Lecture Courses in Geometry

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.

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.

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).

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.

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.

David Lindemann: Differential Geometry

20th April, 2020

In the lectures we will cover the basic concepts of modern differential geometry. Differential geometry studies smooth manifolds, that is, geometric objects that, roughly speaking, locally look like ℝn (and whose global topological properties are not too weird, see for example the so-called 'long line'). You already know examples such as the n-sphere Sn, or smooth surfaces in ℝ3 from analysis. On smooth manifolds we will study a number of constructions and structures, such as vector fields, metrics and various curvature concepts. In addition, smooth manifolds are suitable as spaces for ordinary and partial differential equations, which allow different global topological properties compared to regions in ℝn (e.g. PDEs on the Klein bottle or on the real-projective spaces ℝPn). A focus of this lecture will be submanifolds and induced geometric structures. We will also look at some topics from the perspective of the calculus of variations, e.g. geodesics as critical points of the energy functional. This lecture is also expressly suitable for students of physics courses, as differential geometry represents a fundamental theoretical basis for many modern theories in physics (especially ART, gauge theories such as Yang-Mills -Theory, SuSy, SuGra,...).