María Chara: An introduction to algebraic geometry codes

6th December, 2021

Error-correcting codes play an important role in many areas of science and engineering, as they safeguard the integrity of data against the adverse effects of noise in communication and storage. On the most basic level, good error-correcting codes are able to both transmit data efficiently and correct a large number of errors relative to their length. As observed by V. D. Goppa in 1975, one can use algebraic function fields over 𝔽q to construct a large class of interesting codes. Properties of these codes are closely related to properties of the corresponding function field, and the Riemman-Roch Theorem provides estimates, sharps in many cases, for their main parameters. In this short course we will study Goppa's construction of codes by means of an algebraic function field after a brief introduction of the theory of error-correcting codes, some classical bounds for the parameters of these codes and their detection and error-correction capabilities.

Pierre Raphaël: Non-linear waves

1st October, 2021

A series of talks about non-linear waves, energy supercritical problems, solitons, defocusing Non-Linear Schrödinger equation and shock waves.

Bertrand Eynard: (Mixed) topological recursion and the two-matrix model

17th August, 2021

In this series of lectures we will introduce the 2-matrix model and the issue of mixed traces, then we shall give the answers as formulas. Some formulas will be proved during the lectures, but the main goal is to explain how to use the formulas for practical computations. We shall largely follow the Chapter 8 of the book Counting surfaces, B. Eynard, Birkhäuser 2016.

Roland Speicher: Free probability theory

16th August, 2021

Usual free probability theory was introduced by Voiculescu in the context of operator algebras. It turned out that there exists also a relation to random matrices, namely it describes the leading order of expectation values of the trace for multi-matrix models. Higher order versions of free probability were later introduced by Collins, Mingo, Sniady, Speicher in order to capture in the same way the leading order of correlations of several traces. A prominent role in free probability theory is played by “free cumulants” and “moment-cumulant formulas”, and the underlying combinatorial objects are “non-crossing partitions” and, for the higher order versions, “partitioned permutations”. I will give in my talks an introduction to free probability theory, with special emphasis on the higher order versions, and an eye towards possible relations to topological recursion. In particular, it seems that the problem of symplectic invariance in topological recursion has, at least in the planar sector, something to do with the transition between moments and free cumulants.

Walter van Suijlekom: Non-commutative geometry and spectral triples

16th August, 2021

Our starting point is a spectral approach to geometry, starting with the simple question ’can one hear the shape of a drum?’ This was phrased by Mark Kac in the 1960s, and led to many developments in spectral geometry. For us, it is the motivation for considering spectral triples, which is the key technical device used to describe non-commutative Riemannian spin manifolds. We will give many motivating examples, and also explain how gauge symmetries naturally arise in this context. The connection to the other main theme of the workshop is found via the spectral action principle. It allows for a derivation of an action functional from any given spectral triple. This includes the Hermitian matrix model, but more interesting matrix models appear beyond. We will consider some recent developments for such models by deriving a perturbative series expansion for the spectral action.

Elba Garcia-Failde: Introduction to topological recursion

16th August, 2021

In this mini-course I will introduce the universal procedure of topological recursion, both by treating examples and by presenting the general formalism. We will study the classical case of the Hermitian matrix model in detail, which combinatorially corresponds to ribbon graphs, beginning from the loop equations, which correspond to Tutte’s recursion in the combinatorial setting. This will be the starting point to make the connection to free probability, which moreover provides a combinatorial way of exploring the variation of the topological recursion output when applying a symplectic transformation to the input. Apart from the (conjectural) property of symplectic invariance, topological recursion has many other interesting features and, together with its generalizations, has established connections to various domains of mathematics and physics, like intersection theory of the moduli space of curves and integrability. We will explain some of these properties and connections, giving several ideas why this is worth considering, and is the starting or gluing point of an active field of research, and finally hoping to instigate the search of new beautiful connections.

Christopher Schafhauser: On the classification of nuclear simple C*-algebras

5th August, 2021

A conjecture of George Elliott dating back to the early 1990s asks if separable, simple, nuclear C*-algebras are determined up to isomorphism by their K-theoretic and tracial data. Restricting to purely infinite algebras, this is the famous Kirchberg-Phillips Theorem. The stably finite setting proved to be much more subtle and has been a driving force in research in C*-algebras over the last 30 years. A series of breakthroughs were made in 2015 through the classification results of Elliott, Gong, Lin, and Niu and the quasidiagonality theorem of Tikuisis, White, and Winter. Today, the classification conjecture is now a theorem under two additional regularity assumptions: Z -stability and the UCT. In my recent joint work with José Carrión, Jamie Gabe, Aaron Tikuisis, and Stuart White a much shorter and more conceptual proof of the classification theorem in the stably finite setting was provided. I hope to give an overview of the classification problem for C*-algebras and discuss some of the new techniques that led to the new proof.

Jesse Peterson: Von Neumann algebras and lattices in higher-rank groups

3rd August, 2021

An online lecture course by the University of Münster in Von Neumann algebras.

Lecture 1: We'll briskly review basic properties of semi-finite von Neumann algebras. The standard representation, completely positive maps, group von Neumann algebras, the group-measure space construction, and some characterizations of the hyperfinite II1 factor.

Lecture 2: We discuss some approximation properties that are common in "rank 1" groups: Weak amenability and biexactness.

Lecture 3: We discuss properly proximal groups as defined by Boutonnet, Ioana, and myself, and give some applications to group von Neumann algebras associated to higher-rank groups.

Lecture 4: We’ll introduce measure equivalence (ME), W*-equivalence (W*E), and von Neumann equivalence (VNE). We’ll give examples and discuss invariants.

Rose Morris-Wright: Artin groups: algebraic and geometric techniques

28th June, 2021

Artin groups are a broad class of groups whose presentations all follow a particular pattern. They are generalizations of braid groups and are closely related to Coxeter groups. Artin groups provide examples of groups with many interesting properties but there is very little that is known about ALL Artin groups.

In the first lecture, we’ll define Artin groups, talk about different types of Artin groups, and give a summary of known results and open questions. In the second lecture, we’ll focus on algebraic techniques for studying Artin groups, including the Garside structure, parabolic subgroups, and, if time permits, the Artin monoid. In the third lecture, we’ll discuss geometric techniques for studying Artin groups, including the Deligne complex and newer complexes such as the Clique-cube complex and the systolic Artin complex.

Sahana Balasubramanya: Acylindrically and relatively hyperbolic groups

28th June, 2021

The class of acylindrically hyperbolic groups has been of immense interest in recent times. It is an extremely large class of groups, containing many interesting examples. Yet a significant part of the theory of hyperbolic and relatively hyperbolic groups can be generalized in this context. The goal of this mini course is to provide an introduction to this class of groups, and focus on some important techniques.

In the first lecture, we will define acylindrical actions and talk about the motivation to study them. We will then define acylindrically hyperbolic groups and discuss some examples and properties of this class of groups. The second lecture will focus on the notion of hyperbolically embedded subgroups and discuss relative hyperbolicity in this context. In the last lecture, we will discuss the concept of group theoretic Dehn filling and some of its applications. Time permitting, I will also talk a bit about my own research.

Joshua Frisch: The Poisson boundary and bounded harmonic functions on groups

21st June, 2021

The Poisson boundary of a group has two interpretations. Firstly it measures the asymptotic uncertainty of a random walk on a group. Secondly it classifies the possible range of bounded harmonic functions on the group. In this minicourse we will introduce some of the theory of the Poisson boundary and its relationship with group properties such as polynomial growth, the infinite conjugacy class property, and amenability. In particular we will focus on the following question. For which measured groups are all bounded harmonic functions trivial? This will lead us into a, perhaps surprising, incredibly interconnected web of ideas including convex analysis, dynamical systems, information theory, and probability theory. No prior knowledge of random walks on groups or any of the aforementioned fields will be assumed.

Eduardo Oregón-Reyes: Cubulated groups and virtual specialness

21st June, 2021

CAT(0) cube complexes were introduced by Gromov merely as examples of metric spaces of non-positive curvature, but now they play a prominent role in geometric group theory. One reason for this is that many interesting groups are known to act nicely on these spaces, including free and surface groups, small cancellation groups, 1-relator groups with torsion, and many 3 manifold groups. Another reason is that some of these groups are, in addition, virtually special, notion defined by Haglund and Wise that implies being (up to finite index) the subgroup of some right-angled Artin group.

In the first lecture, we will define CAT(0) cube complexes, explore some of their combinatorial structure, and discuss some examples of cubulated groups. For the second lecture, we will introduce the class of virtually special groups, review some of their properties, and mention some criteria for virtual specialness. We will end the mini-course with a discussion of the main techniques for studying cubulated hyperbolic groups, focusing on some theorems of Wise and Agol. If time permits, I will mention a few things about the relatively hyperbolic case.

Macarena Arenas: A crash course on isoperimetry

21st June, 2021

The word problem for a finitely generated group G is the algorithmic problem of deciding whether a word in the generators represents the trivial element of G. When G is finitely presented, one can interpret this problem topologically by constructing a finite 2-complex X whose 1-cells and 2-cells correspond to the generators and relations. In this way, a word w in the generators which represents the trivial element of G will correspond to a loop in the 1-skeleton of the universal cover of X.

Geometry comes into the picture via the study of isoperimetric functions: An isoperimetric function is a function associated to a finite group presentation which bounds the area of a relation in that group in terms of the length of that relation (or, equivalently, the area of a nullhomotopic loop in the complex described above in terms of the length of that loop). A Dehn function is an optimal isoperimetric function. Dehn functions can be understood as quantifying the complexity of the word problem.

This mini-course will survey what is known about Dehn functions for various classes of groups.

In the first lecture, we will discuss the precise connection between the solvability of the word problem for a group and the growth of its Dehn function. We will compute the Dehn functions of various classes of groups (introducing, on the way, hyperbolic groups and CAT(0) groups) and give examples of groups with very large Dehn functions.

In the second lecture, we will dive more deeply into the structure of Dehn functions, exploring the "isoperimetric spectrum", i.e., the set {d | nd is the growth type of a Dehn function}, and computing some more examples.

We will wrap things up by discussing some generalisations and variations of Dehn functions, and how they connect to other algorithmic problems in geometric group theory. Time permitting, I will also tell you a bit about my own work in this direction.

Alex Evetts: Introduction to growth in groups

19th November, 2020

An LMS online lecture course in growth in groups.

For a finitely generated group, the number of elements that can be spelled with words of length n, for any integer n>0, is called the growth function. This can be interpreted as a measure of the size of the group and is a powerful quasi-isometry invariant which has links to many areas of geometric group theory.

In the first lecture I will present the fundamental properties of the growth function and explore some key examples illustrating what kinds of functions can arise. I will also discuss Gromov's important theorem on groups of polynomial growth.

In the second lecture I will discuss the formal power series associated to the growth function, which is known as the growth series. I will explain some ways in which the behaviour of the growth series can provide insight into the asymptotics, and demonstrate this with examples.

Han Yu: Irrational rotations and number theory

19th November, 2020

An LMS online lecture course in number theory and dynamics.

The main goal of this mini-course is to illustrate a proof of Furstenberg's ×2,×3 theorem: The ×2,×3 orbit of any irrational number on the unit interval is dense. Key results that will be needed for the proof are topological properties of irrational rotation on the unit interval. We will discuss those results and provide detailed backgrounds as well as proofs. At the end of the course, I will introduce various results and problems on digit expansions of integers. The following topics will be covered:

   1.  Irrational rotations on torus;
   2.  Diophantine approximation: Dirichlet theorem, Roth's theorem, Baker's theory of linear forms of logarithms;
   3.  Furstenberg's ×2,×3 theorem;
   4.  Results and problems on digit expansions of integers;
   5.  Furstenberg's theorem on 2-dimensional torus (if time permits).

Note: For 2., I will mostly state the results without giving proofs as they are out of the scope of this mini-course.

Alan Logan: Free groups via graphs

5th November, 2020

An LMS online lecture course in free groups and graph theory.

Free groups may be viewed as the fundamental groups of graphs. This observation allows for a very intuitive view of free groups and their subgroups. These lectures combine topological ideas, due to Stallings in the 1980s, with more combinatorial and computational ones to prove many of the fundamental results in free groups. These results include the Nielsen-Schreier Theorem (subgroups of free groups are free), Howson's Theorem (finitely generated subgroups have finitely generated intersection), and the decidability of the subgroup membership problem.