Difference between revisions of "NTSGrad Fall 2015/Abstracts"

From UW-Math Wiki
Jump to: navigation, search
(Nov 04)
m (Blalberts moved page NTSGrad/Abstracts to NTSGrad Fall 2015/Abstracts: Old)
 
(33 intermediate revisions by 4 users not shown)
Line 1: Line 1:
== Sep 02 ==
+
== Sep 08 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Lalit Jain'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Vladimir Sotirov'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | ''Monodromy computations in topology and number theory''
+
| bgcolor="#BCD2EE"  align="center" | ''Untitled''
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
The monodromy of a family of varieties is a measure of how homology classes vary. Surprisingly, many familiar ideas in number theory, such as Galois representations and Cohen-Lenstra heuristics, are closely linked to monodromy of specific families. In this talk I will define monodromy, explain some number theoretic applications, and describe original work of computing monodromy for moduli spaces of covers of the projective line (Hurwitz spaces). This work generalizes previous results of Achter-Pries, Yu and Hall on hyperelliptic families. Only basic knowledge of algebraic topology and number theory is required.   
+
This is a prep talk for Sean Rostami's talk on September 10.   
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
Line 15: Line 15:
 
<br>
 
<br>
  
== Sep 09 ==
+
== Sep 15 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Megan Maguire'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David Bruce'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Infintely many supersingular primes for every elliptic curve over the rationals.
+
| bgcolor="#BCD2EE"  align="center" | ''The Important Questions''
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
In his 1987 Inventiones paper, Dr. Noam Elkies proved that every elliptic curve over <math>\mathbb{Q}</math> has infinitely many supersingular primes. We shall discuss some of the mathematics needed to prove this result and give a proof.
+
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:
|}                                                                       
+
$$y^2 = x^6 + 8x^5 + 22x^4 + 22x^3 + 5x^2 + 6x + 1?$$
</center>
+
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
 
 
<br>
 
  
== Sep 16 ==
+
PS: The number of references to Elijah Wood will be bounded — just like the number of rational points on our curves.
 
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Silas Johnson'''
 
|-
 
| bgcolor="#BCD2EE"  align="center" | Alternate Discriminants and Mass Formulas for Number Fields
 
|-
 
| bgcolor="#BCD2EE"  | 
 
Kedlaya and Wood have explored alternate invariants for number fields, with the idea of replacing the discriminant in standard field-counting questions with one of these alternate invariants.  We further explore the space of “reasonable” invariants, expanding on Kedlaya and Wood’s definition.  We also discuss a theorem on mass formulas for these invariants.
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
Line 47: Line 35:
 
<br>
 
<br>
  
== Sep 23 ==
+
== Sep 29 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Daniel Hast'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Eric Ramos'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Moments of prime polynomials in short intervals.
+
| bgcolor="#BCD2EE"  align="center" | ''Generalized Representation Stability and FI_d-modules.''
 
|-
 
|-
| bgcolor="#BCD2EE"  |   
+
| bgcolor="#BCD2EE"  |  Let FI denote the category of finite sets and injections.
How many prime numbers occur in a typical "short interval" of fixed width, and how are primes distributed among such intervals? We examine the analogue of this problem for polynomials over a finite field. Our approach is geometric: we interpret each moment of the distribution in terms of counting certain points on an algebraic variety, and we use an algebraic analogue of the Lefschetz fixed-point theorem to compute the leading terms of this "twisted" point-count. As a motivating example, we realize the "prime polynomial theorem" as a geometric statement.
+
Representations of this category, known as FI-modules, have been shown
|}                                                                       
+
to have incredible applications to topology and arithmetic statistics.
</center>
+
More recently, Sam and Snowden have begun looking at a more general
 
+
category, FI_d, whose objects are finite sets, and whose morphisms are
<br>
+
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.
  
== Sep 30 ==
+
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
<center>
+
theorems which restrict the growth of these representations in terms
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
+
of certain combinatorial criterion. The talk will be as self contained
|-
+
as possible. It should be of interest to anyone studying
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''SPEAKER'''
+
representation theory or algebraic combinatorics.
|-
 
| bgcolor="#BCD2EE"  align="center" | TITLE
 
|-
 
| bgcolor="#BCD2EE"  | 
 
ABSTRACT
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
Line 79: Line 66:
 
<br>
 
<br>
  
== Oct 07 ==
+
== Oct 20 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Cocke'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Wanlin Li'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | The Trouble with Sharblies
+
| bgcolor="#BCD2EE"  align="center" |  
|-
 
| bgcolor="#BCD2EE"  | 
 
The Sharbly complex provides a generalization of modular symbols and can be used to compute the Hecke eigenvalues on arithmetic cohomology. Such eigenvalues provide useful information pertaining to generalizations of Serre's conjecture. I will introduce the Sharbly complex and examine the necessary reduction techniques needed to compute the Hecke action. A friendly introduction to a new and emerging tool in computational number theory.
 
|}                                                                       
 
</center>
 
 
 
<br>
 
 
 
== Oct 14 ==
 
 
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brandon Alberts'''
 
|-
 
| bgcolor="#BCD2EE"  align="center" | TITLE
 
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
Line 111: Line 82:
 
<br>
 
<br>
  
== Oct 21 ==
+
== Oct 27 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yueke Hu'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Megan Maguire'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | ''Mass equidistribution on modular curve of level N''
+
| bgcolor="#BCD2EE"  align="center" | ''How I accidentally became a topologist: a cautionary tale''
 
|-
 
|-
| bgcolor="#BCD2EE"  |   
+
| bgcolor="#BCD2EE"  |  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.
It was shown in previous works that the measure associated to holomorphic newforms of weight k and level q will tend weakly to the Haar measure on modular curve of level 1, as qk goes to infinity. In this talk I will show that this phenomenon is also true on modular curves of general level N.
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
Line 127: Line 97:
 
<br>
 
<br>
  
== Oct 28 ==
+
== Nov 3 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David Bruce'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Solly Parenti'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Intro to Complex Dynamics
+
| bgcolor="#BCD2EE"  align="center" | ''Golod-Shafarevich or: How I learned to Stop Worrying and Love Cohomology''
 
|-
 
|-
| bgcolor="#BCD2EE"  |   
+
| bgcolor="#BCD2EE"  |  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
Given a polynomial f(z) with complex coefficients, we can ask for which complex numbers p is the set {f(p), f(f(p)), f(f(f(p))),...} bounded, that is to ask which complex numbers have bounded forward orbit under f(z)? Alternatively we can turn the question around and ask for a fixed complex number p, for which (complex) polynomials is the forward orbit of p bounded? Finite? Periodic? These questions give the interesting fractal pictures many of you have probably seen. Amazingly many of the tools needed to approach these questions, arose well before computers allowed us to generate images like the one above. In this talk we will explore some of the basic tools and results of complex dynamics paying particular attention relations to number theory. The goal being to present some of the background material need for Laura DeMarco’s talk later in the week. (Also getting to see a really cool area of mathematics!)
+
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.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
Line 143: Line 118:
 
<br>
 
<br>
  
== Nov 04 ==
+
== Nov 24 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Vlad Matei'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Peng Yu'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Modular forms for definite quaternion algebras
+
| bgcolor="#BCD2EE"  align="center" | ''Introduction to Singular Moduli''
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
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.
+
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.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
Line 159: Line 134:
 
<br>
 
<br>
  
== Nov 11 ==
+
== Dec 01 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''SPEAKER'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Daniel Ross'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | TITLE
+
| bgcolor="#BCD2EE"  align="center" | Number theory and modern cryptography
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
ABSTRACT
+
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. 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
Line 175: Line 150:
 
<br>
 
<br>
  
== Nov 18 ==
+
== Dec 08 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''SPEAKER'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zachary Charles'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | TITLE
+
| bgcolor="#BCD2EE"  align="center" | ''Generating random factored numbers and ideals, easily''
 
|-
 
|-
| bgcolor="#BCD2EE"  |   
+
| bgcolor="#BCD2EE"  |  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.
ABSTRACT
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
Line 191: Line 165:
 
<br>
 
<br>
  
== Nov 25 ==
+
== Dec 15 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''SPEAKER'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jiuya Wang'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | TITLE
+
| bgcolor="#BCD2EE"  align="center" | ''Introduction to linear code and algebraic geometry code''
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
ABSTRACT
+
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.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
Line 207: Line 181:
 
<br>
 
<br>
  
== Dec 02 ==
+
== Organizer contact information ==
 
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''SPEAKER'''
 
|-
 
| bgcolor="#BCD2EE"  align="center" | TITLE
 
|-
 
| bgcolor="#BCD2EE"  | 
 
ABSTRACT
 
|}                                                                       
 
</center>
 
 
 
<br>
 
 
 
== Dec 09 ==
 
 
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''SPEAKER'''
 
|-
 
| bgcolor="#BCD2EE"  align="center" | TITLE
 
|-
 
| bgcolor="#BCD2EE"  | 
 
ABSTRACT
 
|}                                                                       
 
</center>
 
 
 
<br>
 
  
 +
Megan Maguire (mmaguire2@math.wisc.edu)
  
== Organizer contact information ==
+
Ryan Julian (mrjulian@math.wisc.edu)
  
Sean Rostami (srostami@math.wisc.edu)
+
[http://www.math.wisc.edu/~srostami/ Sean Rostami]
  
 
<br>
 
<br>

Latest revision as of 16:09, 4 September 2016

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 24

Peng Yu
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.


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 08

Zachary Charles
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.


Dec 15

Jiuya Wang
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 (mmaguire2@math.wisc.edu)

Ryan Julian (mrjulian@math.wisc.edu)

Sean Rostami



Return to the Number Theory Graduate Student Seminar Page

Return to the Number Theory Seminar Page

Return to the Algebra Group Page