# Seminars & Events for Graduate Student Seminar

##### The Probabilistic Method and A Classical Result of Erdos

The basic idea behind the 'probabilistic method' in combinatorics is as follows: in order to prove the existence of an object with a certain property, one defines an appropriate (non-empty) probability space and shows that a randomly chosen object from this space has the desired property with positive probability. This simple idea (with a few variations) has been applied to prove many beautiful and sometimes surprising results in combinatorics. I will mostly focus on just one such classical result, namely the theorem of Erdos showing that there exist graphs with arbitrarily large girth and arbitrarily large chromatic number. No previous knowledge of graph theory will be assumed.

##### An Outline of the h-Cobordism Theorem

In the first half of the talk, we shall go through the definitions of manifold, cobordism, h-cobordism, Morse functions and gradient-like flows, and demonstrate many of their properties. In the second half of the talk we shall give an outline of the proof. In the process, we shall prove lemmas about commutation and cancellation of critical points, and see the Whitney trick in action. Time permitting, we shall show how the h-cobordism theorem proves the topological Poincare conjecture in high dimensions.

##### Differential Equations and Arithmetic

One of the questions in number theory is to represent integers by quadratic forms. To be more precise, the question asks for a characterization of the integers which can be represented by a given form. In this talk we will deal with representing integers as sums of two squares and if time permits as sums of four squares. The talk will be completely elementary. (If I were to deliver the talk in the 19th century it wouldn't be called so!!)

##### Hyperbolic 3-Manifolds and Arithmetic

Thurston's geometrisation program tells us that we can understand all closed 3-manifolds if we can understand those with a finite volume hyperbolic structure. This is not easy, however, and it is not even clear at the outset that there exist more than a few such manifolds. I will decribe number theoretic methods which construct a rich class of finite volume hyperbolic 3-manifolds, and beautiful converse results which show that all such manifolds have useful arithmetic information associated with them.

##### Prime Splitting Laws

Algebraic number theory seeks to understand and somehow classify all Galois extensions of a number field $K$. One perspective on this basic classification problem emerges from the Cebotarev density theorem, which implies that such extensions $L/K$ are determined by the primes of $K$ that split completely in $L$. When $L/K$ is abelian, associating such a 'splitting law' is classical, and after defining the relevant terms I will illustrate this theory with some representative examples. The corresponding question for non-abelian extensions remains a great mystery, however, but I hope to use this to motivate a very concrete introduction to some of the modern machinery of number theory and the Langlands program. No knowledge of number theory will be assumed.

##### Counting Circles and Other Things

Take a circle of radius one and inscribe in it two circles that are tangent to each other. Now add another circle into the original one so that it's tangent to all three. If we repeat this process over and over, we get an old picture known as the Apollonian circle packing. What radii will we get from one such packing? There are many ways to approach this simple question, and I'll tell you about some in this talk.

##### The Moment Map and Delzant Polytopes

In symplectic geometry, the moment map (or momentum map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic quotients, symplectic cuts and sums. During the lecture I will introduce this concept and I will show you how to construct symplectic manifolds having some specific image for the moment map. Also I'll say something about the relation between combinatorial data in the image of the moment map and geometric data on the manifold

##### Undecidability and Hilbert's 10th Problem

Hilbert's famous 10th problem asks whether there is a general algorithm to decide whether a polynomial in many variables has a solution. I will explain how Robinson, Davis, Putnam, and Matijasevic answered the question in the negative by proving a much more interesting theorem: that any listable set can be listed using a polynomial equation. I'll also give some awesome mind-blowing consequences of this theorem, such as prime-producing polynomials! No background in anything will be required.

##### Matriod Theory

Every maximal acyclic subgraph of a given graph has the same number of edges. Every maximal independent subset of a given set of vectors in $\mathbb{R}^n$ has the same size. The common generalization of these lies in matroid theory, which, roughly speaking, is an abstraction of the notion of linear independence. Despite the fact that matroids "forget" quite a lot of structure, they are immensely useful to graph theorists. In this talk, I'll give an introduction to the theory of matroids and point to some of its highlights, keeping graph theory in mind as our primary motivation and source of examples.

##### Secant Varieties and Applications

The notion of secant varieties in algebraic geometry is classical, but not much is known about these objects, even for many simple cases. Surprisingly, it is possible to translate certain questions in fields as disparate as complexity theory, statistics, and biology into straightforward (but often unsolved) problems about secant varieties. I will introduce secant varieties and present some examples of applications to other fields and to algebraic geometry itself. No knowledge of algebraic geometry, or of these other fields, will be assumed.

##### The sixth-sphere

Kirchhoff's problem (1947) asks whether $S^6$ is a complex manifold. We will review basic notions in complex and almost-complex geometry and discuss a bit of what is known regarding this problem, making contact with the octonions, $G_2$, Lie brackets, and curvature.