School of Mathematics

Emmy Noether: breathtaking mathematics

Georgia Benkart
University of Wisconsin-Madison
May 6, 2016
By the mid 1920s, Emmy Noether had made fundamental contributions to commutative algebra and to the theory of invariants. Her crowning achievement from this period was "Noether's Theorem," establishing deep connections between conserved quantities in physics and mathematical invariants. She then tackled noncommutative algebra and demonstrated its significance for many fields of mathematics, including number theory, representation theory, and even commutative theory.

Symmetry and conservation laws: Noether's contribution to physics

Karen Uhlenbeck
University of Texas, Austin; Visitor, School of Mathematics
May 6, 2016
A single result of Noether's is widely credited in physics papers as fundamental to the modern way of approaching physics. Two basic ideas are those of symmetry on the one side and the notion of quantities such as energy preserved under flows on the other. While parts of the theory were not new at the time, the 1918 paper of Noether related the two subjects in a beautiful, complete and definitive fashion. After giving some background and an outline of her proof, I will talk a little bit about the importance of symmetry in today's mathematics and physics.

Rational curves on elliptic surfaces

Douglas Ulmer
Georgia Institute of Technology
May 5, 2016
Given a non-isotrivial elliptic curve $E$ over $K = \mathbb F_q(t)$, there is always a finite extension $L$ of $K$ which is itself a rational function field such that $E(L)$ has large rank. The situation is completely different over complex function fields: For "most" $E$ over $K = \mathbb C(t)$, the rank $E(L)$ is zero for any rational function field $L = \mathbb C(u)$. The yoga that suggests this theorem leads to other remarkable statements about rational curves on surfaces generalizing a conjecture of Lang.

NonLERFness of groups of certain mixed 3-manifolds and arithmetic hyperbolic $n$-manifolds

Hongbin Sun
University of California, Berkeley
May 3, 2016
I will show that the groups of mixed 3-manifolds containing arithmetic hyperbolic pieces and the groups of certain noncompact arithmetic hyperbolic $n$-manifolds ($n > 3$) are not LERF. The main ingredient is a study of the set of virtual fibered boundary slopes for cusped hyperbolic 3-manifolds, and some specialty of Bianchi manifolds.

Fourier tails for Boolean functions and their applications

Avishay Tal
Member, School of Mathematics
May 3, 2016

The discrete Fourier transform is a widely used tool in the analysis of Boolean functions. One can view the Fourier transform of a Boolean function $f:\{0,1\}^n \to \{0,1\}$ as a distribution over sets $S \subseteq [n]$. The Fourier-tail at level $k$ is the probability of sampling a set $S$ of size at least $k$.

Reed-Muller codes for random erasures and errors

Amir Shpilka
Tel Aviv University
April 26, 2016
Reed-Muller codes encode an $m$-variate polynomial of degree $r$ by evaluating it on all points in $\{0,1\}^m$. Its distance is $2^{m-r}$ and so it cannot correct more than that many errors/erasures in the worst case. For random errors one may hope for a better result. In his seminal paper Shannon exactly determined the amount of errors and erasures one can hope to correct for codes of a given rate. Codes that achieve Shannon's bound are called capacity achieving codes.

Exponential convergence to the Maxwell distribution of solutions of spatially inhomogenous Boltzmann equations

Gang Zhou
California Institute of Technology
April 25, 2016
In this talk I will present a recent proof of a conjecture of C. Villani, namely the exponential convergence of solutions of spatially inhomogenous Boltzmann equations, with hard sphere potentials, to some equilibriums, called Maxwellians.

Standard conjecture of Künneth type with torsion coefficients

Junecue Suh
University of California, Santa Cruz
April 21, 2016
A. Venkatesh asked us the question, in the context of torsion automorphic forms: Does the Standard Conjecture (of Grothendieck's) of Künneth type hold with mod p coefficients? We first review the geometric and number-theoretic contexts in which this question becomes interesting, and provide answers: No in general (even for Shimura varieties) but yes in special cases.