Ribbon Hopf algebras and their representation theory have long played an important role in low-dimensional topology, chiefly in constructing 3-manifold invariants with the additional assumption of modularity. Recently, there has been interest in similar applications in the context of 4-manifolds, where crucially the categories are no longer assumed to be modular, but rather explicitly non-factorizable. In this talk I will present several new families of ribbon Hopf algebras designed for application in the construction of invariants of 4-dimensional 2-handlebodies up to 2-deformations due to Beliakova and De Renzi, as well as 4-dimensional TQFTs introduced by Constantino-Geer-Haioun-Patureau. I will explain the key properties of the resulting Hopf algebras and their representation categories, in particular that the latter have non-semisimple symmetric centres, and attempt to interpret the resulting 4-manifold invariants. Based on https://arxiv.org/abs/2503.19532 (joint with Q. Faes) and https://arxiv.org/abs/2510.24263.

Mirror pairs XY are related by a geometric duality. Does this Geometry (G) produce Enumerative Mirror Symmetry (EMS), namely the prediction that the genus 0 Gromov-Witten invariants of X are computed by period integrals on Y? I will give a positive answer in a few examples, where we take XY to be intrinsic mirror pairs (Gross--Siebert). Based on joint work…

Operads are combinatorial gadgets encoding categories of algebras. In modern homotopy theory, where categories are replaced by infinity-categories, strict operads are replaced by their homotopy-coherent version, infinity-operads. A natural way infinity-operads arise is through the process of localization, the process of freely inverting a class of morphisms in a homotopy-coherent way.

After recalling the above notions, in this talk I will prove that all infinity-operads arise as a localization of a strict one, and moreover that the localization functor establishes an equivalence of homotopy theories. These results generalize analogue ones for infinity-categories proven by Joyal and Stevenson, resp. Barwick and Kan. We will see that essential to prove such results is the dendroidal formalism for infinity-operads, based on a certain category of trees and its combinatorics. If time permits, I will also highlight some open questions. Part of this work is joint with K. Arakawa and V. Carmona.

Categories of representations of groups, or more generally, Hopf algebras and quantum groups, are endowed with tensor structures. The Grothendieck ring of such a category has a basis consisting of classes of simples objects, with structure contants defined as multiplicities of simple representations in the decomposition of tensor products. The resulting rings together with their bases can have a…

Abstract: Biological systems can undergo sudden transitions when parameters change, making stability analysis of steady states essential. We develop an algebraic framework for studying the parameter spaces of ecological ODE models with polynomial or rational rates. Using tools from computational algebra, we identify boundaries separating stability regimes, and using routing functions, we compute the regions in their complement together with the stability behavior in each region. We demonstrate the method on the classical Levins-Culver competition-colonization model and a modern tripartite symbiosis model, where it uncovers rich and previously hidden stability structure.

Abstract: TBA