Georgia Institute of Technology College of Sciences

Let $H$ be a separable Hilbert space and let $\{x_n\}$ be a Bessel sequence or a frame for $H$ which does not contain any zero elements. We say that $\{x_n\}$ is a normalizable Bessel sequence or normalizable frame if the normalized sequence $\{x_n/||x_n||\}$ remains a Bessel sequence or frame. In this talk, we will present characterizations of normalizable and non-normalizable frames . In particular, we prove that normalizable frames can only have two formulations.  Perturbation theorems tailored for normalizable frames will be also presented. Finally, we will talk about some open questions related to the normalizable frames.

The Matroid Minors Project of Geelen, Gerards, and Whittle describes the structure of minor-closed classes of matroids representable by a matrix over a fixed finite field. To use these results to study specific classes, it is important to study the matroids in the class containing spanning cliques. A spanning clique of a matroid M is a complete-graphic restriction of M with the same rank as M.

In this talk, we will describe the structure of dyadic matroids with spanning cliques. The dyadic matroids are those matroids that can be represented by a real matrix each of whose nonzero subdeterminants is a power of 2, up to a sign. A subclass of the dyadic matroids is the signed-graphic matroids. In the class of signed-graphic matroids, the entries of the matrix are determined by a signed graph. Our result is that dyadic matroids with spanning cliques are signed-graphic matroids and a few exceptional cases.

The main results in this talk will come from joint work with Ben Clark, James Oxley, and Stefan van Zwam. This talk will include a brief introduction to matroids.

We'll talk about problems of optimizing a quadratic function subject to quadratic constraints, in addition to a sparsity constraint that requires that solutions have only a few nonzero entries. Such problems include sparse versions of linear regression and principal components analysis. We'll see that this problem can be formulated as a convex conical optimization problem over a sparse version of the positive semidefinite cone, and then see how we can approximate such problems using ideas arising from the study of hyperbolic polynomials. We'll also describe a fast algorithm for such problems, which performs well in practical situations.

Given n points on a disk, we will describe how to build an A-infinity category based on the instanton Floer complex of links, and explain why it is finitely generated. This is based on work in progress with Ko Honda.