Lecture Courses in Algebra

Hadi Salmasian: Lie Groups and Quantization

4th September, 2024

This is a 23-lecture course, with each lecture being around 80 minutes long, given by Hadi Salmasian.

The goal of the course is to first cover the foundational theory of Lie groups and then move on to more advanced topics that expose the audience to areas of active research. The following is the list of topics that are intended to be covered:

  • Foundational theory of Lie groups: Lie groups, the exponential map, Lie correspondence. Homomorphisms and coverings. Closed subgroups. Classical groups: Cartan subgroups, fundamental groups. Manifolds. Homogeneous spaces. General Lie groups.
  • Introduction to quantization: Symplectic manifolds, pre-quantization, the orbit method. Poisson manifolds, Manin triples. Universal enveloping algebras, quantum sl(2) and its representations, quantum symmetric spaces.

Giulio Tiozzo: Introduction to Random Walks on Groups

12th September, 2023

This is a 21-lecture course, with each lecture being either one or two hours, given by Giulio Tiozzo. It gives an introduction to random walks on groups. This class will focus on properties of group actions from a probabilistic point of view, investigating the relations between the dynamics, measure theory and geometry of groups.

We will start with a brief introduction to ergodic theory, discussing measurable transformations and the basic ergodic theorems. Then we will approach random walks on matrix groups and lattices in Lie groups, following the work of Furstenberg. Topics of discussion will be: positivity of drift and Lyapunov exponents. Stationary measures. Geodesic tracking. Entropy of random walks. The Poisson-Furstenberg boundary. Applications to rigidity. We will then turn to a similar study of group actions which do not arise from homogeneous spaces, but which display some features of negatively curved spaces: for instance, hyperbolic groups (in the sense of Gromov) and groups acting on hyperbolic spaces. This will lead us to applications to geometric topology: in particular, to the study of mapping class groups and Out(FN).

Prerequisites: An introduction to measure theory and/or probability, basic topology and basic group theory. No previous knowledge of geometric group theory or Teichmüller theory is needed.

Slawomir Solecki: The dynamics and structure of transformation groups

17th January, 2023

This is a 24-lecture course, with each lecture being 75 minutes, given by Slawomir Solecki. Note that the 2nd lecture was not recorded. The other lectures might still be of significant interest, but this needs to be known.

This course focuses on the interaction between set theory, geometry, group theory, and dynamics. It will present parts of Rosendal’s Coarse Geometry of Topological Groups, Kechris-Pestov-Todorcevic’s Fraïssé Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups, as well as theory of Borel and measurable combinatorics.

Spencer Unger and Assaf Rinot: Set theory, algebra and analysis

9th January, 2023

This is a 23-lecture course, with each lecture being 75 minutes, given by Spencer Unger and Assaf Rinot.

This course will present a rigorous study of advanced set-theoretic methods including forcing, large cardinals, and methods of infinite combinatorics and Ramsey theory. An emphasis will be placed on their applications in algebra, topology, and real and functional analysis.

Thomas Creutzig: Vertex Operator Algebras

1st August, 2022

This is a 20-lecture course, with each lecture being about 45 minutes or so, given by Thomas Creutzig. It gives an introduction to vertex operator algebras from the point of view of quantum mechanics.

Vertex operator algebras (VOAs) first appeared in the 1980s as the rigorous notion of chiral algebras (the symmetry algebras) of two-dimensional conformal quantum field theories. Since then, they have been employed as key ingredients in many modern problems of mathematical physics and pure mathematics, ranging from monstrous moonshine to knot theory and geometry. The older problems have been mostly concerned with the simplest type of VOAs, so‐called rational theories.

In the last few years, it has been realized that VOAs and their representation theories yield rich invariants of three and four‐dimensional supersymmetric quantum field theories. This provides new insights into low‐dimensional topology and the quantum geometric Langlands programme. Involved VOAs are however not rational (often called logarithmic) and so their representation theory is rich and exciting.

These lectures will be a very modern introduction to the theory of VOAs. We will use techniques from representation theory (especially Lie theory), geometry and topology; no knowledge of VOAs is needed. The lectures will be a mix of general theory and illustrating it with the most important examples, that is free field theories, affine and W‐algebras; and the school will end with an exposition of the very recent use and appearance of VOAs in physics, geometry, and low‐dimensional topology.

Richard Borcherds: Rings and Modules

27th September, 2021

This is a 22-lecture course, with each lecture being about 30 minutes or so, given online by Richard Borcherds. It gives an introduction to rings and modules.