On vector bundles, holonomy and curvature in condensed matter.

Abstract: We present a geometric formulation of the integer quantum Hall effect from the perspective of characteristic classes. Starting from a complex vector bundle over the two-dimensional Brillouin torus, we consider the natural Berry connection arising from a family of spectral projectors with a spectral gap. The associated curvature form represents, via the Chern–Weil homomorphism, the first Chern class of the bundle. Its integral over the Brillouin torus is therefore an integer, independent of the choice of connection. We explain how this topological invariant coincides with the quantized Hall conductivity, thus providing a direct bridge between differential geometry and a measurable physical observable. 

On Frieze Patterns. 

Abstract:  Frieze patterns, introduced by H. S. M. Coxeter and John Conway, are arrays of numbers satisfying a simple local determinant rule. Despite their elementary definition, they have deep connections with combinatorics, cluster algebras, and representation theory. In this talk, I will briefly review classical frieze patterns and then discuss infinite friezes. I will describe their basic properties and illustrate how they arise naturally in combinatorial and representation-theoretic settings.

How playing a game gives you the largest collection of numbers (Hackenbush).

Abstract:   Abstract:   John H. Conway was one of the most free-spirited mathematicians of all time. A testament to this can be seen throughout his works and through some funny anecdotes about him told by his peers and students. In particular, this can be felt by going through his collection of books titled Winning Ways for your mathematical plays.

In this one board seminar, I will discuss one particular game described in the first volume of this collection of books and how this led him to the discovery of the largest collection of numbers, and I will also share some remarks on this fascinating construction.

Bases of Cluster Algebras from Surfaces.

Abstract:  This talk provides a concise introduction to cluster algebras and upper cluster algebras arising from surfaces. We explore the structural properties of these algebras and survey several fundamental families of bases, discussing their construction and their significance within the theory of cluster algebras.

Almost complex geometry.

Abstract:  In this talk we will explore the beauties of almost complex geometry, starting at how they generalize complex geometry and seeing all the different structures that emerge in the attempt to overcome the loss of the holomorphic structure, such as the Grey-Hervella classification and the canonical connections.

 C^∞-algebraic geometry and other variations of algebra.

Abstract:  Vector spaces are algebraic structures where we can evaluate linear combinations. Rings are algebraic structures where we know how to evaluate polynomials. We can study many geometric structures using rings -- even manifolds! Nevertheless, we can evaluate more than just polynomials in rings of C^∞-functions on a manifold. By considering a modification of the notion of rings in which we can evaluate C^∞-functions instead of just polynomials, we obtain an algebraic structure suitable for studying C^∞-geometry. I will present some basic ideas to do with these perspectives, and possibly mention some other variations. I am not an expert on this notion of C^∞-algebraic geometry, but I think it is fun, and hope you will too.

 Homotopy classification of 4-manifolds.

Abstract:  It's a classical result of Milnor/Whitehead that simply-connected 4-manifolds are classified up to homotopy equivalence by their intersection forms, but further invariants are needed in the presence of a fundamental group. For many groups the classification is given by a collection of invariants called the quadratic 2-type, though this also fails in general. I will present the invariants and the cases when they classify, as well as counterexamples, and I will mention my own work where I use an additional modified intersection form to obtain the classification for semidirect products of Z and Z/p.