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.

Speakers:

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Steve Brierley (Riverlane)

Quantum Error Correction, Theory -> Technology

In this talk, I’ll give an introduction to quantum error correction and why it’s critical to scaling quantum computers. Then, I’ll talk about what it takes to turn this idea into a practical technology and embed it in the emerging quantum computing industry.

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François Charles (ENS, Paris-Saclay)

Gaussian Measures, Geometry of Numbers and Diophantine Geometry

Euclidean lattices are very basic objects that appear in many areas of mathematics, physics and computer science. I will focus my talk on a very basic tool in the study of lattices: the sum of the Gaussian measures supported at the point of the lattice. This measure interpolates between the discrete geometry of the lattice and its large-scale properties.

The first part of my talk will focus on various applications of Gaussian measures on lattices. I will first survey some applications to geometry of numbers and questions coming from quantum cryptography, and then try to explain how Gaussian masses of lattices look tantalizingly close to dimensions of some cohomology groups.

In the second part of my talk, I will describe joint work with Jean-Benoît Bost explaining how the study of Gaussian measures on lattices makes it possible to study Euclidean lattices in infinite dimension. I will then explain how infinite-dimensional lattices appear naturally in diophantine geometry through the study of spaces of polynomials or power series with integral coefficients. I will give applications of this line of thoughts to several questions diophantine geometry.

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Alison Etheridge (Oxford)

The Forwards and Backwards of Population Models

What can we infer about the history of a population from the patterns of genetic variation that we observe today? There is a long history of mathematical modelling of the demographic dynamics of a population and their effect on the genetic relationships between individuals sampled from that population. Because patterns of genetic variation are laid down over very long timescales, one expects that local details of reproduction and dispersal will not be important, and can be replaced by some ‘average’ behaviour. Very often modellers simply write down a reaction-diffusion equation, without ever considering what is happening at the individual level. Here we lay out a broad class of individual based models that might describe how spatially heterogeneous populations live, die and reproduce. This class is particularly well suited to modelling plant populations. In particular, a novelty of our approach is that we explicitly model a juvenile phase, which, as we illustrate with a toy example, could have important implications for quantities that we might try to infer from genetic data.

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Jessica Fintzen (Bonn)

An Introduction to Representations of p-adic Groups

An explicit understanding of the (category of all smooth, complex) representations of p-adic groups provides an important tool not just within representation theory. It also has applications to number theory and other areas, and in particular enables progress on various different forms of the Langlands program.

In this talk, I will introduce p-adic groups and provide an overview of what we know about the representations of these groups including new developments of the last six months. We will focus on two crucial aspects:

First, I will survey what we know about the construction of all so called supercuspidal representations, which are the building blocks for all representations. For more than 20 years it remained open to extend a general construction to the case p=2, and I will sketch what makes this case so special and how we could overcome the obstacles in my recent joint work with David Schwein.

Second, we will study the structure of the whole category of representations of p-adic groups in terms of these supercuspidal representations, and, if time allows, I will explain how two recent preprints with Jeffrey Adler, Manish Mishra and Kazuma Ohara allow us to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or are at least easier to achieve.

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Svetlana Jitomirskaya (Berkeley)

Fractals, Quasiperiodicity, and the Robust Ten Martini Problem

What do fractals, magnetic fields, and irrational numbers have in common? Surprisingly, they come together in one of the most iconic images in mathematical physics: the Hofstadter butterfly—a visual representation of the energy spectrum of an electron on a two-dimensional lattice in a magnetic field. While this structure emerged from simple physical intuition, it conceals a wealth of deep mathematics: spectral theory, dynamical systems, number theory, and a challenge famously called the Ten Martini Problem.

In this talk, I’ll take you on a guided tour through this landscape—starting with the physical origins, passing through the spectral phenomena they inspired, and arriving at recent mathematical breakthroughs. We’ll explore how a simple-looking operator with a quasiperiodic potential can produce a spectrum with an amazing structure.

Along the way, I’ll describe a new approach that makes it possible to prove the Cantor structure of the spectrum in wide generality, well beyond classical models. This method draws on recent developments in dynamical systems, including dual Lyapunov exponents and partially hyperbolic cocycles.

Though rooted in mathematical physics, the story connects naturally to broad themes across mathematics: how deterministic systems give rise to complex, fractal behavior; how number-theoretic properties of irrational rotations shape analytic spectra; and how ideas from one field can unlock structures in another. My aim is to share not only some results, but the sense of discovery that makes this area such a compelling place to work.

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Kasia Rejzner (York)

Mathematics at the Heart of Quantum Field Theory

Quantum Field Theory (QFT) is the physical theory that describes particle physics. It shows remarkable agreement with experiment, but its mathematical foundations are not completely well-understood, despite several decades of attempts. Some of the big open questions concern situations when QFT meets general theory, e.g. description of quantum matter in the vicinity of a black hole. Some other problems border quantum information theory, for example how to understand measurement and quantum reference frames in QFT setting. One of the mathematically rigorous frameworks to study foundations of QFT is that of algebraic quantum field theory (AQFT) and its perturbative incarnation pAQFT. In this talk I will present the (p)AQFT framework and report on recent progress and challenges.

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Patrick Rubin-Delanchy (Edinburgh)

The Manifold Hypothesis in Science & AI

The manifold hypothesis is a widely accepted tenet of machine learning which asserts that nominally high-dimensional data are in fact concentrated near a low-dimensional manifold, embedded in high-dimensional space. This phenomenon is observed empirically across a large diversity of application domains, has led to the development of a wide range of statistical methods in the last few decades, and has been suggested as a key factor in the success of modern AI technologies. In this talk, I will show some examples of manifold structure occurring in scientific data and in AI (internal representations of LLMs), and discuss associated questions, particularly around how observed topology and geometry might map to some reality or human-understandable concept. I will present statistical models and theory which help explain the efficacy of popular combinations of tools, such as PCA followed by t-SNE, for extracting manifold structure. Finally, I will point to a vast array of unexplored possibilities in representation learning and potential implications for the future role of AI in science.

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Julia Wolf (Cambridge)

When is a Mathematical Object Well-Behaved?

We will approach the question in the title from two different angles: first, from the viewpoint of model theory, a subject in which for over half a century the notion of “stability” has played a central role in describing tame behaviour; secondly, from the perspective of combinatorics, where so-called “regularity decompositions” have enjoyed a similar level of prominence in a range of finitary settings, with remarkable applications in additive number theory and theoretical computer science.

In recent years, these two fundamental motifs have been shown to interact in interesting ways. In particular, it has been shown that mathematical objects that are stable in the model-theoretic sense admit particularly well-behaved regularity decompositions. If time permits, I will explain how higher-order generalisations of regularity have pointed the way towards a long-sought generalisation of stability, opening up a new dimension of the model-theoretic classification picture.

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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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