# Seminars & Events for Graduate Student Seminar

##### Moments of the Riemann Zeta Function

An important area of study in analytic number theory is the average behaviour of the Riemann zeta function on the critical line Re(s) = 1/2. I'll talk a bit about what we know so far, and how random matrix theory helps us predict even more. If time permits, I'll discuss how one can generalise this to the average behaviour of families of L-functions at the critical point s = 1/2.

##### Lambda-Rings

There are many operations that act on representations of a group, such as Sym^n, Wedge^n, and more exotic beasts. We can express these in terms of a simple, elegant set of operations, that involves, surprisingly, elementary number theory! I will explain how this works, prove some fun theorems about it, and, hopefully, answer the best generals question of all time.

##### Sphere Packing

We discuss a beautiful method of Cohn and Elkies which gives bounds on how densely spheres can be packed in Euclidean space of any dimension. Their method has been elaborated by Cohn and Kumar to show that the Leech lattice is the unique densest lattice packing in 24-dimensional space and that the E8 lattice is the unique densest lattice packing in 8-dimensional space.

##### The Eguchi-Hanson metric

In 1978 the physicists Eguchi and Hanson discovered a "gravitational pseudoparticle." It was quickly stolen by mathematicians. I will discuss some topological, analytic, differential-geometric and algebro-geometric properties of the Eguchi-Hanson construction.

##### War and Peace in Modern Mathematics

Wars, both actual and metaphorical, are omnipresent in the history of mathematics. From sixteenth century arithmetic publishers who promoted the value of trigonometry in out-ballisticking one's foes to the game theorists and fluid dynamicists of the Cold War, mathematicians have both actively participated in warfare and actively appropriated the goals, values, and images of warfare to their own (often commercial) ends. Famous disputes in the history of mathematics often turn to war for their metaphorical vocabulary, and in some famous cases even led to not-just-metaphorical death.

##### I got 3 open problems but an application ain't one

I will describe and talk about 3 open math problems related to my work. They involve number theory, algebraic geometry, and random matrices. Would love to hear your thoughts on any of them.

##### Computations in Algebraic Geometry

This will be a fairly expository talk on the role of computation in algebraic geometry. I will discuss basics of Groebner bases and the geometry associated to them. These give an easy way to construct algorithms to solve many computational problems about projective varieties. Finally I will talk about why Groebner bases are actually computable for the nice geometric objects that arise in algebraic geometry.

##### Smooth Structures in Low-Dimensional Topology

Smooth 4-manifolds are without question the least understood topic in low-dimensional topology. In dimensions 5 and higher, the h-cobordism theorem handles most of the difficulties, while in dimensions 3 and below, smooth and topological manifolds coincide. Dimension 4 is special – it is the only low dimension in which the smooth Poincare conjecture is unsolved, and it is the only dimension in which Euclidean space admits exotic smooth structures. We will discuss where the wildness of four-dimensions comes from, the tools available for dealing with it, and show how knot theory allows us to construct some of the exotic smooth structures on R^4.

##### The Van der Corput lemma and equations in Physics

I will prove the Van der Corput lemma and then describe how it applies to partial differential equations arising in Physics. It will provide intuition for a number of qualitative properties of non-relativistic/relativistic massive particles and massless particles. For example, a single picture will show that relativistic massive particles have rest mass and cannot reach the speed of light.

##### Curry-Howard Correspondence

The Curry-Howard correspondence relates formal proofs in logic and computer programs in a typed functional programming language. It is the underlying theory behind Coq, one of the major systems today for verifying correctness of proofs using computers. The theory is further extended by Vladimir Voevodsky in the univalent foundations program. In this talk I will describe the original correspondence and give some of the simpler examples.

##### Curry-Howard Correspondence

The Curry-Howard correspondence relates formal proofs in logic and computer programs in a typed functional programming language. It is the underlying theory behind Coq, one of the major systems today for verifying correctness of proofs using computers. The theory is further extended by Vladimir Voevodsky in the univalent foundations program. In this talk I will describe the original correspondence and give some of the simpler examples.

##### The polynomial method in Kakeya-type problems

PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT. In the past several years, the so-called "polynomial method" have been used to prove several problems related to analysis and combinatorics.

##### The Tracy-Widom distribution

The Tracy-Widom distributions are a family of probability distributions that were described explicitly by Craig Tracy and Harold Widom, and shown to govern the maximal eigenvalue of large random matrices. Interestingly, they also appear in a number of other, seemingly unrelated problems such as the longest increasing subsequence of a random finite permutation, certain percolation-type models and interacting particle systems such as the totally asymmetric simple exclusion process (TASEP). I will give an overview of some results involving this distribution and, time permitting, discuss its relation to a set of conjectures concerning the so-called KPZ universality class.

##### Answering questions about graphs

Answering questions about graphs has many uses in computer science. Particularly important is finding ways to answer questions about graphs in a way that scales well with the size of the graph. Some questions can be answered in time that is bounded by a polynomial in the number of vertices in the graph; those problems belong to class P. Other questions are decision problems (with a yes or no answer) where if the answer is yes, there is a solution we can verify in polynomial time; those problems are in class NP. The class of NP-complete problems consists of decision problems to which any decision problem in NP can be reduced in polynomial time.

##### Szemeredi's Theorem

In this talk we will discuss Szemeredi's theorem, which states that any set of integers with positive density contains arbitrarily long arithmetic progressions. The focus will be Gowers' approach to proving this theorem where he introduced higher order Fourier analysis and the Gowers uniformity norms.

##### The Riemann hypothesis is valid for at least one third of the zeros of the Riemann zeta function

Let N(T) be the number of the zeros of the Riemann zeta function with imaginary part less than T, and let N_0(T) be the number of such zeros which lie on the critical line. The Riemann hypothesis says that we have N(T) = N_0(T). In this talk we will show that N_0(T) > 1/3 N(T).

##### The Gauss Circle Problem

How many lattice points are contained in the circle of radius R centered at the origin? Gauss used elementary geometric arguments to show that the answer is approximately πR^2. He was able to bound the error of this estimate by O(R). Using some techniques from Fourier analysis and some properties of oscillatory integrals, we can decrease the exponent of R in the error from 1 to 2/3; this will be the main subject of this talk. We will also discuss some ideas that have been used in attempts to lower the exponent further.

##### Volumes of Hyperbolic Manifolds

The Mostow Rigidity Theorem implies that Volume is actually a topological invariant for hyperbolic manifolds. In this talk we give an outline of Gromov’s proof of Mostow Rigidity for hyperbolic 3-manifolds. The proof uses the Gromov norm, which is a pseudo-norm on the homology of a topological space. If time permits, we will discuss theorems about the set of volumes of hyperbolic manifolds.

##### Formation of Trapped Surfaces in General Relativity

An interesting question in general relativity is the dynamical formation of black holes for Einstein vacuum equation. In this talk, we will show that a trapped surface can dynamically form from arbitrary dispersed initial data in vacuum.

##### The Probabilistically Checkable Proofs Theorem

The class NP consists of types of problems for which there is always a reasonably short proof that the answer is yes when it is. While it intuitively seems like one would have to read the entirety of an alleged proof to determine whether or not it is valid, this is actually false, provided one is willing to accept a small probability of accepting a “proof” of a false statement. In fact, one only needs to read a constant number of bits to differentiate with high probability between true proofs and claimed proofs of false statements. This can be used to show that unless P=NP, there are certain problems where not only is the solution hard to calculate, it is even hard to approximate.