Speaker: Phillip Wesolek (Weslyan U.)
Title: An invitation to totally disconnected locally compact groups
Abstract: Locally compact groups appear across mathematics; they arise as Galois groups in algebra, isometry groups in geometry, and full groups in dynamics. The study of locally compact groups splits into two cases: the connected groups and the totally disconnected groups. There is a rich and deep theory for the connected groups, which was developed over the last century. On the other hand, the study of the totally disconnected groups only seriously began in the last 30 years, and moreover, these groups today appear to admit an equally rich and deep theory. In this talk, we will begin by motivating the study of totally disconnected locally compact groups and presenting several examples. We will then discuss a natural dividing line in the theory and a fundamental decomposition theorem.
Title: Computing matrix eigenvalues
Speaker: Yuji Nakatsukasa, National Institute for Informatics, Japan
The numerical linear algebra community solves two main problems: linear
systems, and eigenvalue problems. They are both vastly important; it
would not be too far-fetched to say that most (continuous) problems in
scientific computing eventually boil down to one or both of these.
This talk focuses on eigenvalue problems. I will first describe some of
its applications, such as Google’s PageRank, PCA, finding zeros and
poles of functions, and nonlinear and global optimization. I will then turn to
algorithms for computing eigenvalues, namely the classical QR
algorithm—which is still the basis for state-of-the-art. I will
emphasize that the underlying mathematics is (together with the power
method and numerical stability analysis) rational approximation theory.
Title: Active matter invasion of a viscous fluid and a no-flow theorem
Abstract: Suspensions of swimmers or active particles in fluids exhibit incredibly rich behavior, from organization on length scales much longer than the individual particle size to mixing flows and negative viscosities. We will discuss the dynamics of hydrodynamically interacting motile and non-motile stress-generating particles as they invade a surrounding viscous fluid, modeled by equations which couple particle motions and viscous fluid flow. Depending on the nature of their self-propulsion, colonies of swimmers can either exhibit a dramatic splay, or instead a cascade of transverse concentration instabilities, governed at small times by an equation which also describes the Saffman-Taylor instability in a Hele-Shaw cell, or Rayleigh-Taylor instability in two-dimensional flow through a porous medium. Analysis of concentrated distributions of particles matches the results of full numerical simulations. Along the way we will prove a very surprising “no-flow theorem”: particle distributions initially isotropic in orientation lose isotropy immediately but in such a way that results in no fluid flow anywhere and at any time.