## Heilbronn Annual Conference 2023

07 Mar 2023, by Events in**7 – 8 September 2023**

School of Mathematics, Fry Building, Woodland Road, Bristol BS8 1UG

**Venue:** Wolfson Lecture Theatre, 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. This year we welcome eight distinguished speakers, to deliver lectures intended to be accessible to a general audience of mathematicians.

**Invited Speakers**

**Petter Brändén** (KTH Royal Institute of Technology, Sweden): **Lorentzian Polynomials and Negative Dependence**

Negative dependence models repelling particles in statistical physics, and repelling random variables in probability theory. Negative dependence had for a long time resisted attempts of building a working theory around it. However in recent years two different successful approaches to prove negative dependence inequalities have been developed. One using Hodge theory and the other using the geometry of zeros of multivariate polynomials. The theory of Lorentzian polynomials merges these two approaches. In this talk we will introduce Lorentzian polynomials and give applications to matroid theory and the random cluster model.

**Alessio Corti **(Imperial, UK): **The Classification of Fano Varieties and Mirror Symmetry**

I will introduce the problem of classification of Fano varieties, which is one of the basic questions in algebraic geometry. I will explain a recent point of view coming from mirror symmetry that translates the problem into combinatorics of lattice polytopes. I will summarise progress, including the development of computing infrastructure to work for example with the Kreuzer-Skarke database of nearly half billion 4-dimensional reflexive lattice polytopes, and describe some of the challenges going forward. (Work with several collaborators including Coates, Kasprzyk and others.)

**Ivan Corwin** (Columbia, USA): **Extreme Diffusion **

In a wide range of physical systems large numbers of particles diffuse together in a common environment. In this talk we will consider how the nature of the hidden environment impacts the motion of these diffusing particles and how the motion can in turn be used to interrogate the environment. In particular we will see that the impact is most pronounced in the behaviour of extreme particles that move the furthest and fastest. Surprisingly, the mathematics behind this theory will bring us into the realm of integrable probability and quantum integrable systems, most notably relying on a non-commutative binomial theorem.

**Minhyong Kim** (ICMS, UK): **Diophantine Equations in Two Variables and the Arithmetic Shapes of Solutions**

The equation y^3 = x^6 + 23x^5 + 37x^4 + 691x^3 − 631204x^2 + 5169373941 has the solution (1, 1729). Are there any other solutions in rational numbers? The study of integral or rational solutions to polynomial equations, sometimes known as the theory of Diophantine equations, is among the oldest pursuits in mathematics. This lecture will give an idiosyncratic survey of the remarkable advances made in the 20th and 21st century for the special case of equations of two variables. The emphasis will be on the techniques of arithmetic topology, where we combine the study of numbers with the study of shapes, often in intricate and surprising ways.

**James Newton** (Oxford, UK): ** Some Examples of Equidistribution in Number Theory**

Number theorists are often interested in the distribution of families of arithmetic objects. The most important example is the set of prime numbers, and some of the first information obtained about their distribution was Dirichlet’s theorem on primes in arithmetic progressions. In essence, this says that prime numbers are evenly distributed over the congruence classes (coprime to N) modulo a fixed integer N. I’ll discuss a related question, on which there has been recent progress: the Sato-Tate conjecture on the distribution of the number of points on reductions of elliptic curves modulo primes.

**Françoise Tisseur** (Manchester, UK): **Exploiting Tropical Algebra in Numerical Linear Algebra**

The tropical semiring consists of the real numbers and infinity along with two binary operations: addition defined by the max or min operation and multiplication. Tropical algebra is the tropical analogue of linear algebra, working with matrices with entries on the extended real line. There are analogues of eigenvalues and singular values of matrices, and matrix factorizations in the tropical setting, and when combined with a valuation map these analogues offer `order of magnitude’ approximations to eigenvalues and singular values, and factorizations of matrices in the usual algebra. What makes tropical algebra a useful tool for numerical linear algebra is that these tropical analogues are usually cheaper to compute than those in the conventional algebra. They can then be used in the design of preprocessing steps to improve the numerical behaviour of algorithms. In this talk I will review the contributions of tropical algebra to numerical linear algebra and will present recent results obtained with Akian, Gaubert, and Marchesini on tropical scaling of matrix polynomials.

**Karen Vogtmann **(Warwick, UK): **Moduli Spaces of Finite Metric Graphs**

Finite graphs are used to model many phenomena in mathematics and science. When equipped with a metric one can often deform the metric and sweep out a moduli space that parameterizes all graphs of interest in a particular model. I will give some examples and discuss what is known about the topology and geometry of these moduli spaces, including recent advances.

**Tamar Ziegler** (Hebrew University, Israel): **Sign Patterns of the Mobius Function**

The Mobius function is one of the most important arithmetic functions. There is a vague yet well known principle regarding its randomness properties called the “Mobius randomness law”. It basically states that the Mobius function should be orthogonal to any “structured” sequence. P. Sarnak suggested a far reaching conjecture as a possible formalization of this principle. He conjectured that “structured sequences” should correspond to sequences arising from deterministic dynamical systems. Sarnak’s conjecture follows from Chowla’s conjecture – which is the mobius version of the prime tuple conjecture. I will describe progress in recent years towards these conjectures, building on major advances dynamics, additive combinatorics, and analytic number theory.

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

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