This One-Day Meeting in Combinatorics, which will be held in the School of Mathematics at the University of Bristol on the afternoon of Wednesday 10th June 2026, features four internationally-known speakers in Combinatorics.

Venue: Fry Building, LG02.

1pm: António Girão (UCL): Linear arboricity and related problems.

2pm: Candida Bowtell (Birmingham): Andersen’s rainbow path conjecture.

3pm-3:30pm: Coffee break.

3:30pm: Jessica Enright (Glasgow): TBA.

4:30pm: Shoham Letzter (UCL): An exact minimum codegree threshold for tight Hamilton cycles.

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Abstracts.

António Girão (UCL). Linear arboricity and related problems.

Abstract: In 1980, Akiyama, Exoo, and Harary conjectured that the edges of every graph with maximum degree d can be decomposed into at most ceiling(d+1/2) linear forests, where a linear forest is a collection of vertex-disjoint paths. This conjecture was confirmed asymptotically by Alon in 1988 who showed that (1+o(1))d/2 linear forests suffice. The current best general bound is due to Lang and Postle who showed that there is a decomposition into at most  d/2+ O(d^{1/2} log(d)^4) linear forests. We show that any graph on n vertices can be decomposed into at most \Delta(G)/2+O(log(n)) linear forests. This improves the previous upper bounds for \Delta(G)=\Omega(log(n)^2). Along the way, we show that any d-regular graph on n vertices has a spanning linear forest with at most  2 n/(d+1) paths. This resolves a conjecture of Feige and Fuchs and confirms a well-known conjecture of Magnant and Martin up to a factor of 2. This talk is based on joint work with Micha Christoph, Nemanja Draganić, Eoin Hurley, and Alp Müyesser.

Candida Bowtell (Birmingham). Andersen’s rainbow path conjecture.

Abstract: In 1989, Andersen conjectured that every properly edge-coloured complete graph on n vertices contains a rainbow path covering at least n-1 vertices. That is, a path which uses any colour at most once. We confirm this conjecture for sufficiently large n, moreover, showing that when at least n colours are used, one can find a rainbow Hamilton path. Our proof combines results of Montgomery on rainbow matchings in bipartite graphs with a novel relaxation of a sorting network. This is joint work with Richard Montgomery, Alp Müyesser and Alexey Pokrovskiy.

Jessica Enright (Glasgow). TBA.

Shoham Letzter (UCL). An exact minimum codegree threshold for tight Hamiltonian cycles.

Abstract: A tight Hamilton cycle in a k-uniform hypergraph is a cyclic ordering of the vertices such that every k consecutive vertices form an edge. We solve the following problem (for large n): what is the minimum m such that if G is a k-uniform hypergraph on n vertices whose every k-1 vertices are in at least m edges (i.e. G has minimum codegree at least m) then G has a tight Hamilton cycle? We show that this minimum m is (n–k+2)/2, confirming a conjecture of Katona and Kierstead from 1999. This is joint work with Richard Lang, Arjun Ranganathan, and Nicolás Sanhueza-Matamala.

The Heilbronn Institute for Mathematical Research is delighted to welcome Martin Hairer (Imperial College), who will deliver the next Heilbronn Distinguished Lecture Series (DLS) on 10 – 12 February 2026

Martin Hairer, Professor of Pure Mathematics at EPFL and at Imperial College London, is globally renowned for his breakthrough work at the intersection of analysis and probability. In 2014 he won the Fields Medal for visionary work which introduced a radical new way of constructing solutions for certain nonlinear stochastic partial differential equations which had been intractable before, and which are of great importance in particular to physics.

Please register here  – registration is essential.

For practical information please email the Heilbronn events team at heilbronn-coordinator@bristol.ac.uk

Lecture 1: Tuesday 10th February 2026, 15:30 – 16:30 (Colloquium), followed by a drinks reception

Venue: LG.02, Fry Building

Yang-Mills and the mass gap

We will give an introduction to the probabilistic formulation of the Yang-Mills millenium problem and discuss some of the reasons why it is so hard. We will then discuss some of the progress made in this direction over the past five years or so and some hurdles that still need to be overcome.

Lecture 2: Wednesday 11th February 2026, 16:00 – 17:00 

Venue: G.10, Fry Building

On renormalisation and symmetries

In this lecture, we will delve deeper into the notion of ‘renormalisation’. We will present a typical result from the theory of regularity structures showing how, despite of the presence of renormalisation, one can obtain surprisingly robust continuity statements for singular stochastic PDEs. We will then exploit these to show why theories admitting a “formal” symmetry (like gauge covariance for Yang-Mills) can often be renormalised in such a way that the resulting finite theory still admits that symmetry even though the counterterms arising from renormalisation could potentially break it.

Lecture 3: Thursday 12th February 2026, 14:30 – 15:30

Venue: G.10, Fry Building

Spectral gap for the infinite-volume $\Phi^4_3$ measure at high temperature

We consider the natural (Langevin) dynamic for the $\Phi^4_3$ measure in infinite volume as a toy model for the Yang-Mills dynamic. It is shown that, at high enough temperature, this dynamic is exponentially ergodic and obeys a “one force, one solution” principle. This provides a particularly strong form of uniqueness and stability for the corresponding invariant measure. Some consequences are rotation invariance and a “mass gap” for the $\Phi^4_3$ measure.

A regional network to promote research in operator algebras across the South of the United Kingdom

The next meeting will be held on 21 November at the School of Mathematics, University of Bristol.

We welcome all interested mathematicians. If you would like to attend the workshop, please register [here].

Event webpage: [link]

Hosted by: The Heilbronn Institute, University of Bristol, UK

Jointly funded by the Clay Mathematics Institute and the Heilbronn Institute for Mathematical Research.

Organisers:

Laura Monk (Bristol)

Emma Bailey (Bristol)

Influential conjectures from quantum chaos state that the spectral properties of chaotic systems are universal and given by random matrix theory. Startling heuristics have been provided by mathematical physics, but many are yet to be realised as a formal proof. These past few years have seen significant breakthroughs in the study of random regular graphs where random matrix statistics have been obtained, as well as promising developments in the field of random hyperbolic surfaces. This school will present the rich history of random matrix theory, communicate some of the key ideas behind these exciting advances, and provide an opportunity for new collaborations.

Short course lecturers:

Ofir Gorodetsky (Technion – Israel Institute of Technology)

Michael Magee (Durham)

Theo McKenzie (Stanford)

Guest lecturers:

Igor Wigman (KCL)

Julien Moy (Paris-Saclay)

Bram Petri (Sorbonne)

Maurizia Rossi (Università degli Studi di Milano-Bicocca)

Laura Monk (University of Bristol)

For more information please visit the event website here

Please note that registration for this event is now closed.

A one day meeting on dynamical systems and ergodic theory. There will be connections to number theory via Diophantine approximation. The three speakers will be

Anish Ghosh (Tata institute)   Title of talk: Levy-Khintchine theorems: a brief history and recent progress

Abstract: I will describe some beautiful limiting theorems on continued fractions. I will then introduce a method of interpreting these theorems using dynamical systems on homogeneous spaces. This interpretation allows us to provide a new perspective on this classical question, and to prove new results.

Dmitry Kleinbock (Brandeis) title of talk: Averaging over dilated submanifolds and Diophantine approximation

Abstract: Consider an ergodic R^d-action on a probability space, and take a smooth k-dimensional submanifold M of R^d. When can one prove an ergodic theorem for averages over dilated copies of M? A well-studied special case is that of spherical averages. We prove a quantitative theorem of this kind assuming exponential multiple mixing of the action. This is applied to the diagonal action on the space of unimodular lattices in R^{d+1} to produce a zero-one law for uniform multiplicative Diophantine approximation. In particular we will discuss a uniform version of Littlewood’s conjecture (stronger than the original one) and conclude that its set of exceptions has measure zero. Joint work with Prasuna Bandi and Reynold Fregoli.

Mike Todd (St. Andrews) Title of talk: Almost sure orbits closeness

Abstract:

Given a dynamical system $f$ with initial conditions $x, y$ and a sequence $(r_n)_n$, we define the set $E_n$ as the pairs $x, y$ where there is some pair $1\leq i, j\leq n$ such that the distance between the iterates $f^i(x)$ and $f^j(y)$ is less than $r_n$.  If $(r_n)_n$ shrinks sufficiently slowly, almost every pair $x, y$ will meet this condition for all large enough $n$, i.e., $(\mu\times \mu)(\limsup_n E_n)=1$.  On the other hand, if $(r_n)_n$ shrinks too quickly then the measure of this set is zero.  We are interested in the transition between these behaviours: for simple maps we have a condition on $(r_n)_n$ which gives a dichotomy on the measure of $\limsup_n E_n$ being 0 or 1, depending on the condition being satisfied or not.  For more general systems, we get close to such a dichotomy, depending on the system’s properties.  In this talk I will outline these and related results, joint work with Kirsebom, Kunde and Persson.

The talks will be from 13:30 to 17:30 in LG02 (Fry building) (including breaks),  schedule:

12:00 Lunch

13:30-14:30  Anish Ghosh

15:00-16:00 Mike Todd

16:15-17:15 Dmitry Kleinbock

We are delighted to welcome Nalini Joshi, Payne-Scott Professor of Mathematics & the Chair of Applied Mathematics at the University of Sydney, who will deliver the next Distinguished Lecture Series (DLS) on 3 – 5 March 2025.

Nalini will be hosting three talks during her visit, including a colloquium-style lecture on Monday 3^rd March which will be followed by a drinks reception. Talk titles and abstracts can be found below.

Registration is necessary, so please ensure to complete the registration form to secure your place.

Lecture 1: Monday 3 March 2025 at 16:15 [Colloquium] – followed by drinks reception.

Venue: Seminar Room 2.04, School of Mathematics, Fry Building, Woodland Road, Bristol BS8 1UG

Dynamics on and off elliptic curves (I)

Felix Klein said that the study of new transcendental functions defined by differential equations was “the central problem of the whole of modern mathematics”. The beginnings of this study lay in elliptic functions, which were generalised by the Painlevé transcendents. The analytic theory of their governing differential equations has a counterpart in the theory of the algebraic curves. The most famous examples are elliptic curves. What is not widely known is that the evolution of a solution of a Painlevé equation changes the underlying elliptic curve and points move from one such curve to another under its time evolution.  I will give an introductory overview of how this connects with their asymptotic behaviours and a simple model to describe how Boutroux (1913) initiated such asymptotic descriptions.

Lecture 2: Tuesday 4 March 2025 at 15:00

Venue: Seminar Room 2.04, School of Mathematics, Fry Building, Woodland Road, Bristol BS8 1UG

Dynamics on and off elliptic curves (II)

Starting with the simple model from the last lecture, I will describe behaviours in the limit as the independent variable approaches infinity, which are analogous to those seen in the Painlevé equations. Next, we give an overview of asymptotic results for the first Painlevé equation before describing how we deduced global results from a geometric description of the regularized projective space of initial values. The latter were carried out in collaboration with many co-authors: Duistermaat and Joshi (2011), Howes and Joshi (2014), Joshi and Radnovic (2016-2019), and Heu, Joshi and Radnovic (2023).

Lecture 3: Wednesday 5 March 2025 at 15:00

Venue: Seminar Room 2.04, School of Mathematics, Fry Building, Woodland Road, Bristol BS8 1UG

On q-difference Painlevé equations and their Riemann-Hilbert problems

A widely used method of studying such transcendental functions is through their formulation as Riemann-Hilbert problems, i.e., given functions in certain domains and jumps across their common boundaries, the problem of finding a global function that agrees with the given information. The formulation of a Riemann problem for difference equations was initiated by Birkhoff in 1913. In this talk, I will outline some recent results for q-Riemann-Hilbert problems and their ramifications for special functions that solve q-difference Painlevé equations.

About the Speaker: Nalini Joshi received her PhD from Princeton University with Martin Kruskal as her advisor. Her research focuses on integrable systems,  including the Painlevé equations, lattices and geometric asymptotics. Nalini was elected to the Australian Academy of Science as a Fellow in 2008, awarded a Georgina Sweet Australian Laureate Fellowship in 2012 and appointed Officer of the Order of Australia in 2016. She was President of the Australian Mathematical Society, Vice-President of the International Mathematical Union and is currently a member of its Executive Committee.

In the 2016 Queen’s Birthday honours, Nalini was appointed an Officer of the Order of Australia for distinguished service to mathematical science and tertiary education as an academic, author and researcher, to professional societies, and as a role model and mentor of young mathematicians.

Title: Split Hamiltonian Monte Carlo revisited

Biography

J.M. Sanz-Serna is an emeritus professor at Universidad Carlos III de Madrid. He has contributed to numerical analysis, approximation theory, functional analysis, Monte Carlo methods and other areas. His main interest has been in the numerical analysis of stochastic and deterministic, ordinary or partial differential equations. He served as Universidad of Valladolid Vicechancellor 1998-2006 and as President of the Royal Academy of Sciences of Spain 2018-2024.