I periodically give talks at the Wee Math Seminar (our research group seminar). In 2023 and 2022 I periodically gave talks at GAPS (a junior algebra seminar) and the Postgraduate seminar.

TBA

Given an object in a Frobenius exact category satisfying mild conditions, we construct a derived autoequivalence of the object's endomorphism algebra as a composition of equivalences induced by tilting modules. In fact, we show that this autoequivalence can be characterised as a twist around a functor. The key idea is that this autoequivalence exists when suspension functor in the stable category acts periodically on the object, and this is automatically satisfied when the stable endomorphism algebra of the object satisfies certain "relatively spherical" properties. The properties of this stable endomorphism algebra actually afford some control over the autoequivalence. For example, when this algebra is finite dimensional over a field, then the cotwist is a shift of the Nakayama functor of the stable endomorphism algebra and, moreover, the autoequivalence is a spherical twist. As an application, we obtain new non-trivial derived autoequivalences for very singular varieties.

An important technology in cluster theory is that of mutation, which provides a way of producing more rigid and cluster tilting objects from an initial one. Iyama and Wemyss introduced a concept which is closely related to this mutation: the mutation of modifying modules. In this talk, I will address the question of specifying precisely when the mutation of a modifying module is involutive. When this is the case, we are able to construct interesting autoequivalences called the noncommutative twist and cotwist functors. I will discuss their construction and their properties.

Stashef introduced A-infinity algebras to study "group-like" topological spaces in the begin- ning of the 1960s. In more recent years, these algebras have become relevant for algebra, geometry and mathematical physics. In this talk, I will introduce A-infinity algebras from an algebraic point of view, draw examples from geometry, and discuss their applications in algebraic geometry.

Cluster algebras were introduced by Formin and Zelevinsky in 2002 in order to study canoncial bases of quantum groups. Since then, connections between cluster algebras and several areas of mathematics have been discovered. Notably, these algebras find applications in Poisson geometry, integrable systems, algebraic geometry, and the representation theory of quivers. The link between cluster algebras and the representation theory of quivers passes through a categorification of these algebras. In this talk, I will present an introduction to cluster algebras, and discuss their categorification via cluster categories.

Sheaves are key tools in algebraic geometry. The key idea is that we can study a space by equipping it with extra data, and sheaves present a way to associate a collection of data to a space and organize it in such a way that we can study the local and global properties of the space, as well as the connections between these properties. A motivating example of a sheaf is the sheaf of functions on the real line. In this talk, we will explore the properties of this sheaf in order to understand the abstract definition of a sheaf