## Heilbronn Annual Conference 2021

11 Dec 2020, by Events in**9 – 10 September 2021**

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:**

**Caucher Birkar** (Cambridge): Higher Dimensional Algebraic Geometry

The classification problem of algebraic varieties is at the heart of algebraic geometry. With roots in the ancient world the theory saw great advances in dimensions one and two in the 19th century and the first half of 20th century. It was only in the 1970-80’s that a general framework was formulated, and by the early 1990’s a satisfactory theory was developed in dimension 3. The last 30 years has seen great progress in all dimensions. In this talk I will try to give a historical perspective and discuss the theory in general terms. I will explain how the theory is based on birational transformations and moduli considerations.

**Jon Brundan** (Oregon): Braids, Ribbons and Webs in Representation Theory

Thirty five years after the discovery of quantum invariants for knots and links, graphical methods have become increasingly important for working with monoidal categories. I will tell a little of this story from an algebraist’s perspective, from symmetric groups and the HOMFLY skein category, to webs and Heisenberg categories.

**Ana Caraiani **(Imperial College London): Reciprocity Laws and Torsion Classes

The Langlands program is a vast network of conjectures that connect many areas of pure mathematics, such as number theory, representation theory, and harmonic analysis. At its heart lies reciprocity, the conjectural relationship between Galois representations and modular, or automorphic forms. A famous instance of reciprocity is the modularity of elliptic curves over the rational numbers: this was the key to Wiles’s proof of Fermat’s last theorem. I will give an overview of some recent progress in the Langlands program, with a focus on new reciprocity laws over imaginary quadratic fields.

**Heather Harrington** (Oxford): Algebraic Systems Biology

Signalling pathways in molecular biology can be modelled by polynomial dynamical systems. I will present models describing two biological systems involved in development and cancer. I will overview approaches to analyse these models with data using computational algebraic geometry, differential algebra and statistics. Finally, I will present how topological data analysis can provide additional information to distinguish wild-type and mutant molecules in one pathway. These case studies showcase how computational geometry, topology and dynamics can provide new insights to better understand model parameter values as well as biological systems, specifically how changes at the molecular scale (e.g. molecular mutations) result in kinetic differences that are observed as phenotypic changes (e.g. mutations in fruit fly wings).

**Gil Kalai **(Hebrew University of Jerusalem): A World Without Quantum Computers

The lecture will briefly discuss my theory explaining why quantum computers are not possible. I will address the question if Google’s Sycamore and other recent experiments falsify my theory, and discuss consequences of my theory to classical philosophical questions of predictability, free will, and reductionism. On the mathematical side, the connection between noise and Fourier expansion will play a role.

**Peter Keevash** (Oxford): Hypergraph Decompositions and their Applications

Many combinatorial objects can be thought of as a hypergraph decomposition, i.e. a partition of (the edge set of) one hypergraph into (the edge sets of) copies of some other hypergraphs. For example, a Steiner Triple System is equivalent to a decomposition of a complete graph into triangles. In general, Steiner Systems are equivalent to decompositions of complete uniform hypergraphs into other complete uniform hypergraphs (of some specified sizes). The Existence Conjecture for Combinatorial Designs, which I proved in 2014, states that, bar finitely many exceptions, such decompositions exist whenever the necessary ‘divisibility conditions’ hold. I also obtained a generalisation to the quasirandom setting, which implies an approximate formula for the number of designs; in particular, this resolved Wilson’s Conjecture on the number of Steiner Triple Systems. A more general result that I proved in 2018 on decomposing lattice-valued vectors indexed by labelled complexes provides many further existence and counting results for a wide range of combinatorial objects, such as resolvable designs (the generalised form of Kirkman’s Schoolgirl Problem), whist tournaments or generalised Sudoku squares. In this talk, I plan to review this background and then describe some more recent and ongoing applications of these results and developments of the ideas behind them.

**Tatiana Nagnibeda** (Geneva): Spectra and Spectral Measures on Cayley Graphs of Finitely Generated Groups and their Actions

We will be interested in Laplacians on graphs associated with infinite finitely generated groups: Cayley graphs and more generally, Schreier graphs corresponding to some natural group actions. The spectrum of such an operator is a compact subset of the interval [0,1], but not much more can be said about it in general. Little is known about the associated spectral measures either. We will discuss various spectral problems arising in this context and stemming from the famous question “Can one hear the shape of a drum?”

**Jeremy Quastel** (Toronto): Integrable Fluctuations in 1+1 Dimensional Random Growth

Random growth off a one dimensional substrate is often described by the Kardar-Parisi-Zhang stochastic partial differential equation. It is a member of a large universality class characterized by unusual fluctuations, some of which appeared earlier in random matrix theory. On large space time scales, the fluctuations in the class turn out to be described by a special scaling invariant Markov process — the KPZ fixed point — obtained through the solution of a special discrete model, TASEP. Both turn out to be new integrable systems, leading to new connections between random growth and classical integrable partial differential equations.

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

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