School of Mathematics

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...

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...

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...

A distinct covering system of congruences is a finite collection of arithmetic progressions to distinct moduli \[ a_i \bmod m_i, 1 m_1 m_2 \cdots m_k \] whose union is the integers. Answering a question of Erdős, I have shown that the least...

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...

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...

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...

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...

The dichotomy between overtwisted and tight contact structures has been central to the classification of contact structures in dimension 3. Ozsvath-Szabo's contact invariant in Heegaard Floer homology proved to be an efficient tool to distinguish...

In this talk we will discuss further the existence of knot complements with essential surfaces of unbounded Euler characteristics. More precisely, we show the existence of a knot with an essential tangle decomposition for any number of strings. We...