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Series: SIAM Student Seminar

This joint SIAM student conference is organized by the SIAM Student Chapter at School of Mathematics, Georgia Tech together with SIAM chapters at Clemson University, Emory University and University of Alabama at Birmingham. Detailed schedule and information can be found at jssc.math.gatech.edu.

Series: SIAM Student Seminar

Series: SIAM Student Seminar

Series: Math Physics Seminar

Localization properties of quantum many-body systems have been a very active subject in theoretical physics in the most recent decade. At the same time, finding rigorous approaches to understanding many-body localization remains a wide open challenge. We will report on some recent progress obtained for the case of quantum spin chains, where joint work with A. Elgart and A. Klein has provided a proof of several manifestations of MBL for the droplet spectrum of the disordered XXZ chain.

Series: Math Physics Seminar

Series: ACO Student Seminar

We study the $A$-optimal design problem where we are given vectors $v_1,\ldots, v_n\in \R^d$, an integer $k\geq d$, and the goal is to select a set $S$ of $k$ vectors that minimizes the trace of $\left(\sum_{i\in S} v_i v_i^{\top}\right)^{-1}$. Traditionally, the problem is an instance of optimal design of experiments in statistics (\cite{pukelsheim2006optimal}) where each vector corresponds to a linear measurement of an unknown vector and the goal is to pick $k$ of them that minimize the average variance of the error in the maximum likelihood estimate of the vector being measured. The problem also finds applications in sensor placement in wireless networks~(\cite{joshi2009sensor}), sparse least squares regression~(\cite{BoutsidisDM11}), feature selection for $k$-means clustering~(\cite{boutsidis2013deterministic}), and matrix approximation~(\cite{de2007subset,de2011note,avron2013faster}). In this paper, we introduce \emph{proportional volume sampling} to obtain improved approximation algorithms for $A$-optimal design.Given a matrix, proportional volume sampling involves picking a set of columns $S$ of size $k$ with probability proportional to $\mu(S)$ times $\det(\sum_{i \in S}v_i v_i^\top)$ for some measure $\mu$. Our main result is to show the approximability of the $A$-optimal design problem can be reduced to \emph{approximate} independence properties of the measure $\mu$. We appeal to hard-core distributions as candidate distributions $\mu$ that allow us to obtain improved approximation algorithms for the $A$-optimal design. Our results include a $d$-approximation when $k=d$, an $(1+\epsilon)$-approximation when $k=\Omega\left(\frac{d}{\epsilon}+\frac{1}{\epsilon^2}\log\frac{1}{\epsilon}\right)$ and $\frac{k}{k-d+1}$-approximation when repetitions of vectors are allowed in the solution. We also consider generalization of the problem for $k\leq d$ and obtain a $k$-approximation. The last result also implies a restricted invertibility principle for the harmonic mean of singular values.We also show that the $A$-optimal design problem is$\NP$-hard to approximate within a fixed constant when $k=d$.

Series: ACO Student Seminar

Series: ACO Student Seminar

Series: Algebra Seminar

The nerve complex of an open covering is a well-studied notion. Motivated by the so-called Lyubeznik complex in local algebra, and other sources, a notion of higher nerves of a collection of subspaces can be defined. The definition becomes particularly transparent over a simplicial complex. These higher nerves can be used to compute depth, and the h-vector of the original complex, among other things. If time permits, I will discuss new questions arises from these notions in commutative algebra, in particular a recent example of Varbaro on connectivity of hyperplane sections of a variety. This is joint work with J. Doolittle, K. Duna, B. Goeckner, B. Holmes and J. Lyle.

Series: Algebra Seminar