Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

Numerical algebraic geometry provides a collection of novel methods to treat the solutions of systems of polynomial equations. These hybrid symbolic-numerical methods based on homotopy continuation technique have found a wide range of applications in both pure and applied areas of mathematics. This talk gives an introduction to numerical algebraic geometry and outlines directions in which the area has been developing. Two topics are highlighted: (1) computation of Galois groups of Schubert problems, a recent application of numerical polynomial homotopy continuation algorithms to enumerative algebraic geometry; (2) numerical primary decomposition, the first numerical method that discovers embedded solution components.

After introducing and reviewing the situation for rational and integral points on curves, I will discuss various aspects of integral points on higher-dimensional varieties. In addition to discussing recent higher-dimensional results, I will also touch on connections with the value distribution theory of holomorphic functions and give some concrete open problems.

It is now increasingly common in statistical practice to encounter datasets in which the number of observations, n, is of the same order of magnitude as the number of measurements, p, we have per observation. This simple remark has important consequences for theoretical (and applied) statistics. Namely, it suggests on the theoretical front that we should study the properties of statistical procedures in an asymptotic framework where p and n both go to infinity (and p/n has for instance a finite non-zero limit). This is drastically different from the classical theory where p is held fixed when n goes to infinity. Since a number of techniques in multivariate statistics rely fundamentally on sample covariance matrices and their eigenvalues and eigenvectors, the spectral properties of large dimensional covariance matrices play a key role in such "large n, large p" analyses. In this talk, I will present a few problems I have worked on, concerning different aspects of the interaction between random matrix theory and multivariate statistics. I will discuss some fluctuation properties of the largest eigenvalue of sample covariance matrices when the population covariance is (fairly) general, talk about estimation problems for large dimensional covariance matrices and, time permitting, address some applications in a classic problem of mathematical finance. The talk will be self-contained and no prior knowledge of statistics or random matrix theory will be assumed.