I will discuss several results on abstract homomorphisms between the groups of rational points of algebraic groups. The main focus will be on a conjecture of Borel and Tits formulated in their landmark 1973 paper.
Our results settle this conjecture in several cases; the proofs make use of the notion of an algebraic ring. I will mention several applications to character varieties of finitely generated groups and representations of some non-arithmetic groups.
The Random Fourier Features (RFF) method (Rahimi, Recht, NIPS 2007) is one of the most practically successful techniques for accelerating computationally expensive nonlinear kernel learning methods. By quickly computing a low-rank approximation for any shift-invariant kernel matrix, RFF can serve as a preprocessing step to generically accelerate algorithms for kernel ridge regression, kernel clustering, kernel SVMs, and other benchmark data analysis tools.
This talk will be an introduction to the methods used in the study of spectral properties of Schroedinger operators with a potential defined via the action of an ergodic transformation. Open problems relating to Lyapunov exponents over a skew shift base will be discussed.
We develop efficient algorithms for estimating low-degree moments of unknown distributions in the presence of adversarial outliers. The guarantees of our algorithms improve in many cases significantly over the best previous ones, obtained in recent works. We also show that the guarantees of our algorithms match information-theoretic lower-bounds for the class of distributions we consider. These better guarantees allow us to give improved algorithms for independent component analysis and learning mixtures of Gaussians in the presence of outliers.
Given a measure preserving dynamical system, a real-valued observable determines a random process (by composing the observable with the iterates of the transformation). An important topic in ergodic theory is the study of the statistical properties of the corresponding sum process.