Georgia Institute of Technology College of Sciences

This presentation is based on the papers by D. Paul and I. Johnstone (2007) and V.Q. Vu and J. Lei (2012). Here is the abstract of the second paper. We study the sparse principal components analysis in the high-dimensional setting, where $p$ (the number of variables) can be much larger than $n$ (the number of observations). We prove optimal, non-aymptotics lower bounds and upper bounds on the minimax estimation error for the leading eigenvector when it belongs to an $l_q$ ball for $q\in [0,1]$. Our bound are sharp in $p$ and $n$ for all $q\in[0,1]$ over a wide class of distributions. The upper bound is obtained by analyzing the performance of $l_q$-constrained PCA. In particular, our results provide convergence rates for $l_1$-constrained PCA.

The Neumann-Zagier equations are well-understood objects of classical hyperbolic geometry. Our discovery is that they have a nontrivial quantum content, (that for instance captures the perturbation theory of the Kashaev invariant to all orders) expressed via universal combinatorial formulas. Joint work with Tudor Dimofte.

The study of transport is an active area of applied mathematics of interest to fluid mechanics, plasma physics, geophysics, engineering, and biology among other areas. A considerable amount of work has been done in the context of diffusion models in which, according to the Fourier-­‐Fick’s prescription, the flux is assumed to depend on the instantaneous, local spatial gradient of the transported field. However, despiteits relative success, experimental, numerical, and theoretical results indicate that the diffusion paradigm fails to apply in the case of anomalous transport. Following an overview of anomalous transport we present an alternative(non-­‐diffusive) class of models in which the flux and the gradient are related non-­‐locally through integro-­differential operators, of which fractional Laplacians are a particularly important special case. We discuss the statistical foundations of these models in the context of generalized random walks with memory (modeling non-­‐locality in time) and jump statistics corresponding to general Levy processes (modeling non-­‐locality in space). We discuss several applications including: (i) Turbulent transport in the presence of coherent structures; (ii) chaotic transport in rapidly rotating fluids; (iii) non-­‐local fast heat transport in high temperature plasmas; (iv) front acceleration in the non-­‐local Fisher-­‐Kolmogorov equation, and (v) non-­‐Gaussian fluctuation-­‐driven transport in the non-­‐local Fokker-­‐Planck equation.

The outer automorphism group Out(F) of a non-abelian free group F of finite rank shares many properties with linear groups and the mapping class group Mod(S) of a surface, although the techniques for studying Out(F) are often quite different from the latter two. Motivated by analogy, I will present some results about Out(F) previously well-known for the mapping class group, and highlight some of the features in the proofs which distinguish it from Mod(S).