# Seminars & Events for Graduate Student Seminar

##### Nonlinear Waves

Given a nonlinear wave equation whose nonlinearity contains derivatives of the unknown function: $-\partial_t^2\phi+\sum_{i=1}^ 3 \partial_{x_i}^2\phi=N(\partial\phi)$ in $3+1$ dimensions, what can we say about the global existence of solutions? What if the initial conditions are small and compactly supported? If time permits, I will discuss the case of a wave map.

##### Differential equations arising from group actions on manifolds

We consider ODE's arising from the action of Lie groups on manifolds and discuss how solvability of corresponding lie algebra helps to find explicit solution

##### Strong multiplicity one and $l$-adic Galois representations

The strong multiplicity one theorem and its refinements amount to a local-to-global principle in the theory of automorphic representations of $GL(N)$. I will discuss a Galois-theoretic analogue with a surprisingly elementary proof, along with some related questions about the images of $l$-adic Galois representations.

##### Harmonic measure

Harmonic measure is a measure on the boundary of domains in the complex plane whose associated integral generates the solution of the Dirichlet problem. I will explain Nevanlinna's "principle of harmonic measure," and give a number of examples of its use. Time allowing, I will discuss recent geometric results about harmonic measure, and its relation to other conformal invariants such as Beurling's extremal length. The talk will be completely elementary!

##### The Congruence Subgroup Problem for $SL(n,Z)$

The group $SL(n, k)$ ($k$ a field) is simple modulo its center, but for a ring $A$, $SL(n, A)$ is not: the kernels of the maps $SL(n,A)\rightarrow SL(n,A/I)$ ($I<A$ an ideal) give many normal subgroups. The congruence subgroup problem asks whether all finite index normal subgroups of $SL(n,A)$ are obtained in this way. I will discuss the case of $SL(n,Z)$ and more generally, when $A$ is the ring of integers of a number field, with some applications.

##### The Reflector Antenna Problem and its Connection to Mass Transport

The reflector antenna problem is the problem of constructing a reflective surface which directs a specified energy distribution on the sphere to another specified energy distribution on the so-called far-field sphere. I will discuss some of the basic analytic and geometric facts of this problem, and a connection that has been discovered in recent years to Monge's optimal mass transportation problem.

##### Extend Your Function Now! Results Guaranteed or Your Money Back

A question that is often asked in Extension Theory is the following: Given $E \subset \mathbb{R}^n$, and $f:E \rightarrow \mathbb{R}$. Is it possible to extend $f$ to a function lying in the space $X(\mathbb{R}^n)$ ?. This question has been answered in the case when $X$ is the space $C^m$ of functions continuously differentiable through order $m$. I will prove the relevant theorem in the special case $m=2$ for finite sets $E$, and time permitting discuss some other interesting variants.

##### Bigger is Better

Originally, Hardy and Littlewood developed their "circle method" to study Waring's problem on the representation of numbers as the sums of $k^th-powers$. In the circle method, one decomposes the circle into "major" and "minor" arcs. Some rough estimates on the minor arcs give a power saving, and the work is then to study the major arcs. The guiding principle is "Bigger is better", i.e. the best estimates arise from making the major arcs as large as possible. Recently, the circle method has been applied to discrete analogues in harmonic analysis. I will discuss the classical circle method, the spherical maximal function and higher degree analogues, and then discuss how these combine to give a discrete spherical maximal function.

##### Infinite Ergodic Theory and Continued Fractions

I will compare (classical) Ergodic Theory and Infinite Ergodic Theory, i.e. when the space has infinite measure. In particular I will describe how to modify the Birkhoff Ergodic Theorem in the infinite setting. As examples, I will discuss the (classical) Euclidean continued fractions and the (less classical) continued fractions with even partial quotients. Time permitting, I will show the connection of such continued fraction expansions with theta sums.

##### Magma

We give a quick tour of some features of the Magma computer algebra system. These will include: modular forms, algebraic geometry (sheaf cohomology and Groebner bases), computing with L-functions, machinery for function fields, lattices, and some group/representation theory. No experience with Magma will be assumed.

##### Fixed Point Theorems and Applications in PDEs

I'll begin with Banach's fixed point theorem in which strict contraction is required and then give examples to show how this simple theorem implies the local and global existence to vary kinds of evolution equations. I'll also introduce Schauder and Schaefer's fixed point theorem which would be of importance in elliptic theory and verify this by several examples if time permitted.

##### Iwasawa Theory

I'll talk about Iwasawa theory. I'll start with the structures of the class groups of number fields, and see how these generalize to various main conjectures in Iwasawa theory. If there's time, I'll also sketch the main idea of the proof and review what's known so far, including the recent result of Skinner-Urban.

##### Representation Varieties in 3-manifold Topology

An important technique in 3-manifold topology is to study representations of the fundamental group of a 3-manifold into a Lie group. Under appropriate circumstances, a collection of such representations can be used to cut out an algebraic variety that is well defined up to birational equivalence. This is useful not only because it produces algebro-geometric invariants of 3-manifolds, but also because it allows one to reinterpret certain topological questions as (hopefully easier) questions in algebraic geometry. In this talk I'll describe the basic ingredients of celebrated work of Culler and Shalen that uses these techniques together with Bass-Serre theory to find special surfaces in 3-manifolds.

##### Representation Varieties in 3-manifold Topology

I will start with Dirichlet's Theorem in arithmetic progressions and then I will talk about Chebotarev Density Theorem (which is a generalization) and how it is related with class field theory.

##### Smooth Poincare Conjecture: Is There an Exotic Four-sphere?

After briefly reviewing the history of (smooth) Poincare conjecture, we shall focus on the case of four-dimensional sphere. We would like to discuss a construction due to Gluck that provides many candidates of exotic four-spheres, i.e., manifolds homeomorphic but not diffeomorphic to the standard four-sphere. We will prove that Gluck's manifolds are indeed homeomorphic to four-sphere and then sketch an argument by Freedman et al. that could potentially lead to the discovery of exotic four-spheres.

##### Transcendence

We all know that e and π are transcendental. How about numbers like $e+π$, $e^π$, or $sqrt(2)^[sqrt(2)]$ If $2n$, $3n$, and $5n$ are all integers, must $n$ be an integer as well? What if only $2n$ and $3n$ are integers? In this talk, I will talk about these and related questions. In particular, I hope to prove the well-known fact above: $e$ and $π$ are transcendental. I will also mention the applications of transcendence theory to other subjects, like logic and integration.

##### Regularity of the Hardy Littlewood Function

$Lp$ boundedness of the Hardy-Littlewood function is a classic result in harmonic analysis. But not much is understood about the regularity of it. For instance, if your function $f$ is in the Sobolev W1,1 space, is its maximal function in$L1$? The answer to this question is unknown, but I will discuss our partial understanding.

##### Stationary Phases and Spherical Averages

In this talk we will give an expository account of the following theorem of Stein about spherical averages, which asserts that if f is a function in Lp on Rn, with n≥3 and p>n/(n-1), then for almost every x in Rn, the average of f over a sphere of radius r centered at x is well-defined, and converges to f(x) as r tends to 0. Along the way we will see some beautiful ideas in harmonic analysis, and their connections to other subjects.