Monday, October 15, 2018
10/15/2018 - 4:15pm
S&DS Joint Statistics & Data Science and Applied Mathematics Colloquium
In this talk, I will give two vignettes on the theme of sparse matrices in sparse analysis. The first vignette covers work from compressive sensing in which we want to design sparse matrices (i.e., matrices with few non-zero entries) that we use to (linearly) sense or measure compressible signals. We also design algorithms such that, from these measurements and these matrices, we can efficiently recover a compressed, or sparse, representation of the sensed data. I will discuss the role of expander graphs and error correcting codes in these designs and applications to high throughput biological screens. The second vignette flips the theme; suppose we are given a distance or similarity matrix for a data set that is corrupted in some fashion, find a sparse correction or repair to the distance matrix so as to ensure the corrected distances come from a metric; i.e., repair as few entries as possible in the matrix so that we have a metric. I will discuss generalizations to graph metrics, applications to (and from) metric embeddings, and algorithms for variations of this problem. I will also touch upon applications in machine learning and bio-informatics.
10/15/2018 - 4:30pm
A meromorphic open-string vertex algebra (MOSVA for short) is an algebraic structure formed by vertex operators that satisfy associativity but do not necessarily satisfy commutativity. It was introduced by Yi-Zhi Huang in 2012. I will start by recall the definitions and give a brief summary on the current progress of our joint studies on these algebras and their representations. Then I will introduce the cohomology theory and explain the proof of the following reductivity theorem: if for every bimodule for the MOSVA, the first cohomology is given by the zero-mode derivations, then for every left module for the MOSVA that is of finite length and satisfies a composability condition, it is completely reducible.
10/15/2018 - 5:30pm
One way to obtain a Kleinian group is to consider the group generated by reflections in the faces of a hyperbolic polyhedron with dihedral angles submultiples of \pi. Andreev’s theorem guarantees the existence of hyperbolic polyhedra with non-obtuse dihedral angles satisfying certain combinatorial conditions. In this talk, I will construct a one dimensional family of hyperbolic polyhedra with the help of Andreev’s theorem, so that the Kleinian groups generated by reflections in some of the faces realize a one dimensional deformation of a convex cocompact acylindrical hyperbolic 3-manifold.