We consider the finite element solution of the vector Laplace equation on a
domain in two dimensions.  For some choices of boundary conditions, there is a
theory, making use of finite element differential complexes and bounded
cochain projections, that shows that a mixed finite element method using
appropriate choices of finite element spaces, and in which the rotation
of the solution is introduced as a second unknown, leads to a stable,
optimally convergent discretization. However, the theory that leads to these
conclusions does not apply to the case of Dirichlet boundary conditions, in
which both components of the solution vanish on the boundary.  We present
computational examples that demonstrate that such a mixed finite element method
does not perform optimally in this case, and an analysis which theoretically
confirms the suboptimal convergence that does occur and indicates the source
of the problem.  These results also have implications for the solution of the
biharmonic equation and of the Stokes equations using a mixed formulation
involving the vorticity.

Recent years have seen an increasing number of applications of descriptive set theory in ergodic theory and dynamical systems. We present some set theoretic background and survey some of the applications.

Slides for this series of talks can be found here:

https://www.dropbox.com/sh/om8efuv6ez10ysb/AADOA4SPbdjXKoDajEftFb2pa?dl=0

I will discuss recent investigations of various reducibility notions between Pi^1_2 principles of second-order arithmetic, the most familiar of which is implication over the subsystem RCA_0. In many cases, such an implication is actually due to a considerably stronger reduction holding, such as a uniform (a.k.a. Weihrauch) reduction. (Here, we say a principle P is uniformly reducible to a principle Q if there are fixed reduction procedures Phi and Gamma such that for every instance A of P, Phi(A) is an instance of Q, and for every solution S to Phi(A), Gamma(A + S) is a solution to A.) As an example, nearly all the implications between principles lying below Ramsey's theorem for pairs are uniform reductions. In general, the study of when such stronger implications hold and when they do not gives a finer way of calibrating the relative strength of mathematical propositions, and has led to the development of a number of new forcing techniques for constructing models of second-order arithmetic with prescribed combinatorial properties. In addition, this analysis sheds light on several open questions from reverse mathematics, including that of whether the stable form of Ramsey's theorem for pairs (SRT^2_2) implies the cohesive principle (COH) in \omega (standard) models of RCA_0.

Program:

3:00-3:50 PM    Xin Zhou (UC Santa Barbara), `Min-max minimal hypersurfaces with free boundary'

Abstract: I will present a joint work with Martin Li. Minimal surfaces with free boundary are natural critical points of the area functional in compact smooth manifolds with boundary. In this talk, I will describe a general existence theory for minimal surfaces with free boundary. In particular, I will show the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. The minimal surfaces with free boundary were constructed using the min-max method. I will explain the basic ideas behind the min-max theory as well as our new contributions.

4:00-4:50 PM    Vladimir Markovic (Caltech), `Harmonic maps and heat flows on hyperbolic spaces'

Abstract: We prove that any quasi-isometry between hyperbolic manifolds is homotopic to a harmonic quasi-isometry.

A lattice in a Euclidean space is called extremal if it is a local maximum of the packing density function in its dimension. An old theorem of Voronoi gives a beautiful characterization of extremal lattices in terms of their geometric properties. We will review Voronoi's criterion, and then apply it to exhibit families of extremal lattices coming from some algebraic and arithmetic constructions.