Short Courses in Topology

Mohammed Abouzaid: Theory of bordisms

6th August, 2024

In this introductory lecture, which should be accessible to a general mathematical audience, I will review the classical bordism theory of manifolds, from its origin in Poincare's work, to the subsequent development by Pontryagin, Thom, Milnor, Wall, and Quillen among others.

Lecture 2: Bordism of orbifolds

An orbifold is a space with additional structure that describes it locally as the quotient of a manifold by a finite group. I will describe Pardon's recent result which reduces the study of orbifolds to the study of manifolds with Lie group actions. Then I will explain the relationship between equivariant and orbifold bordism, and formulation some structural properties of this theory.

Lecture 3: Bordism of derived orbifolds

The notion of a derived orbifold arises naturally in pseudo-holomorphic curve theory, and plays a central role in the emerging field of Floer homotopy. I will explain how it is related to the notion of "homotopical bordism" due to tom Dieck in the 1970s, and formulate some conjectures about its structure in the complex oriented case.

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.

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.

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.