Aberdeen Math PGR Seminar 2025

04/02/2025 Baylee Schutte

• Title: Real and complex line fields on manifolds.

• Abstract: The projective span of a smooth manifold is the maximal number of linearly independent line fields. In this talk, I will explore how algebraic invariants can govern the geometric behavior of certain real and almost-complex manifolds. To begin, I will define the numerical invariant projective span and motivate its study. Next, I will explain the calculation of the projective span of all of the Wall manifolds (which are real manifolds of longstanding interest in differential topology) carried out in joint work with Mark Grant [Bol. Soc. Mat. Mex. 30(3):75]. Finally, from recent work joint with Nikola Sadovek [arXiv:2411.14161], I will identify complete obstructions to the existence of 1, 2, or 3 linearly independent complex line fields on certain classes of almost-complex manifolds and then explain how these results interact with complex geometry. 

18/02/2025 Ludovico Dziecielski

• Title: An infinity categorical approach to Thom spectra.

• Abstract: The study of spectra is an important part of modern algebraic topology. In some ways, spectra form a category close to topological spaces but with better proprieties. In particular, with spectra, the connection between algebra and topology becomes even stronger and many purely algebraic constructions have been generalized to spectra during the years. Between these constructions, one of great importance is Thom spectra; which can be informally described as a "stabilization" of principal G-bundles of topological spaces. In my talk, following the work of  M. Ando et. al., I will use the machinery of infinity categories to give a simple definition of Thom spectra and present their main proprieties. 

04/03/2025 Gabriel Corrigan

• Title: Outer spaces and finiteness properties of automorphism groups of right-angled Artin groups

• Abstract: The rich class of right-angled Artin groups (RAAGs) is well-liked in geometric group theory because it provides a natural interpolation between the classes of free and free abelian groups. The study of automorphisms of free groups was invigorated by 1986 by in the invention of 'Outer space' - a certain nice complex upon which the outer automorphism group of the free group acts. This construction has recently been generalised to all RAAGs. I will present work which uses this construction to study various groups as automorphisms of RAAGs. In one direction, I show that the gap between an algebraic notion of dimension of the 'untwisted' subgroup and the dimension of the corresponding Outer space can be (surprisingly) arbitrarily large. In another direction, I introduce the 'symmetric automorphism group' and build and Outer space for it; along the way, we prove that many subgroups of the outer automorphism group of a RAAG satisfy a very strong finiteness condition. This will all be preceded by a gentle introduction to RAAGs, and I also aim to give you some idea of what Outer space looks like. I will assume no prerequisites beyond some undergraduate group theory and topology. 

18/03/2025 Matthew Di Meglio

• Title: Categorical Hilbert Theory

• Abstract: During my PhD, I unravelled a deep connection between limits in analysis and directed limits in category theory, enabling the first characterisation of a category of finite-dimensional Hilbert spaces. I also developed abstract categorical frames - R*-categories and M* categories - that capture algebraic and analytic aspects common to Hilbert spaces and other similar mathematical structures like Hilbert W*-modules and unitary representations, enabling a unified high-level account of their theory. In this talk, I will give an accessible overview of these aspects of my PhD research. 

15/04/2025 Lucy Spouncer

• Title: Cone-structing Deformations: a Categorical Approach

• Abstract: In this talk, I will present a categorical approach to deformation of a closed immersion Spec A → Spec B to the normal cone, an algebro-geometric analogue of tubular neighbourhoods in differential topology. I'll begin by reviewing the geometric intuition behind this deformation and then reframe it in terms of an elegant adjunction between filtered and graded objects in derived categories. 

We will focus on accessible examples and an illustrative result of Brantner and Mathew, and time permitting, sketch how this categorical setup naturally extends to higher algebraic structures. This includes a brief overview of our ongoing work applying these techniques to operads, specifically En, Pn and BDn​​, which promises to provide new insights into deformation quantization and shifted Poisson structures in the spirit of Beilinson-Drinfeld, Costello, and Livernet-Loday's foundational work.

06/05/2025 Malthe Sporring

• Title: Twofold monoidal structures

• Abstract: A twofold symmetric monoidal category is a category with two symmetric monoidal products, one of which laxly distributes over the other. Just as ordinary symmetric monoidal structures are controlled by the category of finite sets, twofold structures are controlled by the category of reflexive cographs, a certain subcategory of the category of graphs. This result is due to forthcoming work by Barwick. In this talk, I will give an overview of twofold structures and explain why you might care. Although the focus will be on infinity-categories, no prior knowledge about infinity-categories is expected.

13/05/2025 Matthew Turner

• Title: Harnessing the power of AI for biodiversity forecasting

• Abstract: Species traits are the morphological, demographic, physiological, phenological, and genetic features of a given species, and are used in studies such as biodiversity forecasting, parameterising process-based models, and investigating species evolutionary histories, as well as in conservation projects. These data often display high degrees of missingness, and as such effective data handling techniques are essential to maximise data availability. One such approach is data imputation, in which a most probable value is determined for missing data. In my talk, I will be presenting my work to date on developing deep learning models for species trait data imputation, as well as my current plans for the utilisation of multimodal foundational models for improving model performance.