Abstract: A D-set is a relational structure on the leaves of a tree, consisting of a quaternary relation defined such that we have D(x,y,z,w) if and only if the path from x to y is disjoint from the path from z to w. In this talk I will introduce several model-theoretic notions, namely ultrahomogeneity, indiscernible sequences, dp-minimality and distality, and I will discuss how they arise during the study of D-sets and what they tell us about structures in general.
It has long been known that some important spin systems on lattices, such as the spin-O(N) model, have representations in terms of discrete structures (loops, paths etc) on the lattice. This has been a fruitful connection, however many discrete models on lattices, such as percolation, the dimer model, spatial random permutations, and others, are also of considerable interest. In these cases there was no precise link to spin systems, either because the motivation came from elsewhere, or because these are toy models that have taken on a life of their own. In this talk I will present a spin system where spins take continuous complex values. By choosing the measures appropriately, virtually any discrete system of interest can be obtained as a special case. These spin systems then lend themselves naturally to several of the well developed tools of classical spin systems. This is joint work with Volker Betz.
Joint work with Daniel Baldwin (Leeds) and Alan McKane (Manchester emeritus)
Traditionally, stochastic processes are modelled using either a Fokker-Planck PDE approach, or a Langevin SDE approach. There is also a third way: the functional or path integral. Originally developed by Wiener in the 1920s to model Brownian motion, path integrals were famously applied to quantum mechanics by Feynman in the 1950s. However, they also offer much to classical stochastic processes. In this talk I will introduce the formalism, focussing on the one-dimensional case when the noise is weak. There exists a remarkable correspondence between the most-probable stochastic paths and Hamiltonian mechanics in an effective potential [1,2,3]. It turns out that in some cases, the paths that dominate the integral, and hence quantities like the potential barrier crossing (Kramers) rate, depart from the real line. This is in some sense analogous to the way the residues at complex poles control ordinary integrals along the real line.
[1] Ge, Hao, and Hong Qian. Int. J. Mod. Phys. B 26.24 1230012 (2012)
[2] SPF et al. J. Chem. Phys. 158.12 (2023)
[3] Honour, Tom and SPF. J. Phys. A 57 175002 (2024)
The goal of the talk is to give an overview of the metric theory of currents by Ambrosio-Kirchheim, together with some recent progress. Metric currents are a generalization to the metric setting of classical currents. Classical currents are the natural generalization of oriented submanifolds, as distributions play the same role for functions. We present a structure result for metric 1-currents as superposition of 1-rectifiable sets in complete and separable metric spaces, which generalizes a previous result by Schioppa. This is based on an approximation result of metric 1-currents with normal 1-currents and a more refined analysis in the Banach space setting. This is joint work with D. Bate, J. Takáč, P. Valentine, and P. Wald (Warwick).
The game of cops and robbers is played on a fixed graph, with the cop choosing a vertex to start at, then the robber chooses his, and then they take turns in moving to adjacent vertices. The game ends if the cop captures the robber (lands on its vertex). What graphs allow the cop to have a winning strategy, and how long does the game typically last, assuming optimal play? For finite graphs, the situation is very well understood — the cop-win graphs are precisely constructible graphs (constructed from a single vertex by repeatedly adding dominated vertices), and the capture time can be any finite ordinal (attained for example by finite paths).
In the infinite case, not much is known. In particular, there is no structural characterisation of cop-win graphs. What about the capture time? Is there an ordinal such that for any cop-win graph the sequence of moves of an optimal game is never that ordinal?
In this talk we will explore this question by showing that the answer is surprisingly 'no'.
In this presentation, I will draw on reflections from a collaborative project with students at Monterrey University, developed as part of a Collaborative Online International Learning (COIL) course. This experience provided a valuable opportunity to explore the impact of varied teaching styles and assessment methods in supporting student engagement across different educational contexts. Through this collaboration, I examined how flexibility in teaching, incorporating discussion-based sessions, guided problem-solving, and reflective tasks, can help students transition towards greater independence and confidence in their learning. I will discuss how the use of varied assessments, including formative and assessment approaches, encourages deeper understanding, promotes self-regulation, and helps students connect mathematical ideas. The session will share insights from both staff and student perspectives, highlighting how intentional diversity in teaching and assessment design can improve motivation, and meaningful engagement in mathematics education.
Abstract: Therapist-led trials are central to evaluating interventions in psychotherapy, physiotherapy, and rehabilitation. These are typically designed as individually randomised, parallel-group trials, where patients are assigned to interventions at enrollment. The primary aim is to estimate the effect of the intervention, defined as a specific therapy or procedure of interest.
A persistent methodological challenge in such trials is the confounding of therapist effects with intervention effects. One proposed solution is to randomise patients to therapists, but this introduces significant practical constraints, including therapist availability, capacity, and turnover. Neglecting these constraints during trial design can lead to inflated costs and biased estimates of intervention effects. In this talk, we shall explore key logistical challenges in therapist randomisation and present strategies to address them at both the design and analysis stages. In particular, different randomisation methods will be examined, along with their implications for trial structure, statistical inference, and operational feasibility.
