The Heilbronn Annual Conference is the Heilbronn Institute for Mathematical Research flagship event. It takes place over two days, and it covers a broad range of mathematics, including algebra, combinatorics, data science, geometry, number theory, probability, and quantum information. It brings together members of the Institute, distinguished visiting speakers, and other members of the UK mathematical community. We have been fortunate to attract excellent speakers to our Annual Conference since the Institute’s inception in 2005. This year we welcome eight distinguished speakers, to deliver lectures intended to be accessible to a general audience of mathematicians.

Confirmed speakers

Sougato Bose, UCL
Jens Marklof, University of Bristol
Kathryn Hess Bellwald, EPFL
Karin Baur, Ruhr-University Bochum
Timothy Trudgian, UNSW

Registration is expected to open March 2027. Please check back at a later date.

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

Join the Heilbronn Event mailing list to keep up to date with our upcoming events.

Location: The International Centre for Mathematical Sciences, Edinburgh, UK

Jointly funded by: the Clay Mathematics Institute

and the Heilbronn Institute for Mathematical Research

Organisers

Alex Bartel (University of Glasgow)
Rachel Newton (King’s College London)

Topic description

Questions about rational points on varieties are among the oldest in mathematics, but also are at the centre of vibrant areas of modern research. Two of the remaining six Clay 1 Million Dollar problems concern points on varieties. In recent years, seemingly entirely unrelated problems, such as counting number fields, have been reformulated as problems about points on geometric objects. Moreover, several widely used cryptographic schemes are based on arithmetic properties of varieties.

The summer school will offer research students a rapid entry into the area by exposing them to some of the major fruitful research directions in arithmetic geometry. A mixture of lectures and exercise sessions will ensure that students develop intuition for and familiarity with the technical tools that they will be learning. In addition, there will be several guest lectures on topics of current interest that are not covered by the courses.

The summer school will consist of the following four courses, with one lecturer each:

Course 1: Brauer–Manin and/or other obstructions (Damián Gvirtz, University of Glasgow)
Course 2: The modular method (Samir Siksek, University of Warwick)
Course 3: Galois image on torsion, Mazur’s Programme B (Andrew Sutherland, MIT)
Course 4: Hyperelliptic curves and cluster pictures (Céline Maistret, University of Bristol)

Guest speakers

Tim Browning (IST Austria)
Jennifer Balakrishnan (Boston University)

How to apply

Registration is expected to open in November 2026. Please check back at a later date for details on how to apply.

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): Solving problems on (parameterised) temporal graphs.

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, Lukas Michel 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). Solving problems on (parameterised) temporal graphs.

Abstract: Temporal graphs are graphs with edges that are only active at some specified timesteps and are useful for modelling systems where contact between agents changes over time.  As for classic graphs, many of the combinatorial optimisation problems we might want to solve are intractable in general, and parameterised algorithms are one approach that will let us attack these problems when there is some useful restriction on the temporal graph.  I will discuss several of the parameters we have found to be algorithmically useful and outline some general approaches to using them for e.g. firefighting, dominating set, temporal Hamiltonian path.

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