Georgia Institute of Technology College of Sciences

The set of knots up to a four-dimensional equivalence relation can be given the structure of a group, called the (smooth) knot concordance group. We will discuss how to compute concordance invariants using Heegaard Floer homology. We will then introduce the idea of a "reduced" knot Floer complex, see how it can be used to simplify computations, and give examples of how it can be helpful in distinguishing knots which are not concordant.

We finish our discussion on Oseledets Theorem by proving the convergence of the filtration. This is part of a reading seminar geared towards understanding of Smooth Ergodic Theory. (The study of dynamical systems using at the same time tools from measure theory and from differential geometry)It should be accesible to graduate students and the presentation is informal. The first goal will be a proof of the Oseledets multiplicative ergodic theorem for random matrices. Then, we will try to cover the Pesin entropy formula, invariant manifolds, etc.

Given a closed subvariety X of affine space A^n, there is a surjective map from the analytification of X to its tropicalisation. The natural question arises, whether this map has a continuous section. Recent work by Baker, Payne, and Rabinoff treats the case of curves, and even more recent work by Cueto, Haebich, and Werner treats Grassmannians of 2-spaces. I will sketch how one can often construct such sections when X is obtained from a linear space smeared around by a coordinate torus action. In particular, this gives a new, more geometric proof for the Grassmannian of 2-spaces; and it also applies to some determinantal varieties. (Joint work with Elisa Postinghel)

In this age of high-dimensional data, many challenging questions take the following shape: can you check whether the data has a certain desired property by checking that property for many, but low-dimensional data fragments? In recent years, such questions have inspired new, exciting research in algebra, especially relevant when the property is highly symmetric and expressible through systems of polynomial equations. I will discuss three concrete questions of this kind that we have settled in the affirmative: Gaussian factor analysis from an algebraic perspective, high-dimensional tensors of bounded rank, and higher secant varieties of Grassmannians. The theory developed for these examples deals with group actions on infinite-dimensional algebraic varieties, and applies to problems from many areas. In particular, I will sketch its (potential) relation to the fantastic Matroid Minor Theorem.