Georgia Institute of Technology College of Sciences

In this talk, I will present two examples of the application of wavelet analysis to the understanding of mild Traumatic Brain Injury (mTBI). First, the talk will focus on how wavelet-based features can be used to define important characteristics of blast-related acceleration and pressure signatures, and how these can be used to drive a Naïve Bayes classifier using wavelet packets. Later, some recent progress on the use of wavelets for data-driven clustering of brain regions and the characterization of functional network dynamics related to mTBI will be discussed. In particular, because neurological time series such as the ones obtained from an fMRI scan belong to the class of long term memory processes (also referred to as 1/f-like processes), the use of wavelet analysis guarantees optimal theoretical whitening properties and leads to better clusters compared to classical seed-based approaches.

Sampling permutations from S_n is a fundamental problem from probability theory. The nearest neighbor transposition chain M_n is known to converge in time \Theta(n^3 \log n) in the uniform case and time \Theta(n^2) in the constant bias case, in which we put adjacent elements in order with probability p \neq 1/2 and out of order with probability 1-p. In joint work with Prateek Bhakta, Dana Randall and Amanda Streib, we consider the variable bias case where the probability of putting an adjacent pair of elements in order depends on the two elements, and we put adjacent elements x < y in order with probability p_{x,y} and out of order with probability 1-p_{x,y}. The problem of bounding the mixing rate of M_n was posed by Fill and was motivated by the Move-Ahead-One self-organizing list update algorithm. It was conjectured that the chain would always be rapidly mixing if 1/2 \leq p_{x,y} \leq 1 for all x < y, but this was only known in the case of constant bias or when p_{x,y} is equal to 1/2 or 1, a case that corresponds to sampling linear extensions of a partial order. We prove the chain is rapidly mixing for two classes: Choose Your Weapon,'' where we are given r_1,..., r_{n-1} with r_i \geq 1/2 and p_{x,y}=r_x for all x < y (so the dominant player chooses the game, thus fixing his or her probability of winning), and League Hierarchies,'' where there are two leagues and players from the A-league have a fixed probability of beating players from the B-league, players within each league are similarly divided into sub-leagues with a possibly different fixed probability, and so forth recursively. Both of these classes include permutations with constant bias as a special case. Moreover, we also prove that the most general conjecture is false. We do so by constructing a counterexample where 1/2 \leq p_{x,y} \leq 1 for all x < y, but for which the nearest neighbor transposition chain requires exponential time to converge.

Self-adjoint $n$-by-$n$ matrices have a natural partial ordering, namely $A \leq B$ if the matrix $B - A$ is positive semi-definite. In 1934 K. Loewner characterized functions that preserve this ordering; these functions are called $n$-matrix monotone. The condition depends on the dimension $n$, but if a function is $n$-matrix monotone for all $n$, then it must extend analytically to a function that maps the upper half-plane to itself. I will describe Loewner's results, and then discuss what happens if one wants to characterize functions $f$ of two (or more) variables that are matrix monotone in the following sense: If $A = (A_1, A_2)$ and $B = (B_1,B_2)$ are pairs of commuting self-adjoint $n$-by-$n$ matrices, with $A_1 \leq B_1$ and $A_2 \leq B_2$, then $f(A) \leq f (B)$. This talk is based on joint work with Jim Agler and Nicholas Young.

One of the most outstanding problems in differential geometry is concerned with flexibility of closed surface in Euclidean 3-space: Is it possible to continuously deform a smooth closed surface without changing its intrinsic metric structure? In this talk I will give a quick survey of known results in this area, which is primarily concerned with convex surfaces, and outline a program for studying the general case.