Wednesday, March 8, 2023
03/08/2023 - 1:00pm
Boundary Value Problems (BVPs) are ubiquitous in engineering and scientific applications. One of the most robust and accurate methods for solving BVPs is the Boundary Integral Equation Method, which has the great advantage of dimensionality reduction: all of the unknowns reside on the boundary surface instead of in the volume. A key challenge when solving integral equations is that special quadrature methods are required to discretize the underlying singular and near-singular integral operators. Accurate discretization of these operators is especially important in, for example, problems that involve close structure-structure or fluid-structure interactions. In this talk, we present some recent advancements on singular and near-singular numerical integration based on one of the simplest quadrature methods -- the Trapezoidal rule.
03/08/2023 - 4:15pm
Ramsey theory is a branch of combinatorics which seeks to find patterns in disorganized situations. One of its main achievements, Szemeredi’s theorem on arithmetic progressions, got an ergodic theoretic proof in 1977 when Furstenberg created a Correspondence Principle to encode combinatorial information about sets of integers into a dynamical system. Since then ergodic methods have been very successful in obtaining new Ramsey theoretic results, some of which still have no purely combinatorial proof.
I will survey some of the history of how ergodic theory and Ramsey theory are interconnected, leading to a recent result involving infinite sumsets.