Simon Fraser University

We investigate the long-time behaviour of solutions to a nonlocal partial differential equation on smooth Riemannian manifolds of bounded sectional curvature. The equation models self-collective behaviour with intrinsic interactions that are modeled by an interaction potential. Without the diffusion term, we consider attractive interaction potentials and establish sufficient conditions for a consensus state to form asymptotically. In addition, we quantify the approach to consensus, by deriving a convergence rate for the diameter of the solution’s support. With the diffusion term, the attractive interaction and the diffusion compete. We provide the conditions of the attractive interaction for each part to win.

Steklov eigenproblems and their variants (where the spectral parameter appears in the boundary condition) arise in a range of useful applications. For instance, understanding some properties of the mixed Steklov-Neumann eigenfunctions tells us why we shouldn't use coffee cups for expensive brandy. 

In this talk I'll present a high-accuracy discretization strategy for computing Steklov eigenpairs. The strategy can be used to study questions in spectral geometry, spectral optimization and to the solution of elliptic boundary value problems with Robin boundary conditions.

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Driven by its numerous applications in computational science, the approximation of smooth, high-dimensional functions via sparse polynomial expansions has received significant attention in the last five to ten years.  In the first part of this talk, I will give a brief survey of recent progress in this area.  In particular, I will demonstrate how the proper use of compressed sensing tools leads to new techniques for high-dimensional approximation which can mitigate the curse of dimensionality to a substantial extent.  The rest of the talk is devoted to approximating functions defined on irregular domains.  The vast majority of works on high-dimensional approximation assume the function in question is defined over a tensor-product domain.  Yet this assumption is often unrealistic.  I will introduce a method, known as polynomial frame approximation, suitable for broad classes of irregular domains and present theoretical guarantees for its approximation error, stability, and sample complexity.  These results show the suitability of this approach for high-dimensional approximation through the independence (or weak dependence) of the various guarantees on the ambient dimension d.  Time permitting, I will also discuss several extensions.

Mazur observed that in many cases where an elliptic curve E has a non-trivial element C in its Tate-Shafarevich group, one can find another elliptic curve E' such that ExE' admits an isogeny that kills C. For elements of order 2 and 3 one can prove that such an E' always exists. However, for order 4 this leads to a question about rational points on certain K3-surfaces. We show how to explicitly construct these surfaces and give some results on their rational points.

This is joint work with Tom Fisher.
 

I will describe how a search for the answer to an old question about the existence of monotone arithmetic progressions in permutations of positive integers led to the study of infinite words with bounded additive complexity. The additive complexity of a word on a finite subset of integers is defined as the function that, for a positive integer $n$, counts the maximum number of factors of length $n$, no two of which have the same sum.

Over the last several decades, many mesh generation methods and a plethora of adaptive methods for solving differential equations have been developed.  In this talk, we take a general approach for describing the mesh generation problem, which can be considered as being in some sense equivalent to determining a coordinate transformation between physical space and a computational space.  Our description provides some new theoretical insights into precisely what is accomplished from mesh equidistribution (which is a standard adaptivity tool used in practice) and mesh alignment.  We show how variational mesh generation algorithms, which have historically been the most common and important ones, can generally be compared using these mesh generation principles.  Lastly, we relate these to a variety of moving mesh methods for solving time-dependent PDEs.

This is joint work with Weizhang Huang, Kansas University