We consider coalescing random walks in 1+1 dimensional space-time, with a jump kernel that has finite moments up to order alpha. We view this system as a random set of coalescing paths, with one path starting from each point of space-time. When alpha>3, the diffusive scaling limit is known to be the Brownian web. We study the regime in which alpha is between 2 and 3. In this regime tightness fails (in the sense of continuous paths) due to erratic behaviour near the start times of some of the paths. We show that a surprising transition in behaviour occurs at alpha=9/4; when alpha>9/4 a diffusive scaling limit exists in which paths are essentially Brownian but some paths possess jumps at their initial times, whilst when alpha<9/4 tightness "truly" fails.
Abstract - Dehn’s famous decision problems for finitely presented groups have been studied for over a century by combinatorial and geometric group theorists. In recent years, a variant of one of these classical problems, namely the twisted conjugacy problem, has been studied. The motivation for this problem comes from Bogopolski, Martino and Ventura who, in 2009, proved an equivalence between conjugacy in group extensions and twisted conjugacy.
In this talk I will give a brief survey of this lesser-known decision problem, and discuss some of the latest results in this area. This includes a framework which can be applied to dihedral Artin groups.
Rotwisted calorons are self-dual Yang-Mills connections on R^4 invariant under a glide rotation, and were first introduced in the context of rotating quark-gluon plasmas. Much of the success in the study of self-dual Yang-Mills has been via the existence of a nonlinear transform called the ADHMN (Atiyah-Drinfeld-Hitchin-Manin-Nahm) construction. In this talk, after reviewing the many facets of ADHMN constructions, we shall discuss the formulation of a Nahm transform for rotwisted calorons, identifying them with solutions of an integrable delayed-differential equation. We will also describe some solutions in the simplest non-trivial case. This is based on joint work with Derek Harland.
Two days. Two industry-backed challenges. One chance to make a real difference.
If you can code, analyse data, and think critically, this is your arena!
Free Event
Work in Teams
Registration Necessary (deadline: 16 March)
Refreshments Provided
Organised by the DS & AI Educators group, the AI Hub, and the Leeds Institute for Data Analytics (LIDA)
Choose your challenge track
TRACK 01 — Healthcare & Digital Pharmacy
Industry partner: Pharmacy2U
Work with real digital health data
Apply ML & data analysis to live pharmacy challenges
TRACK 02 — Earth, Environment & Climate
Research partner: Earth & Environment – Dr Jim McQuaid
Analyse carbon footprint datasets
Build models for evidence-based climate action
Open to
MSc level & PhD students at the University of Leeds and University of Leeds staff with relevant analytical skills. All disciplines welcome, domain expertise is a bonus: not required!
What we're looking for
Python, R, or equivalent
Data wrangling with real datasets
ML, statistics, or visualisation
Clear problem framing & communication
How you'll be judged
Quality of technical approach
Soundness of evaluation methodology
Clarity of presentation to industry panel
Real, well-reasoned output — not perfect code
Recognition
Winners will receive formal recognition from the University and industry partners
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.
We propose a joint modeling framework that integrates zero-inflated longitudinal count data with time-to-event outcomes, explicitly accounting for a cure fraction. The longitudinal process is modeled using flexible mixed-effects Hurdle models to handle excess zeros and overdispersion, while the survival component combines a Cox model with a mixture cure formulation to distinguish susceptible and cured individuals. The two processes are linked through current longitudinal information, enabling dynamic risk prediction. Inference is performed using Hamiltonian Monte Carlo for robust estimation. We validate the approach through simulations and apply it to an HIV cohort, demonstrating its value for personalized risk assessment and clinical decision-making.
We study a family of balls-in-bins models with a power-law feedback and a local
interaction determined by an underlying graph on the bins. Specifically, for a fixed
graph on $d$ bins, and fixed positive real numbers $\beta_1, \dots, \beta_d$, at each time
step the model allocates a new ball to bin $i$ with probability proportional to
$U_i^{\beta_i}$, where $U_i$ is the total number of balls currently allocated to all bins
in the graph neighbourhood of bin $i$ (including bin $i$ itself).
In this talk, we focus attention on the case of a path graph on 3 bins, studying the
asymptotic behaviour of $X_n$, the vector of the number of balls allocated to each bin
after $n$ steps. Despite its apparent simplicity, the model exhibits a variety of
behaviours, depending on the parameters $\beta_1, \beta_2, \beta_3$. We analyse both the
symmetric ($\beta_i$ equal) and asymmetric ($\beta_i$ distinct) cases, presenting a
complete classification of the growth rates of the coordinates of $X_n$ when $\beta_i > 1$
for all $i$. In each case, we identify when the asymptotic behaviour is
deterministic, and when it is random.
Our analysis employs the method of stochastic approximation for the symmetric case,
semimartingale methods for the asymmetric case, and liberal use of L\'evy's extension of
the Borel--Cantelli lemma. This is joint work with Mikhail Menshikov and Vadim
Shcherbakov.
As cancer advances, cells often spread from the primary tumor to other parts of the body and form metastases. I'll present a conceptually elegant model of metastasis formation where each primary cell can initiate metastatic lesions which lesions then evolve as independent branching processes. We assume that the primary tumor is resected upon detection. Of fundamental importance is whether synchronous (detectable) or metachronous (undetectable) mets are present at this detection time, the distribution of their numbers and sizes. If there are only metachronous mets at detection, how long until these mets become detectable, leading to the relapse of the disease? We'll extend this model to cancers in which cells first need to evolve the ability to metastasize. Using sequence data from primary and corresponding met samples we'll propose that these intermediate cells are indeed present for certain primary-met pairs, why not there for others.
