Georgia Institute of Technology College of Sciences

The Anderson model on a discrete graph is given by the graph Laplacian perturbed by a random potential. I study spectral properties of this random Schroedinger operator on a random regular graph of fixed degree in the limit where the number of vertices tends to infinity.The choice of model is motivated by its relation to two important and well-studied models of random operators: On the one hand there are similarities to random matrices, for instance to Wigner matrices, whose spectra are known to obey universal laws. On the other hand a random Schroedinger operator on a random regular graph is expected to approximate the Anderson model on the homogeneous tree, a model where both localization (characterized by pure point spectrum) and delocalization (characterized by absolutely continuous spectrum) was established.I will show that the Anderson model on a random regular graph also exhibits distinct spectral regimes of localization and of delocalization. One regime is characterized by exponential decay of eigenvectors. In this regime I analyze the local eigenvalue statistics and prove that the point process generated by the eigenvalues of the random operator converges in distribution to a Poisson process.In contrast to that I will also show that the model exhibits a spectral regime of delocalization where eigenvectors are not exponentially localized.

For a group G, stable G-equivariant homotopy theory studies (the stabilizations of) topological spaces with a G-action up to G-homotopy. For a field k, stable motivic homotopy theory studies varieties over k up to (a stable notion of) homotopy where the affine line plays the role of the unit interval. When L/k is a finite Galois extension with Galois group G, there is a functor F from the G-equivariant stable homotopy category to the stable motivic homotopy category of k. If k is the complex numbers (or any algebraically closed characteristic 0 field) and L=k (so G is trivial), then Marc Levine has shown that F is full and faithful. If k is the real numbers (or any real closed field) and L=k[i], we show that F is again full and faithful, i.e., that there is a "copy" of stable C_2-equivariant homotopy theory inside of the stable motivic homotopy category of R. We will explore computational implications of this theorem.This is a report on joint work with Jeremiah Heller.

We will outline the proof that gives a positive solution to Weaver's conjecture $KS_2$. That is, we will show that any isotropic collection of vectors whose outer products sum to twice the identity can be partitioned into two parts such that each part is a small distance from the identity. The distance will depend on the maximum length of the vectors in the collection but not the dimension (the two requirements necessary for Weaver's reduction to a solution of Kadison--Singer). This will include introducing a new technique for establishing the existence of certain combinatorial objects that we call the "Method of Interlacing Polynomials." This talk is intended to be accessible by a general mathematics audience, and represents joint work with Dan Spielman and Nikhil Srivastava.

We study the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in p-dimensional spaces as the number of points n goes to infinity, while the dimension p is either fixed or growing with n. For both settings, we derive the limiting empirical distribution of the random angles and the limiting distributions of the extreme angles. The results reveal interesting differences in the two settings and provide a precise characterization of the folklore that ``all high-dimensional random vectors are almost always nearly orthogonal to each other". Applications to statistics and connections with some open problems in physics and mathematics are also discussed. This is a joint work with Tony Cai and Jianqing Fan.