SCU Mathematics Colloquium Series Schedule
Title: A Cousin Problem formulation of Domain Decomposition
We present an overlapping domain decomposition procedure for solving the Poisson equation with Dirichlet boundary values. The procedure uses a single set of local solutions, and casts the problem of resolving the mismatch between the local solutions as a Cousin Problem that can be solved directly without iteration. The overall order of computational complexity is given by the local solver. The procedure requires minimal overlap and minimal data transfer between the subdomains, and can be extended to more general geometries and more general elliptic equations. We will discuss this work in the context of two classical papers from the late ninetheenth century, one by P. Cousin the other by A. Schwarz. We will present the theoretical underpinnings of our work and conclude with several examples.
Title: Small Gaps Between Primes: My 45 Minutes of Fame
I will describe my recent joint work with Janos Pintz and Cem Yildirimon small gaps between primes. One surprising result is that if the primes are well distributed in arithmetic progressions then one can prove results not too far from the twin prime conjecture, for example, if the Elliott-Halberstam conjecture is true then there are infinitely many pairs of primes with difference 16 or less. Unconditionally we prove that there are pairs of primes mu
Title: Determinants For Elliptic Operators and Their Many Uses
Abstract: The notion of a determinant of an elliptic operator arises naturally in a variety of contexts. I will discuss one such context and develop the notion of a zeta regularized determinant for certain elliptic operators. I will survey the use of this determinant in geometry and topology, sketching new proofs of classic theorems and emphasizing the properties of zeta regularized determinants which make them such a powerful tool. Finally, I will discuss limitations of zeta regularization, techniques for addressing the limitations, and what is to be gained from addressing the limitations. The talk is intended for a general audience and will begin with familiar exercises from calculus and linear algebra. The talk will strive to avoid technicalities and focus on simple examples.
Unless noted otherwise, talks will be at 4:00pm Tuesdays. Room O'Connor 206. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.