5 – 6 September 2024

The Heilbronn Institute, School of Mathematics, University of Bristol, UK

Venue: G.10, Fry Building, School of Mathematics, Woodland Road, Bristol BS8 1UG

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. We have been fortunate to attract excellent speakers to our Annual Conferences since the Institute’s inception in 2005. You can find a list of past speakers in this link. This year we welcome a number of distinguished speakers, to deliver lectures intended to be accessible to a general audience of mathematicians.

Invited Speakers:

Tim Austin (Warwick) | Chair: Ben Krause (Bristol)

Notions of Entropy in Ergodic Theory and Representation Theory

Entropy has its origins in thermodynamics and statistical mechanics. It gained mathematical rigour in Shannon’s work on the foundations of information theory, and quickly found striking applications to ergodic theory in work of Kolmogorov and Sinai. Many variants and other applications have appeared in pure mathematics since, connecting probability, combinatorics, dynamics and other areas. I will survey a few recent developments in this story, with an emphasis on some of the basic ideas that they have in common. I will focus largely on (i) Lewis Bowen’s “sofic entropy”, which helps us to study the dynamics of “large” groups such as free groups, and (ii) a cousin of sofic entropy in the world of unitary representations, which leads to new connections with random matrices.

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Tara Brendle (Glasgow) | Chair: Luke Jeffreys (Bristol)

Symmetries of Surfaces

The mapping class group of a surface is its group of symmetries. In the 1930s, Dehn demonstrated that this inherently topological object could also be understood from a purely algebraic viewpoint. Much later, in the 1980s, N. Ivanov proved that the mapping class group could also be encoded via a combinatorial object known as the curve graph. In this talk we will give an introduction to mapping class groups, and explain Ivanov’s “metaconjecture” describing combinatorial models for mapping class groups. We will also describe work with Margalit that sheds light on Ivanov’s metaconjecture, by emphasizing the connections between these three viewpoints (topological, algebraic, and combinatorial).

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Christophe Breuil (C.N.R.S. & Paris-Saclay) | Chair: Miriam Norris (Manchester)

On the Mod p Langlands Programme for GL2

Let p be a prime number and F a finite extension of the field of p-adic numbers. I will give a survey of recent results on modular representations of GL2(F) in characteristic p (same p as for F!) and of their relations with 2-dimensional representations of the absolute Galois group of F in characteristic p. These recent results are joint work with F. Herzig, Y. Hu, S. Morra and B. Schraen. The talk is not intended for specialists only.

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Chaim Goodman-Strauss (Arkansas) | Chair: Edward Crane (Bristol)

How the World Fits Together

Dave Smith’s recent discovery of the “Hat” monotile reminds us that small gadgets, fitting together by local interactions can give rise to fantastic complexity, even in recreational mathematics. Across the sciences and mathematics, as fragments of formal logic, symbols in a computer language, or proteins, all producing irreducible complexity when combined by simple proscribed rules. We’ll discuss the source of this in combinatorial tiling puzzles.

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Dipendra Prasad (Indian Institute of Technology Bombay) | Chair: Fred Diamond (KCL)

Branching Laws for Representations of Groups

Representation theory of groups are studied in a variety of contexts. In most cases, restriction of representations to well chosen subgroups plays a very important role. We discuss some of the very studied special cases, ending with some questions of much current interest for padic groups.

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Richard Samworth (Cambridge) | Chair: Nick Whiteley (Bristol)

Nonparametric Inference Under Shape Constraints: Past, Present and Future

Traditionally, we think of statistical methods as being divided into parametric approaches, which can be restrictive, but where estimation is typically straightforward (e.g. using maximum likelihood), and nonparametric methods, which are more flexible but often require careful choices of tuning parameters. The area of nonparametric inference under shape constraints sits somewhere in the middle, seeking in some ways the best of both worlds. I will give an introduction to this active area, providing some history, recent developments and a future outlook.

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Barbara M. Terhal (TU Delft) | Chair: Niko Breuckmann (Bristol)

Classical, Quantum and Fermionic Satisfiability Problems

We will review how classical satisfiability problems can be generalized to quantum satisfiability problems. We will discuss what is known about the computational complexity of these classical and quantum problems. We also introduce a new generalization, fermionic satisfiability, and mention some first results on (particle number conserving) fermionic 2-SAT.

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Josephine Yu (Georgia Institute of Technology) | Chair: James Maxwell (Bristol)

Tropical Geometry, Positivity, and Applications

Tropicalization is a degeneration process that transforms algebraic geometric objects into polyhedral combinatorial objects. Tropical geometry is used to study the limiting behavior of (semi)algebraic sets near zero and infinity. It can reveal hidden combinatorial structures and aid with computations. In this talk I will introduce tropical geometry, explore its applications, and discuss recent results on tropicalization of principal minors of positive definite matrices and resulting matroidal structures.

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

Information on past Heilbronn Annual Conference is available here

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