Abstract: Although Évariste Galois died at the age of 20 after a duel, the ideas he introduced continue to influence modern mathematics nearly two centuries later. In this talk, we will begin with a short refresher on classical Galois theory before moving to differential Galois theory, introducing Picard–Vessiot theory to study linear differential equations. We will then discuss the difference-differential setting, where differential equations are considered together with additional endomorphism(s) acting on the base field, and briefly outline my current research on extending this theory to several commuting endomorphisms..
Abstract: In sub-Riemannian geometry, Carnot groups play a role analogous to that of Euclidean spaces in the Riemannian setting. Their special structure allows one to define an intrinsic notion of differentiability, namely Pansu differentiability, which in turn gives rise to the Pansu pullback on differential forms. In this talk, I will discuss how pullback operators interact with the differentials of the de Rham and Rumin complexes, focusing on commutativity properties. If time permits, I will also present a recent result on the commutativity of the Pansu pullback with the differentials arising in the Spectral complexes associated with the de Rham complex.
Abstract: In 1968, N. Jacobson proposed the problem of characterizing finite-dimensional Lie algebras that have a primitive universal enveloping algebra. Later, in 1976, A. Ooms demonstrated that these can be characterized exactly as Frobenius Lie algebras. In this talk, I will provide an introduction to this class of Lie algebras. We will discuss their fundamental properties and explore their connections with other algebraic structures, such as contact Lie algebras and pre-Lie structures.
Abstract: We show that for every graph $H$, there is a hereditary weakly sparse graph class $\mathcal C_H$ of unbounded treewidth such that the $H$-free (i.e., excluding $H$ as an induced subgraph) graphs of $\mathcal C_H$ have bounded treewidth. This refutes several conjectures and critically thwarts the quest for the unavoidable induced subgraphs in classes of unbounded treewidth, a wished-for induced counterpart of the Grid Minor theorem. We actually show a stronger result: For every positive integer $t$, there is a hereditary graph class $\mathcal C_t$ of unbounded treewidth such that for any graph $H$ of treewidth at most $t$, the $H$-free graphs of $\mathcal C_t$ have bounded treewidth. Our construction is a variant of so-called \emph{layered wheels}. We also introduce a framework of abstract layered wheels, based on their most salient properties. In particular, we streamline and extend key lemmas previously shown on individual layered wheels. We believe that this should greatly help develop this topic, which appears to be a very strong yet underexploited source of counterexamples.
Abstract: Gödel’s incompleteness theorems state that any recursively axiomatized theory which can interpret sufficient arithmetic is incomplete and cannot prove its own consistency. I will give an overview of their proofs, how they relate to recursion theory, and go over some examples of statements which are undecidable in the theory of Peano arithmetic.
Abstract: A semigroup is a set equipped with an associative binary operation. The definition is simple, but semigroups exhibit rich internal structure — one that is surprisingly different to groups. In this talk I will introduce the notion of a semigroups along with central notions to semigroup theory including Green's relations, key tools to understanding the structure of semigroups which partition elements according to how they generate ideals. This talk will give an overview of the area aimed at building intuition rather than deep technical knowledge.
Abstract: Set theory is often accepted as a foundation of mathematics because of our ability to embed mathematical objects as sets, and prove classical theorems from the axioms of set theory. However, this answer does not examine what ‘job’ we want a foundation of mathematics to do, what it is that makes a theory foundational. We will discuss the response of the philosopher Penelope Maddy to this question, and time permitting also discuss category theory and homotopy type theory through this lens.
Abstract: In order to scandalise the growing number of Italians in this department, I will present an unhinged rant rethinking this well-loved dish. In contrast to more standard recipes, the main ingredient will be a functorial link homology theory in the thickened 3-ball. In fact, my talk will be primarily concerned with unpacking, as much as time allows, the following sentence: “skein lasagna modules are four manifold invariants defined using functoriality of Khovanov-Rosansky homology, which is a categorification of the Jones polynomial”.
Abstract: Coxeter groups are groups which can be generated by involutions and certain simple relations. We will discuss how they relate to geometric reflection groups and we'll go over as many remarkable properties as possible. For example, the word problem is solvable: there is an algorithm to decide if two words in the group represent the same element (in general this is not possible!).
Abstract: Solitons are solutions to field equations arising from classical field theory that are localised in some sense, topological solitons are those which are stabilised by topological features of the target space. They are used to model a wide variety of physical phenomena, and often have a rich geometric and dynamical structure. In this talk we will take a tour through soliton theory, describing some important topological solitons and their key features. Time permitting, we will discuss some connections between the existence of solitons and the geometry of the underlying space.
