Fiona Torzewska: Mapping class groupoids and motion groupoids

4th November, 2020

An LMS online course in mapping class groupoids.

A topological phase of matter is a physical system whose behaviour may be effectively described via a topological quantum field theory i.e. functor from cob to vect. The study of topological quantum field theories has applications in quantum computing but also involves a lot of beautiful mathematics which is interesting in its own right. A central role in the description of topological phases of matter in 2 (spatial) dimensions is played by the representations of braid groups. A natural generalisation to study the statistics of higher (spatial) dimensional phases of matter is then to look for generalisations of the braid group. Braid groups can be equivalently defined as the mapping class groups or as the motions groups of points in a disk, as well as in several other equivalent ways. In these lectures we will introduce generalisations of these two definitions. In each case we will show first that these give us groupoids and then that we can get back to the classical definitions by considering the endomorphisms of a single object. The mapping class groupoid is a simpler construction but is not in general the right notion to take when considering particles moving through space. We will construct a functor from the motion groupoid to the mapping class groupoid and hence see which cases we can study only the mapping class groupoid. We will use lots of examples to aid intuition and intend this talk to be accessible to those with minimal knowledge of topology.

Dave Sixsmith and Vasiliki Evdoridou: Lectures on Holomorphic Dynamics

27th October, 2020

An LMS online lecture course in holomorphic dynamics.

The series will consist of 6 one-hour lectures which will focus on the iteration of entire functions. We explore, among other things, some famous fractal Julia sets and the well-known Mandelbrot set. In particular, we will cover the following topics:

   1.  Equicontinuity, normal families, Montel's theorem, Riemann mapping theorem, the Riemann sphere.
   2.  Iteration of polynomials. Definition of the Fatou set and the Julia set for a polynomial. Examples.
   3.  The filled Julia set. Fixed and periodic points.
   4.  An introduction to the properties of the Fatou set and the Julia set.
   5.  The Mandelbrot set: its definition and properties.
   6.  Introduction to the iteration of transcendental entire functions.
   7.  Similarities and differences between polynomials and transcendental entire functions.
   8.  The escaping set: definition, properties, and its important role.
   9.  Examples of the Fatou, Julia and escaping sets for transcendental entire functions.

The lecture series is addressed to PhD students from diverse mathematical backgrounds. We shall assume a basic knowledge of complex analysis and a little topology. Some more advanced background in complex analysis will be covered in the first lecture. No knowledge of dynamics will be assumed.

Marialaura Noce: Groups of automorphisms of rooted trees

22nd October, 2020

An LMS online lecture course in groups acting on trees.

Groups of automorphisms of rooted trees have been studied for years as an important source of groups with interesting properties. For instance, the Grigorchuk group (that is a group acting on the binary tree) is the first example of a finitely generated group with intermediate growth (this answered an open question posed by Milnor) and the first example of an amenable but not elementary amenable group. Furthermore, this group provides a counterexample to the General Burnside Problem.

In these lectures we will first introduce the basic theory of groups of automorphisms of rooted trees and their subgroups. Then we will give examples and main properties of such groups, including the aforementioned Grigorchuk group, and the GGS groups.

Various Speakers: An Introduction to Quantum Field Theory in Curved Spacetime

12th October, 2020

An LMS online lecture course in quantum field theory.

In this mini-course we will give an introduction to quantum field theory in curved spacetime, one of the dominant research areas in modern algebraic quantum field theory. It is aimed at postgraduate students unfamiliar with the topic, although some preliminary knowledge of elementary differential geometry (notions of manifold, (co)tangent bundle, covariant derivative), as well as some familiarity with quantum mechanics and standard quantum field theory would be helpful.

The course will cover some of the underlying concepts of the algebraic approach to quantum field theory in curved spacetime and discuss several physically relevant examples. The focus will be on the linear (i.e., non-interacting) scalar field, favouring depth over breadth.

Topics for the lectures are:

   •  Lorentzian geometry, causality and the Klein–Gordon equation;
   •  Algebraic quantum field theory and the quantised scalar field;
   •  Aspects of QFT in curved spacetimes;
   •  Introduction to spacetimes;
   •  An introduction to Hadamard states.

Markus Land: L-theory of rings via higher categories

21st September, 2020

An online lecture course by the University of Münster in L-theory of rings.

We will introduce Witt groups and various flavours of L-groups and discuss some examples. We will then discuss a process called algebraic surgery. This process permits, under suitable assumptions, to simplify representatives in L-groups, and we will touch on two flavours (surgery from below and surgery from above). We will indicate how these can be used to show that various comparison maps between different L-theories are isomorphisms (in suitable ranges). Then we will go on and discuss three methods that allow for more calculations: Localisation sequences, a dévissage theorem, and an arithmetic fracture square. Using those, we will calculate the L-groups of Dedekind rings whose fraction field is a global field.

Yonatan Harpaz: New perspectives in hermitian K-theory

21st September, 2020

An online lecture course by the University of Münster in K-theory of forms.

In this lecture series we will describe an approach to hermitian K-theory which sheds some new light on classical Grothendieck-Witt groups of rings, especially in the domain where 2 is not assumed to be invertible. Our setup is higher categorical in nature, and is based on the concept of a Poincaré ∞-category, first suggested by Lurie. We will explain how classical examples of interest can be encoded in this setup, and how to define the principal invariants of interest, consisting of the Grothendieck-Witt spectrum and L-theory spectrum, within it. We will then describe our main abstract results, including additivity, localization and universality statements for these invariants and their relation to each other and to algebraic K-theory via the fundamental fibre sequence.

Various Speakers: Planar random growth and scaling limits

14th September, 2020

An LMS online lecture course in random growth.

Conformal growth models are motivated by some real-world growth processes, and are constructed using conformal maps. We will introduce the one-parameter Hastings–Levitov model, which is used to describe Laplacian growth and allows us to vary between off-lattice versions of many well studied models. Then we investigate the "small particle" scaling limit, which often entails finding a martingale and relating its behaviour to its analogue for the proposed continuum limit.

Tiago Fonseca: A crash course on modular forms and cohomology

11th September, 2020

An LMS online lecture course in modular forms.

This is a geometrically flavoured introduction to the theory of modular forms. We will start with a standard introduction to some basic analytic aspects concerning modular forms and to their interpretation as sections of line bundles on modular curves.

Then, our main goal will be to explain how one can attach certain 2-dimensional cohomology groups to Hecke eigenforms. In this course, we will only deal with algebraic de Rham and Betti cohomology, but this can also serve to build geometric intuition on the l-adic setting, which gives rise to the famous l-adic representations attached to modular forms.

We will finish with a discussion on the Eichler-Shimura isomorphism, periods of modular forms, and, depending on time, Manin's theorem on the critical values of L-functions of modular forms.

Chak Hei Lo: Foster–Lyapunov methods for Markov chains

10th September, 2020

An LMS online lecture course in Markov chains.

We will start the course by presenting various results using the semimartingale approach for Markov chains. These results include Foster–Lyapunov criteria by which a suitable Lyapunov function can determine whether a process is transient or recurrent. We will then move on to some applications on these methods, including to some random walks on strips and some interacting particles systems, such as voter models.

   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.