11 – 13 April 2022 (Hybrid)

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

Lauren Williams, Harvard University, USA

Lauren K. Williams is the Robinson Professor of Mathematics at Harvard University and the Seaver Professor at the Radcliffe Institute. She is a world leading mathematician working in the general areas of algebraic, enumerative, and topological combinatorics, and their connections with algebraic geometry, representation theory, and physics. Lauren’s research has attracted numerous awards and prizes. For example, she is invited speaker at the 2022 ICM, she was LMS Hardy Lecturer in 2018 and Simons Fellow in 2014.  She is also the recipient of the AWM-Microsoft Research Prize in algebra and number theory.

Lecture 1: Colloquium Style

Tropical Geometry, Cluster Algebras, and Shallow Water Waves

Algebraic geometry studies geometric objects associated to polynomials. Tropical geometry is a newer field which studies geometric objects associated to “tropical” polynomials, in which the operations of multiplication and addition are replaced by addition and the minimum operation. Meanwhile, cluster algebras are a class of commutative rings with a rich combinatorial structure, designed as a framework for answering questions in total positivity. Surprisingly, tropical geometry and cluster algebras arise in some real life applications. In particular, I will explain how tropical curves and cluster algebras can be used to answer questions about soliton solutions of the KP equation, which in turn model shallow water waves.

Lecture 2: Combinatorics of Hopping Particles

The asymmetric simple exclusion process (ASEP) is a model for translation in protein synthesis and traffic flow; it can be defined as a Markov chain describing particles hopping on a one-dimensional lattice. I will give an overview of some of the connections of the stationary distribution of the ASEP to combinatorics (tableaux and multiline queues) and special functions (Askey-Wilson polynomials, Macdonald polynomials, and Schubert polynomials). I will also make some general observations about positivity in Markov chains

Lecture 3: The Positive Grassmannian and the Amplituhedron

The positive Grassmannian is the subset of the real Grassmannian where all Plucker coordinates are nonnegative.  It has a beautiful combinatorial structure as well as connections to statistical physics and integrable systems.  The amplituhedron is the image of the positive Grassmannian of k-planes in n-space under a positive linear map from R^n to R^{k+m}.  Its “volume” computes scattering amplitudes in N=4 Super Yang Mills theory.  I’ll explain how ideas from oriented matroids, tropical geometry, and cluster algebras shed light on the structure of the positive Grassmannian and the amplituhedron.

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