NTSGrad/Abstracts

Sep 06

Brandon Alberts

Introduction to the Cohen-Lenstra Measure

The Cohen-Lenstra heuristics describe a conjectured probability distribution for the class group of quadratic fields. In this talk, I will give a brief introduction to the heuristic and how it is related to random groups. The remainder of the talk will focus on the Cohen-Lenstra probability measure for choosing a random p-group. This talk is based on a similar talk given by Bjorn Poonen.

Sep 13

Vlad Matei

Overview of the Discrete Log Problem

The discrete logarithm problem (DLP) was first proposed as a hard problem in cryptography in the seminal article of Diffie and Hellman. Since then, together with factorization, it has become one of the two major pillars of public key cryptography. In its simplest version for $\mathbb{Z}/p\mathbb{Z}$, the problem is given $b\hspace{2mm} ( \text{mod}\hspace{2mm} p)$ and $b^n \hspace{2mm}(\text{mod} \hspace{2mm} p)$ to find $n$. As far as we know, this problem is VERY HARD to solve quickly. Nobody has admitted publicly to having proved that the discrete log can't be solved quickly, but many very smart people have tried hard and not succeeded. This is a prep talk for the Thursday seminar 9/15/2016

Sep 20

Wanlin Li

Gauss's Genus Theory and 2-Selmer Groups of Elliptic Curves

I'm going to talk about how Gauss's genus theory characterized 2 torsion elements of the class group for imaginary quadratic fields and how to use 2 decent to compute the 2-Selmer group for an elliptic curve.

Sep 27

Ewan Dalby

Modular forms of half integral weight

Usually when we think of modular forms we think of functions that gain an integral power of the automorphic factor under a group action. However we can generalize this notion to allow half integral weights. I will describe how this generalization works and it's connections to the theory of integral weight modular forms.

Oct 4

Daniel Hast

Introduction to arboreal Galois representations

Arboreal Galois representations are representations of Galois groups as automorphism groups of certain trees. We'll introduce the main definitions, see how iterating polynomial functions gives an abundant source of arboreal representations, and survey some of the major theorems and conjectures about these representations.

Oct 11

Peng Yu

Modular Forms and Elliptic Curves

I will give a quick review of the basics for the classical case of Galois representations associated to modular forms, including the definition of modular forms, Hecke operators, elliptic curves, Galois representations arising from the Tate modules of elliptic curves, and the Modularity Theorem (i.e. the Taniyama-Shimura-Weil conjecture) that every elliptic curve defined over Q is modular.

Oct 18

Soumya Sankar

Cohen Lenstra Heuristics for p=2, or the lack thereof

I will introduce Cohen and Lenstra's Heuristics and talk about why they don't work for p=2. I will also talk about the alternative route to understanding the two-power torsion of the class group, in particular the work of Fouvry-Kluners and Gerth.

Oct 25

Solly Parenti

Complex Multiplication

As a youth, I had recurring nightmares about abelian extensions of number fields. Kronecker and Weber tell us about abelian extensions of Q, but even in the next simplest case, quadratic fields, what the Hecke can we say about these extensions? Deuring this talk, we will see what elliptic curves can tell us about extensions of imaginary quadratic fields.

Nov 1

Brandon Boggess

Splitting Varieties for Cup Products

I will define splitting varieties, and then show a method for constructing a large class of them. To convince you that this is not a tremendous waste of time, an automatic realization result for Galois groups will be given as an application of our splitting varieties.

Nov 8

none

Nov 15

David Wagner

Apollonian circle packings

To form an Apollonian gasket, first nest two circles of integer curvature inside a third so that the three circles are mutually tangent and the points of tangency are distinct. By continuing to place tangent circles, one obtains an intricate packing in which all circles drawn have integer curvature. Using strong approximation and facts about thin groups (plus cool analytic tools), we give a partial answer the non-trivial question of which integers can occur as curvatures in a given gasket. Similar results continue to hold when we introduce a natural generalization of the gasket (to be discussed in greater depth on Thursday) in which geometric features of the packing correspond to arithmetic properties of a chosen imaginary quadratic field.

Nov 22

none

Happy Thanksgiving!

Nov 29

Daniel Hast

Introduction to p-adic Hodge theory

This is a survey of the motivation and main concepts of p-adic Hodge theory. I will discuss criteria for good reduction of elliptic curves, classical (complex) Hodge theory, types of p-adic representations (Hodge-Tate, de Rham, semistable, crystalline), and comparison isomorphisms.

Dec 6

Wanlin Li

Supersingular Elliptic Curves

As its name suggests, supersingular elliptic curves are super special. And being special made them useful. A few weeks ago I gave a talk at GAGS about bounding gonality of modular curves. A result I used there without proof is a consequence of special properties of supersingular elliptic curves. And I'm going to explain that in this talk. I will define what supersingular elliptic curves are and hopefully convince you that they are cute.

Dec 13

Jiuya Wang

Chebotarev Density Theorem

I don't have anything particularly interesting, since I will just talk about the proof of the theorem.

Dec 20

Daniel Ross

A Primer on the Main Conjecture of Iwasawa Theory

This talk will introduce a basic case of a theorem that establishes a surprising relationship between L-functions and class groups. This talk will attempt to convey the structure of the proof as well as two key ideas that boil down to a clever uses of p-adic analysis and of ramification in number fields.

Organizer contact information

Brandon Alberts (blalberts@math.wisc.edu)

Megan Maguire (mmaguire2@math.wisc.edu)

Bobby Grizzard

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