# Seminars & Events for Graduate Student Seminar

##### Asymptotic Representation Theory

I will offer a glimpse of asymptotic representation theory, focusing on the typical and extremal behavior of representations of S_n as n approaches infinity.

##### The Bieberbach Conjecture

The Bieberbach conjecture was a mirage, but it has become solid reality. Who knows when the next mirage will become an oasis of heavenly delight?" -- Lars Ahlfors

##### Five hundred years of differentiation in fifty minutes

We will cover the history of the derivative and its many historical variants from the late Renaissance through early twentieth century generalized derivatives, highlighting the problems, politics, and institutions that drove changing mathematical conceptions of differentiation in the modern world.

##### Belyi's Theorem

Belyi's Theorem provides a surprising and fairly recently-proved connection between the analytic and arithmetic theory of Riemann surfaces/algebraic curves. It states that a Riemann surface/algebraic curve has a holomorphic map to CP1 that is unramified except over 3 points if and only if it has a projective embedding where it is defined by polynomial equations with coefficients in a number field. I will explain what that means, provide a proof, and discuss consequences, such as the fact that the deep and mysterious absolute Galois group of Q acts faithfully on things you can draw on a blackboard.

##### Percolation on Z^d

Percolation is a stochastic process defined on graphs which was introduced by Broadbent and Hammersley as a model for the flow of a fluid through a porous medium. I will define independent bond percolation on the lattice Z^d and explain the percolation phase transition in this case. I will then give some results which describe the subcritical and supercritical phases. Time permitting, I will end with some theorems about percolation at the critical point in two dimensions.

##### Analytic Number Theory in Function Fields

Function fields are a good toy model for number fields. I'll give an easy proof of the prime number theorem for the function field F_q(t), along the way proving the Riemann hypothesis, and talk a little bit about number theory in field extensions. If time permits, I'll discuss the Mertens conjecture in both number fields and function fields. No algebraic geometry will be harmed in this talk.

##### Open problems in mathematical general relativity

Introduced by Einstein in 1915, the theory of general relativity is probably one of the most well-tested theories in physics. And yet a basic mathematical understanding of this theory is lacking (for no other

reason than being difficult). I would like to introduce some of the major open questions.

##### On Fluid Mechanics and its Difficulties

I'll spend the first half (but hopefully less) of the talk giving an abridged introduction to the Euler and Navier-Stokes equations, covering the basic derivations and some of the known results. Afterwards, I will provide two examples to illustrate some of the difficulties in analyzing these equations. The first example is a 1-dimensional model problem for the 3-dimensional Euler equations that mimics turbulent behavior. This model is simpler, but fails global well-posedness in a bad way. The second example is an Eulerian-Lagrangian recasting of Navier-Stokes that fails global well-posedness for somewhat topological reasons.

##### Higher Composition Laws

In 1798, Gauss described a composition law on the space of SL_2(Z) equivalence classes of binary quadratic forms with fixed discriminant D. This law turned this set into a group, which turned out to be the narrow class group of the field Q(\sqrt{D}). However, this composition law was complicated and seemed unmotivated. 200 years later, an interpretation was found, which led to the higher composition laws. I will discuss these spaces, where they come from and what has been done with them.

##### Honeycombs

We will discuss Horn's conjecture and a visually appealing way to understand it.

##### Singularity Theory

Pioneered by Whitney in the 1950's, singularity theory deals with the local behavior of smooth mappings between manifolds. We will discuss some basics of singularity theory, including what a "generic" smooth map looks like, and hopefully hint at some applications in topology as well as the real world.

##### Topological Invariants in Knot Theory

The primary motivation for studying knot theory is the immediate applications to 3 and 4 dimensional topology. This has led to the development of homology theories and other invariants meant to extract information in this context. However, there are much simpler topological invariants that arise naturally and, despite their simplicity, are incredibly difficult to calculate. We will discuss several such invariants including Seifert genus, slice genus, and unknotting number, as well as the methods used in proving things about them.

##### The Gauss-Bonnet Theorem

The Gauss-Bonnet Theorem states that a certain integral of a two-dimensional Riemannian manifold's curvature is equal to the manifold's Euler characteristic. I shall prove this theorem in three (or maybe more) ways, using three different definitions of curvature, and three different definitions of the Euler characteristic. But are these "really" all the same definition? And are these "really" all the same proof?

##### From Picard-Lefschetz to quantum singularity theory

Picard-Lefschetz theory can be viewed as a holomorphic analogue of Morse theory, which studies isolated singularities of holomorphic functions $W: {\mathbb C}^n \to {\mathbb C}$. In this talk, I will discuss the topological aspects about isolated singularities, and discuss the ADE-classification of simple elliptic singularities. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Algebraic hypergeometric functions

A power series F(T) is called hypergeometric if the ratio between the coefficient of T^n and the coefficient of T^{n+1} is a rational function of n. I will discuss the question of when such a series is also algebraic, i.e. p(T,F(T)) = 0 for some nonzero polynomial p. After considering some elementary examples, I will briefly describe the classification of all algebraic hypergeometric functions by Beukers and Heckman. I'll finish with a beautiful observation of Rodriguez-Villegas based on this classification.

##### Large product spaces and sharp thresholds

I will discuss the behavior of boolean functions depending on a large number of independent random variables, in particular, the phenomenon of "sharp thresholds".

##### Univalent Foundations of Mathematics

The Univalent Foundations of Mathematics is a new foundational system, proposed by Vladimir Voevodsky, building on a recently discovered deep connection between logic (more precisely, type theory) and homotopy theory. Many logical methods can be then applied to problems from homotopy theory and, conversely, homotopy-theoretic viewpoint is useful in understanding some logical constructions. In the talk, I will explain this connection and survey the main results of the Univalent Foundations program.

##### Nakajima Quiver varieties

A Nakajima variety is an algebraic variety that is associated to a quiver and some extra data. In this talk I will give an overview of the construction of quiver varieties and of moduli spaces of quiver representations. After reviewing some basic results from GIT, I will talk about stability conditions for representations of quivers, then I will use these facts to define quiver varieties and show that in some cases one can arrange things so that the quiver varieties are smooth and symplectic. I will describe in detail the example of the quiver with one vertex and one loop, and show that how this case relates to the Hilbert scheme of points on \mathbb{C}^2.

##### Modularity lifting theorems

Modularity lifting is a deep and technical methodology introduced by Wiles and Taylor-Wiles in the proof of Fermat's Last Theorem. Modularity lifting theorems have played a central role in many subsequent developments, including the proofs of the Sato-Tate conjecture for elliptic curves over totally real fields and Serre's conjecture. We explain why modularity lifting theorems are important, describe what the statement of such a theorem roughly looks like, and sketch the several types of arguments that are involved in their proof.

##### Quantum groups from Calculus 3

Where do quantum groups arise in nature? For one, from integrals of multivalued functions on configuration spaces of points. I will describe a beautiful construction of the simplest quantum group from such integrals,

due to Felder and Wieczerkowski: no prior knowledge or interest in quantum groups required.