Lecture Courses

Test

Test

Nikolay Bogachev: Geometry, Arithmetic, and Dynamics of Discrete Groups

10th September, 2024

This is a 22-lecture course, with each lecture being between one and two hours, given by Nikolay Bogachev.

Modern research in the geometry, topology, and group theory often combines geometric, arithmetic and dynamical aspects of discrete groups. This course is mostly devoted to hyperbolic manifolds and orbifolds, but also will deal with the general theory of discrete subgroups of Lie groups and arithmetic groups. Vinberg's theory of hyperbolic reflection groups will also be discussed, as it provides a lot of interesting examples and methods that turn out to be very practical. One of the goals of this course is to sketch the proof of the famous Mostow rigidity theorem via ergodic methods. Another goal is to talk about very recent results, giving a geometric characterization of arithmetic hyperbolic manifolds through their totally geodesic subspaces, and their applications. Throughout the course we will consider many examples from reflection groups and low-dimensional geometry and topology. In conclusion, I am going to provide a list of open problems related to this course.

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.

Chris Godsil: Algebraic graph theory and quantum computing

8th January, 2024

This is a 32-lecture course, with each lecture being about 45 minutes, given by Chris Godsil. Note that the 17th lecture was not recorded, but slides are at least available for it. The other 31 lectures are still of interest, but this needs to be known.

This course will provide an introduction to problems in quantum computing that can be studied using tools from algebraic graph theory. The quantum topics will relate to quantum walks and to quantum homomorphisms, automorphisms and colouring. The tools from algebraic graph theory include graphs automorphisms and homomorphisms, spectral decomposition and generating functions.

Prerequisites: I will assume a solid background in linear algebra and knowledge of what a permutation group is. Other topics will be covered in class, or in the notes. I will assume the knowledge of physics I had when I started on this topic, that is, no knowledge.

Nigel Higson: The tangent groupoid in non-commutative geometry

12th September, 2023

This is a 22-lecture course, with each lecture being 90 minutes, given by Nigel Higson.

The tangent groupoid is a geometric construction that can be applied to any smooth manifold. Alain Connes pointed out its relevance to the Atiyah-Singer index theorem, and ever since he did so the tangent groupoid has appeared regularly in noncommutative geometry, often in ways related to index theory but usually illuminating other issues at the same time. Good examples of this are the elegant and simple ways of understanding pseudodifferential operators that have been developed recently by Claire Debord and Georges Skandalis, and by Erik van Erp and Bob Yuncken. I shall start with pseudodifferential operators, then introduce the tangent groupoid through them, and go on to examine applications in representation theory, hypoelliptic partial differential equations and elsewhere.

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.

George Elliott: K-theory and C*-algebras

8th September, 2023

This is a 35-lecture course, with each lecture being an hour, given by George Elliott. Note that the 32nd lecture was not recorded. The first 31 lectures are still of great interest, but this needs to be known.

The theory of operator algebras was begun by John von Neumann eighty years ago. In one of the most important innovations of this theory, von Neumann and Murray introduced a notion of equivalence of projections in a self-adjoint algebra (*-algebra) of Hilbert space operators that was compatible with addition of orthogonal projections (also in matrix algebras over the algebra), and so gave rise to an abelian semigroup, now referred to as the Murray-von Neumann semigroup.

Later, Grothendieck in geometry, Atiyah and Hirzebruch in topology, and Serre in the setting of arbitrary rings (pertinent for instance for number theory), considered similar constructions. The enveloping group of the semigroup considered in each of these settings is now referred to as the K-group (Grothendieck's terminology), or as the Grothendieck group.

Among the many indications of the depth of this construction was the discovery of Atiyah and Hirzebruch that Bott periodicity could be expressed in a simple way using the K-group. Also, Atiyah and Singer famously showed that K-theory was important in connection with the Fredholm index. Partly because of these developments, K-theory very soon became important again in the theory of operator algebras. (And in turn, operator algebras became increasingly important in other branches of mathematics.)

The purpose of this course is to give a general, elementary, introduction to the ideas of K-theory in the operator algebra context. (Very briefly, K-theory generalizes the notion of dimension of a vector space.)

The course will begin with a description of the method (K-theoretical in spirit) used by Murray and von Neumann to give a rough initial classication of von Neumann algebras (into types I, II, and III). It will centre around the relatively recent use of K-theory to study Bratteli's approximately finite-dimensional C*-algebras, both to classify them (a result that can be formulated and proved purely algebraically), and to prove that the class of these C*-algebras - what Bratteli called AF algebras - is closed under passing to extensions (a result that uses the Bott periodicity feature of K-theory).

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.

Spiro Karigiannis: A Second Course in Riemannian Geometry

7th September, 2022

This is a 24-lecture course, with each lecture being about 80 minutes, given by Spiro Karigiannis.

This is a second course in Riemannian geometry. The emphasis will be on the intimate relationship between curvature and geodesics.

Nicola Gigli: Introduction to the Riemannian Curvature Dimension condition

5th September, 2022

This is a 22-lecture course, with each lecture being about 90 minutes, given by Nicola Gigli. Note that the 14th and 18th lectures were not recorded.

Created by Professor Nicola Gigli, the aim of the course is to provide an introduction to the world of synthetic description of lower Ricci curvature bounds, which has seen a tremendous amount of activity in the last decade: by the end of the lectures the student will have a clear idea of the backbone of the subject and will be able to navigate through the relevant literature.

We shall start by studying Sobolev functions on metric measure spaces and the notion of heat flow. Then following, and generalizing, the intuitions of Jordan-Kinderlehrer-Otto we shall see that such heat flow can be equivalently characterized as gradient flow of the Cheeger-Dirichlet energy on L2 and as gradient flow of the Boltzmann-Shannon entropy w.r.t. the optimal transportation metric W2. This provides a crucial link between the Lott-Villani-Sturm (LSV) condition and Sobolev calculus on metric measure spaces and, in particular, it justifies the introduction of 'infinitesimally Hilbertian' spaces as those metric measure structures for which W1;2(X) is a Hilbert space. By further developing calculus on these spaces we shall see that on infinitesimally Hilbertian spaces satisfying the LSV condition (these are called Riemannian curvature dimension spaces, or RCD for short) the Bochner inequality holds. 

We shall then discuss more sophisticated calculus tools, such as the concept of differential of a Sobolev function, that of vector field on a metric measure spaces and the notion of Regular Lagrangian Flow on RCD spaces.

We shall finally see how these are linked to the lower Ricci curvature bound - most notably we shall prove the Laplacian comparison theorem - and finally how they can be used to prove a geometric rigidity result like the splitting theorem for RCD spaces. It is worth noticing that such statement gives new information - compared to those available through Cheeger-Colding's theory of Ricci-limit spaces - even about the structure of smooth Riemannian manifolds.

Prerequisites: some familiarity with Riemannian geometry and optimal transport theory in the case cost=distance2 is preferred, but not required: I shall provide the necessary background when needed.

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.

Robert McCann: Optimal Transportation, Geometry and Dynamics

11th January, 2022

This is a 24-lecture course, with each lecture being around 80 minutes, given by Robert McCann. It gives an introduction to optimal transport.

This course is an introduction to the active research areas surrounding optimal transportation and its deep connections to problems in geometry, physics, nonlinear partial differential equations, and machine learning. The basic problem is to find the most efficient structure linking two or more continuous distributions of mass; think of pairing a cloud of electrons with a cloud of positrons so as to minimize average distance to annihilation.

Applications include existence, uniqueness, and regularity of surfaces with prescribed Gauss curvature (the underlying PDE is Monge-Ampère), geometric inequalities with sharp constants, image processing, optimal decision making, long time asymptotics of dissipative systems, and the geometry of fluid motion (Euler's equation and approximations appropriate to atmospheric, oceanic, damped and porous medium flows). The course builds on a background in analysis, including measure theory, but will develop elements as needed from the calculus of variations, game theory, differential equations, fluid mechanics, physics, economics, and geometry. A particular goal will be to expose the developing theories of curvature and dimension in metric-measure geometry, which provide a framework for adapting powerful ideas from Riemannian and Lorentzian geometry to non-smooth settings which arise both naturally in applications, and as limits of smooth problems.

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.

Piotr Kowalski: Model Theory

1st May, 2021

This is a 22-lecture course, with each lecture being about 45 minutes or so, given online by Piotr Kowalski. It gives an introduction to model theory.

We state several classical results about fields (Ax’s theorem, Hilbert's Nullstellensatz), which have easy model-theoretic proofs and then introduce the necessary basic model-theoretic tools to describe those proofs. In this way we motivate and present basic definitions and results from model theory. We will also sketch at the end of the lecture some recent applications of model theory regarding non-existence of classical solutions of some differential equations, which was a problem considered by Painlevé in 1895 and an argument was found only recently.

Ben Webster: Symplectic Geometry

11th January, 2021

This is a 24-lecture course, with each lecture being about 90 minutes or so, given online by Ben Webster.

This class covers the basic theory of symplectic manifolds. Symplectic structures play a key role in modern mathematics and physics. We will discuss their basic local theory (in particular, the Darboux theorem), connections to complex and Kähler geometry, Hamiltonian mechanics, moment maps and symplectic reduction, and some additional topics, such as toric varieties, hyperkähler structures, quantization, Fukaya categories and mirror symmetry.

Prerequisites: Familiarity with the basics of differential geometry: smooth manifolds, tangent vectors and forms. In particular, exterior and Lie derivatives will play an important role. Some knowledge of Lie groups and Lie algebras will also help, though we will briefly discuss the required background.

Kirill Zainoulline: Algebraic Geometry

11th January, 2021

This is a 23-lecture course, with each lecture being around 80 minutes, given online by Kirill Zainoulline. It gives an introduction to algebraic geometry.

A brief overview of commutative algebra: rings and ideals, Nakayama's Lemma, localization, Krull-dimension, direct-limits, integral dependence. Toward algebraic varieties: Regular functions, algebraic sets, Hilbert's Nullstellensatz, Zariski topology, ringed spaces, affine and projective varieties. Toward sheaves and group schemes: functors of points, Grothendieck topologies, representable functors, group schemes, tori, Grassmannians, torsors and twisted forms, quadrics and Severi-Brauer varieties.

Ilijas Farah: Massive C*-Algebras

11th January, 2021

This is a 22-lecture course, with each lecture being around 90 minutes, given by Ilijas Farah.

The route to understanding separable C*-algebras frequently involves a detour via non-separable C*-algebras, such as the Calkin algebra, the asymptotic sequence algebras, ultrapowers, and ultraproducts. Some basic ideas from logic can be used to analyse these massive C*-algebras. Among other things, we will see that the existence of outer automorphisms of the Calkin algebra depends on the set-theoretic axioms.