Georgia Institute of Technology College of Sciences

We extend the theory of infinite matroids recently developed by Bruhn et al to a well-known classical result in finite matroids while using the theory of connectivity for infinitematroids of Bruhn and Wollan. We prove that every infinite connected matroid M determines a graph-theoretic decomposition tree whose vertices correspond to minors of M that are3-connected, circuits, or cocircuits, and whose edges correspond to 2-separations of M. Tutte and many other authors proved such a decomposition for finite graphs; Cunningham andEdmonds proved this for finite matroids and showed that this decomposition is unique if circuits and cocircuits are also allowed. We do the same for infinite matroids. The knownproofs of these results, which use rank and induction arguments, do not extend to infinite matroids. Our proof avoids such arguments, thus giving a more first principles proof ofthe finite result. Furthermore, we overcome a number of complications arising from the infinite nature of the problem, ranging from the very existence of 2-sums to proving the treeis actually graph-theoretic.

We study the problem of testing for the significance of a subset of regression coefficients in a linear model under the assumption that the coefficient vector is sparse, a common situation in modern high-dimensional settings.  Assume there are p variables and let S be the number of nonzero coefficients.  Under moderate sparsity levels, when we may have S > p^(1/2), we show that the analysis of variance F-test is essentially optimal.  This is no longer the case under the sparsity constraint S < p^(1/2).  In such settings, a multiple comparison procedure is often preferred and we establish its optimality under the stronger assumption S < p^(1/4).  However, these two very popular methods are suboptimal, and sometimes powerless, when p^(1/4) < S < p^(1/2).  We suggest a method based on the Higher Criticism that is essentially optimal in the whole range S < p^(1/2).  We establish these results under a variety of designs, including the classical (balanced) multi-way designs and more modern `p > n' designs arising in genetics and signal processing. (Joint work with Emmanuel Candès and Yaniv Plan.)

The Jones polynomial is a link invariant that can be understood in terms of the representation theory of the quantum group associated to sl2. This description facilitated a vast generalization of the Jones polynomial to other quantum link and tangle invariants called Reshetikhin-Turaev invariants. These invariants, which arise from representations of quantum groups associated to simple Lie algebras, subsequently led to the definition of quantum 3-manifold invariants. In this talk we categorify quantum groups using a simple diagrammatic calculus that requires no previous knowledge of quantum groups. These diagrammatically categorified quantum groups not only lead to a representation theoretic explanation of Khovanov homology but also inspired Webster's recent work categorifying all Reshetikhin-Turaev invariants of tangles.