factorization uniformly at random. We will then extend this to the problem of generating ideals in number fields and their factorization uniformly at random, in polynomial time. If time permits, we will discuss how to extend this to arbitrary number fields.|
|Introduction to linear code and algebraic geometry code|
Linear code is an important kind of error correcting code. I will introduce some basic knowledge of linear code and then focus on those linear codes arising from algebraic curves. We will see how the study of algebraic curve over finite field sheds light on coding theory.
Organizer contact information
Megan Maguire (firstname.lastname@example.org)
Ryan Julian (email@example.com)
Sean Rostami (firstname.lastname@example.org)
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