Seminars & Events for Graduate Student Seminar

September 27, 2012
12:30pm - 1:30pm
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.

Speaker: Boris Alexeev , Princeton University
Location:
Fine Hall 314
October 4, 2012
12:30pm - 1:30pm
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

Speaker: Matthew de Courcy-Ireland, Princeton University
Location:
Fine Hall 314
October 11, 2012
12:30pm - 1:30pm
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.

Speaker: Michael Barany , Princeton University
Location:
Fine Hall 314
October 18, 2012
12:30pm - 1:30pm
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.

Speaker: Will Sawin , Princeton University
Location:
Lewis Library 121
October 25, 2012
12:30pm - 1:30pm
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.

Speaker: Phil Sosoe , Princeton University
Location:
Fine Hall 314
November 8, 2012
12:30pm - 1:30pm
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.

Speaker: Peter Humphries, Princeton University
Location:
Fine Hall 314
November 15, 2012
12:30pm - 1:30pm
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.

Speaker: Jonathan Kommemi, Princeton University
Location:
Fine Hall 314
November 29, 2012
12:30pm - 1:30pm
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.

Speaker: Andrei Tarfulea , Princeton University
Location:
Fine Hall 314
December 6, 2012
12:30pm - 1:30pm
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.

Speaker: Sam Ruth , Princeton University
Location:
Fine Hall 314
December 13, 2012
12:30pm - 1:30pm
Honeycombs

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

Speaker: Percy Wong, Princeton University
Location:
Fine Hall 314
February 7, 2013
12:30pm - 1:30pm
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.

Speaker: Béla Racz , Princeton University
Location:
Fine Hall 314
February 14, 2013
12:30pm - 1:30pm
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.

Speaker: Nate Dowlin , Princeton University
Location:
Fine Hall 314
February 21, 2013
12:30pm - 1:30pm
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?

Speaker: Heather Macbeth , Princeton University
Location:
Fine Hall 314
February 28, 2013
12:30pm - 1:30pm
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.

Speaker: Guangbo Xu , Princeton University
Location:
Fine Hall 314
March 7, 2013
12:30pm - 1:30pm
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.

Speaker: Aaron Pixton , Princeton University
Location:
Fine Hall 314
March 14, 2013
12:30pm - 1:30pm
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".

Speaker: Phil Sosoe , Princeton University
Location:
Fine Hall 314
March 28, 2013
12:30pm - 1:30pm
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.

Speaker: Chris Kapulkin , Princeton University
Location:
Fine Hall 314
April 11, 2013
12:30pm - 1:30pm
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.

Speaker: Giulia Sacca, Princeton University
Location:
Fine Hall 314
April 18, 2013
12:30pm - 1:30pm
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.

Speaker: Shrenik Shah , Princeton University
Location:
Fine Hall 314
May 2, 2013
12:30pm - 1:30pm
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.

Speaker: Michael McBreen , Princeton University
Location:
Fine Hall 314