Georgia Institute of Technology College of Sciences

The general linear groups GL_n(A) can be defined for any ring A, and Quillen's definition of K-theory of A takes these groups as its starting point. If A is commutative, one may define symplectic K-theory in a very similar fashion, but starting with the symplectic groups Sp_{2n}(A), the subgroup of GL_{2n}(A) preserving a non-degenerate skew-symmetric bilinear form. The result is a sequence of groups denoted KSp_i(A) for i = 0, 1, .... For the ring of integers, there is an interesting action of the absolute Galois group of Q on the groups KSp_i(Z), arising from the moduli space of polarized abelian varieties. In joint work with T. Feng and A. Venkatesh we study this action, which turns out to be an interesting extension between a trivial representation and a cyclotomic representation.

This is a brief (15 minute) presentation of an undergraduate project that took place in the 2017 Fall semester.

Cars are placed with density p on the lattice. The remaining vertices are parking spots that can fit one car. Cars then drive around at random until finding a parking spot. We study the effect of p on the availability of parking spots and observe some intriguing behavior at criticality. Joint work with Michael Damron, Janko Gravner, Hanbeck Lyu, and David Sivakoff. arXiv id: 1710.10529.

Random effects models are commonly used to measure genetic variance-covariance matrices of quantitative phenotypic traits. The population eigenvalues of these matrices describe the evolutionary response to selection. However, they may be difficult to estimate from limited samples when the number of traits is large. In this talk, I will present several results describing the eigenvalues of classical MANOVA estimators of these matrices, including dispersion of the bulk eigenvalue distribution, bias and aliasing of large "spike" eigenvalues, and distributional limits of eigenvalues at the spectral edges. I will then discuss a new procedure that uses these results to obtain better estimates of the large population eigenvalues when there are many traits, and a Tracy-Widom test for detecting true principal components in these models. The theoretical results extend proof techniques in random matrix theory and free probability, which I will also briefly describe.This is joint work with Iain Johnstone, Yi Sun, Mark Blows, and Emma Hine.