Pure PGR

Abstract: TBA.

Abstract: TBA.

Abstract: TBA.

Abstract: Topological solitons are smooth, localized, finite-energy solutions of nonlinear PDEs arising in field theory. They are topologically stable in the sense that they cannot be continuously deformed into a constant solution without requiring infinite energy, and are characterized by an integer called the topological degree. The simplest examples of topological solitons are kinks in one spatial dimension. I will discuss their construction and properties, including the Bogomolny argument, which gives a lower bound on the energy in terms of the topological degree. For certain special equations, it is possible to generate explicit multi-soliton solutions systematically. In these rare situations, the system is called integrable. As an example of this, I will discuss sine-Gordon kinks.

Abstract: The Schröder problems are a set of 4 combinatorial questions relating to counting different kinds of bracketings. In this talk, we introduce the basic idea of combinatorial species, demonstrate a common species approach which can solve the four of these problems, and look at some of the objects produced by extending this approach.

Abstract: Potts models are relatively easily described models defined on a lattice, which are of interest as they exhibit a physical phenomenon known as a phase transition. For example, on a square lattice we might consider "configurations" where we assign each point a value from {0,1}. Each configuration has an associated "Hamiltonian" and from these we may construct the "partition function", which may be analysed to derive results of physical relevance - one such analysis is looking at the complex zeros of a polynomial derived from the partition function, producing rather striking plots.

We'll discuss all of the details (no knowledge assumed), and go through some interesting complications that arise. Time permitting, we'll demonstrate some code in Julia, muse about LLM tool usage including Codex, and explore connections to knot invariants.
 

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.