We will give an overview of some of the developments in recent years dealing with the description of asymptotic states of solutions to semilinear evolution equations ("soliton resolution conjecture").
New results will be presented on damped subcritical Klein-Gordon equations, joint with Nicolas Burq and Genvieve Raugel.
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.
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.
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.