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In this talk, we will discuss the cohomology groups of the free loop space of a topological space X, which is the space of all continuous maps from S^1 to X. Using the Eilenberg–Moore spectral sequence, we will connect these cohomology groups and Hochschild homology and compute them in specific cases. We will mostly follow the article "On the Characteristic Zero Cohomology of the Free Loop Space" by L. Smith.

We will present the fastest known algorithm for computing discrete cubical homology, a valuable graph invariant with a wide range of applications, including matroid theory, hyperplane arrangements, and topological data analysis. This invariant is capable of detecting certain types of "holes" within a graph, providing insight into its structure.

We will begin by defining discrete cubical homology and outlining the standard approach to its computation. We will then present an algorithm designed to improve efficiency by using techniques such as faster generation of singular cubes, reducing chain complex dimensions through quotients over automorphisms, and preprocessing graphs using results from discrete homotopy theory. These advancements aim to make the invariant more accessible computationally for applications. We are now able to compute examples that were previously considered out of reach by experts. Part of the motivation for this work was a joint project with the research group of R. Laubenbacher (Dept. of Medicine, University of Florida) on analyzing gene regulatory networks.

This talk is based on the paper: Kapulkin, Kershaw, Efficient computations of discrete cubical homology, arXiv:2410.09939.

This talk has been postponed to a later date.

Imagine you don't have perfect knowledge of the entries of a matrix (which is a what happens usually in applications of matrices). What can be said about the eigenvalues of such a matrix?  Is there a pattern to the eigenvalues at all? Can we say anything about them? In this talk I shall start with very simple examples and gradually  examine the question, using some computer calculations and some basic undergraduate mathematics.

From the perspective of a pure mathematician, the application of mathematics to concrete problems (like evolution of the species) is not of primary importance. However, that does not imply that applied mathematics deals with problems that are trivial mathematically speaking, or that there is nothing to learn from applications. In this talk, I will showcase different ways in which mathematics are being used in the study of Evolution, ranging from systems of ODEs and PDEs to "simple" stochastic models, with some sprinkles of numerical analysis. My goal is to spark the interest of the pure math audience not only in applying math to biology, but also in further analyzing the mathematics that arise from biological problems for mathematics' sake.