Georgia Institute of Technology College of Sciences

We show that for a fixed $q$, the number of $q$-ary $t$-error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t \leq (1 - q^{-1})n - 2\sqrt{n \log n}$, where $H_q(n, t) = q^n / V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner and makes progress towards a 2005 question of Sapozhenko. 

Is there a natural way to order data in dimension greater than one? The approach based on the notion of data depth, often associated with the name of John Tukey, is among the most popular. Tukey’s depth has found applications in robust statistics, graph theory, and the study of elections and social choice. 

We will give an introduction to the topic, describe the properties of Tukey’s depth, and introduce some remaining open questions as well as our recent progress towards the solutions.

The talk is based on a joint work with Yinan Shen.

A recent advance by Smith establishes a quantitative converse (conjectured by Smyth and Serre) to Fekete's celebrated theorem for compact subsets of $\mathbb{R}$. Answering a basic question raised by Smith, we formulate and prove a quantitative converse of Fekete for general symmetric compact subsets of $\mathbb{C}$. We highlight and exploit the algorithmic nature of our approach to give concrete applications to abelian varieties over finite fields and to extremal problems in algebraic number theory.

In this paper we propose a general dynamical mechanism that can lead to the failure of the Batchelor's mode-wise power spectrum law in passive scalar turbulence and hyperbolic dynamics, while the cumulative law remains true. Of technical interest, we also employ a novel method of power spectral variance to establish an exponential radial shell law for the Batchelor power spectrum. An accessible explanation of the power spectrum laws via harmonic analysis is also given.