Georgia Institute of Technology College of Sciences

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 shows the existence of bipartite Ramanujan Graphs of any degree as well as some of mixed degrees. Our approach uses the idea of Bilu and Linial to show that there exists a 2-lift of a given Ramanujan graph which maintains the Ramanujan property. 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 computer science audience, and represents joint work with Dan Spielman and Nikhil Srivastava.- See more at: http://www.arc.gatech.edu/events/arc-colloquium-adam-marcus-crisplycom-and-yale-university#sthash.qdZRaV1k.dpuf

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.

We will start by describing some general features of quasilinear dispersive and wave equations. In particular we will discuss a few important aspects related to the question of global regularity for such equations. We will then consider the water waves system for the evolution of a perfect fluid with a free boundary. In 2 spatial dimensions, under the influence of gravity, we prove the existence of global irrotational solutions for suitably small and regular initial data. We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the 3 dimensional case.

Motivated by the ubiquity of signal-plus-noise type models in high-dimensional statistical signal processing and machine learning, we consider the eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Applications in mind are as diverse as radar, sonar, wireless communications, spectral clustering, bio-informatics and Gaussian mixture cluster analysis in machine learning. We provide an application-independent approach that brings into sharp focus a fundamental informational limit of high-dimensional eigen-analysis. Building on this success, we highlight the random matrix origin of this informational limit, the connection with "free" harmonic analysis and discuss how to exploit these insights to improve low-rank signal matrix denoising relative to the truncated SVD.