Georgia Institute of Technology College of Sciences

I will discuss the numerical approximation of differential operators on unknown manifolds where the manifolds are identified by a finite sample of point cloud data. While our formulation is general, we will focus on Laplacian operators whose spectral properties are relevant to manifold learning. I will report the spectral convergence results of these formulations with Radial Basis Functions approximation and their strengths/weaknesses in practice. Supporting numerical examples, involving the spectral estimation of various vector Laplacians will be demonstrated. Applications to solve elliptic PDEs will be discussed. To address the practical issue with the RBF approximation, I will discuss a weak approximation with a higher-order local mesh method that not only promotes sparsity but also allows for an estimation of differential operators with nontrivial Cristoffel symbols such as Bochner and Hodge Laplacians.

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.