The fields of set theory and homological algebra are both centrally concerned with
questions of compactness, regarding the extent to which a structure's global properties are
determined by its local properties. It is thus no surprise that there has been considerable interplay
between these two fields. In this talk we will discuss some recent applications of set-theoretic
techniques to the study of the derived functor of the inverse limit, with further applications to the study
of strong homology and to the developing field of condensed mathematics. We then relate these
applications back to one of the oldest questions in set theory, that of the cardinality of the continuum.
At their core, these applications reduce to simple, purely combinatorial problems that are of interest in
their own right. No prior knowledge of either set theory or homological algebra will be assumed.
Representation theory involves studying mathematical objects by interpreting them in linear algebra: for example, we interpret group elements as matrices. It can be useful to abstract our linear algebra problem using a quiver, which is a directed graph where the vertices correspond to vector spaces and the edges to linear transformations. Gabriel showed that a quiver has finitely many representations precisely when its underlying graph is of ADE Dynkin type and noticed a pattern which Kontsevich later formalised as the fractionally Calabi-Yau property, based on categorical properties occurring in geometry. I will explain work with Mathew Pugh where we show how this is a shadow of a property of the Temperley-Lieb category, formed from non-crossing lines between dots in the plane, when the quantum parameter is a complex root of unity. This involves working with new definitions of Frobenius algebra objects and Nakayama morphisms in monoidal categories.
SubRiemannian geometry represents a vast generalisation of Riemannian geometry and it is meant to model motions that are permitted only along a limited set of directions at any point. The aim of this talk is to give an intuition of how subRiemannian geometry naturally arises from modelling different mathematical and physical problems (e.g. optimal control, image processing, thermodynamics). Surprisingly, for such a setting many fundamental mathematical objects are not yet defined or understood. If time allows, I will present what obstacles appear when trying to extend the most basic tools of geometric analysis and differential geometry to subRiemannian manifolds.
This will be a talk aimed at a wide audience. In the first part, I will recall the notion of an elliptic curve over the complex numbers and I will review the beautiful theory that allows us to classify them. Following this, I will recall what it means for an elliptic curve to have complex multiplication (CM) and I will discuss major advances over the last 30-40 years that describe how tuples of CM elliptic curves are distributed. Finally, I will venture into the realm of "unlikely intersections", which aims to study the distribution of other forms of "additional structure" on elliptic curves and their generalizations.
In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, uncovering a surprising connection with matched pairs of groups. We will introduce all necessary definitions, including the notion of irretractable solutions, and explore how these solutions correspond with matched pairs of groups. Finally, I will show how each irretractable solution lifts to provide the full classification of all bijective solutions.
