Heilbronn Annual Conference 2022

8 – 9 September 2022

The Heilbronn Institute, School of Mathematics, University of Bristol, UK
Organised in collaboration with the School of Mathematics, University of Bristol

The Heilbronn Annual Conference is the institute’s 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, quantum information. It brings together members of the Institute, distinguished visiting speakers, and other members of the UK mathematical community. This year we welcome eight distinguished speakers, to deliver lectures intended to be accessible to a general audience of mathematicians.

Invited Speakers:

Viviane Baladi (LPSM, Paris): Statistical Properties of Dispersive Billiards

Dispersive billiards (or the periodic Lorentz gas) are natural dynamical systems which have been challenging mathematicians for half a century. In the past decade, a new mathematical tool to study them has emerged: Ruelle transfer operators acting on scales of anisotropic Banach spaces. I will survey results obtained in the past decade on statistical properties of two-dimensional dispersive billiards (Sinai billiards), both in the discrete and continuous time setting, for various equilibrium states, including the physical measure and the measure of maximal entropy. Since I will not consider the setting of open billiards, the singularities of the map or flow need to be addressed, and this is a major technical difficulty.

Jennifer Balakrishnan (Boston): Mordell’s Conjecture and the Last 100 Years

Let C be a smooth projective curve with genus at least 2 defined over the rational numbers. It was conjectured by Mordell in 1922 and proved by Faltings in 1983 that C has finitely many rational points. However, Faltings’ proof does not give an algorithm for finding these points, and in practice, given such a curve, provably finding its set of rational points can be quite difficult. I’ll give a survey of what’s happened in the 100 years since Mordell’s conjecture, including computational approaches.

Martin Bridson (Oxford): Aspects of Rigidity for Automorphism Groups of Free Groups

I will begin by reviewing the powerful 3-way analogy between mapping class groups of surfaces, automorphism groups of free groups, and arithmetic lattices such as SL(n,Z). I shall then discuss how geometric rigidity results familiar from the last setting can be transported into the other settings, where techniques of low-dimensional geometry and topology prevail. One such result provides an analogue of the fundamental theorem of projective geometry, with projective subspaces replaced by free factors of a free group. This result, which comes from joint work with Mladen Bestvina, allowed Ric Wade and me to prove that all isomorphisms between finite-index subgroups of Aut(F_n) extend to conjugations of the ambient group. This “commensurator rigidity” is antithetical to arithmetic behaviour, as I shall explain.

Toby Cubitt (UCL): A Mathematical Theory of Hamiltonian Simulation and Duality

“Analogue” Hamiltonian simulation involves engineering a Hamiltonian of interest in the laboratory and studying its properties experimentally. Large-scale Hamiltonian simulation experiments have been carried out in optical lattices, ion traps and other systems for two decades. This is often touted as the most promising near-term application of quantum computing technology, as it is argued it does not require a scalable, fault-tolerant quantum computer. Despite this, the theoretical basis for Hamiltonian simulation is surprisingly sparse. Even a precise definition of what it means to simulate a Hamiltonian was lacking. In my talk, I will explain how we put analogue Hamiltonian simulation on a rigorous theoretical footing, by drawing on techniques from Hamiltonian complexity theory in computer science, and Jordan and C* algebras in mathematics. I will then explain how this proved to be far more fruitful than a mere mathematical tidying-up exercise, leading to the discovery of universal quantum Hamiltonians [Science, 351:6 278, p.1180 (2016); Proc. Natl. Acad. Sci. 115:38 p.9497, (2018)], later shown to have a deep connection back to quantum complexity theory [PRX Quantum 3:010308 (2022)]. It has even found applications in quantum gravity, leading to the first toy models of AdS/CFT duality that encompass energy scales, dynamics, and (toy models of) black hole formation [J. High Energy Phys. 2019:17 (2019); J. High Energy Phys. 2022:52 (2022)].

Nicolas Curien (Paris-Sud Orsay): On the Cheeger Constant of Hyperbolic Surfaces

It is a well-known result due to Bollobas that the maximal Cheeger constant of large d-regular graphs cannot be close to the Cheeger constant of the d-regular tree. We shall prove analogously that the Cheeger constant of closed hyperbolic surfaces of large genus is bounded from above by 2/π ≈ 0, 63…. which is strictly less than the Cheeger constant of the hyperbolic plane. The proof uses a random construction based on a Poisson-Voronoi tessellation of the surface with a vanishing intensity and makes an interesting object appear: the pointless Poisson–Voronoi tessellation of the hyperbolic plane.

Joint work with Thomas Budzinski and Bram Petri

Laure Saint-Raymond (ENS-Lyon): Dynamical Correlations in a Hard Sphere Gas

We consider the hard sphere gas at low density with random initial data away from equilibrium, and study the dynamical correlations. We develop an alternative strategy to the BBGKY hierarchy, focusing on the clustering process of physical trajectories. This method is used to derive a fluctuation theory around the Boltzmann equation.

Laura Schaposnik (Illinois, Chicago): Mirror Symmetry for Higgs Bundles, Generalized Hyperpolygons and More

In this talk we will introduce Higgs bundles and generalized hyperpolygons, and look into different directions that have attracted attention within the area in the last decade. In particular, we shall see how Mirror Symmetry can be seen in terms of branes within the Hitchin fibration and show that, under certain assumptions on flag types for generalized hyperpolygons, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system. Time permitting, we will also see how I have been using my geometric background to answer questions in other areas of science.

Much of the talk follows recent work joint with Steven Rayan

Benny Sudakov (ETH, Zurich): Emergence of Regularity in Large Graphs

“Every large system, chaotic as it may be, contains a well-organized subsystem”. This phenomenon is truly ubiquitous and manifests itself in different mathematical areas. One of the central problems in extremal combinatorics, which was extensively studied in the last hundred years, is to estimate how large a graph/hypergraph needs to be to guarantee the emergence of such well-organized substructures. In the first part of this talk we will give an introduction to this topic, mentioning some classical results as well as a few applications to other areas of mathematics. Then we discuss the recent solution (with Oliver Janzer) of the following fundamental problem, posed by Erdos and Sauer about 50 years ago: “How many edges on n vertices force the existence of an r-regular subgraph (r>2)?” Our proof uses algebraic and probabilistic tools, building on earlier works by Alon, Friedland, Kalai, Pyber, Rödl and Szemerédi.

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