Mathematical Institute University of Oxford

Gromov-hyperbolic groups are classically defined geometrically, by the negative curvature of their Cayley graphs. Interestingly, an algorithmic characterization of hyperbolicity is possible in terms of properties of the formal languages of quasigeodesics (geodesics up to bounded error) in their Cayley graphs. Holt and Rees proved, roughly speaking, that these formal languages are regular in the case of hyperbolic groups. More recently Hughes, Nairne, and Spriano established the converse. In this talk, I will discuss progress towards a conjectured strengthening of the result, where we consider context-free quasigeodesic languages. This is based on my summer project, supervised by Joseph MacManus and Davide Sprianoc

Skein lasanga modules are a smooth 4-manifold invariant that was introduced by Morrison, Walker and Wedrich using Khovanov homology. This invariant was recently used by Ren and Willis to give the first analysis free proof of the existence of exotic 4-manifolds. However, even for simple handlebodies it remains difficult to compute. A generalisation was introduced by Chen using Knot Floer homology, which in principle should be easier to compute due to cabling formulas for knot Floer homology. I will give a general introduction to lasagna modules assuming no knowledge of Khovanov or knot Floer homology, and then explain some methods, from upcoming work, for computing Floer Lasagna modules.

We give an introduction to a rigorous renormalization group analysis of the sine-Gordon model with a focus on deriving the lowest-order beta function.

Understanding how living systems dynamically self-organise across spatial and temporal scales is a fundamental problem in biology; from the study of embryo development to regulation of cellular physiology. In this talk, I will discuss how we can use mathematical modelling to uncover the role of microscale physical interactions in cellular self-organisation. I will illustrate this by presenting two seemingly unrelated problems: environmental-driven compartmentalisation of the intracellular space; and self-organisation during collective migration of multicellular communities. Our results reveal hidden connections between these two processes hinting at the general role that chemical regulation of physical interactions plays in controlling self-organisation across scales in living matter

Underlying many biological models are chemical reaction networks (CRNs), and identifying allowed and forbidden dynamics in reaction networks may 
give insight into biological mechanisms. Algebraic approaches have been important in several recent developments. For example, computational 
algebra has helped us characterise all small mass action CRNs admitting certain bifurcations; allowed us to find interesting and surprising 
examples and counterexamples; and suggested a number of conjectures.   Progress generally involves an interaction between analysis and 
computation: on the one hand, theorems which recast apparently difficult questions about dynamics as (relatively tractable) algebraic problems; 
and on the other, computations which provide insight into deeper theoretical questions. I'll present some results, examples, and open 
questions, focussing on two domains of CRN theory: the study of local bifurcations, and the study of multistationarity.