Georgia Institute of Technology College of Sciences

Using the covering involution on the double branched cover of S3 branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot (concordance) invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots. This is a joint work with A. Alfieri and A. Stipsicz.

A fundamental question in Dynamical Systems is to identify regions of
phase/parameter space satisfying a given property (stability,
linearization, etc).  In this talk, given a family of analytic circle
diffeomorphisms depending on a parameter, we obtain effective (almost
optimal) lower bounds of the Lebesgue measure of the set of parameters
for which that diffeomorphism is conjugate to a rigid rotation.
We estimate this measure using an a-posteriori KAM
scheme that relies on quantitative conditions that
are checkable using computer-assistance. We carefully describe
how the hypotheses in our theorems are reduced to a finite number of
computations, and apply our methodology to the case of the
Arnold family, in the far-from-integrable regime.

This is joint work with Jordi Lluis Figueras and Alejandro Luque.
 

Rank-structured matrix representations, e.g., H2 and HSS, are commonly used to reduce computation and storage cost for dense matrices defined by interactions between many bodies. The main bottleneck for their applications is the expensive computation required to represent a matrix in a rank-structured matrix format which involves compressing specific matrix blocks into low-rank form.
We focus on the study and application of a class of hybrid analytic-algebraic compression methods, called the proxy point method. We address several critical problems concerning this underutilized method which limit its applicability. A general form of the method is proposed, paving the way for its wider application in the construction of different rank-structured matrices with kernel functions that are more general than those usually used. Further, we extend the applicability of the proxy point method to compress matrices defined by electron repulsion integrals, which accelerates one of the main computational steps in quantum chemistry. 

Committee members: Prof. Edmond Chow (Advisor, School of CSE, Georgia Tech), Prof. David Sherrill (School of Chemistry and Biochemistry, Georgia Tech), Prof. Jianlin Xia (Department of Mathematics, Purdue University), Prof. Yuanzhe Xi (Department of Mathematics, Emory University), and Prof. Haomin Zhou (School of Mathematics, Georgia Tech).