**Time**: Thursdays 10am-12pm, starting January 21st 2021, finishing March 11th.

The existence of a coarse embeddings of a group or space into a space with "nice geometry" has important consequences in many areas of mathematics and computer science. The focus of this course is on invariants which behave monotonically with respect to coarse embeddings - such as growth and asymptotic dimension - which can therefore be used to prove that there is no coarse embedding between a pair of spaces.

**Course outline**:

- growth function: definition and coarse invariance; nilpotent groups have polynomial growth; examples of groups with exponential growth; Grigorchuk's first group; open questions.

- asymptotic dimension: definitions and coarse invariance; calculations for trees, Euclidean and real hyperbolic spaces; product formula; Hurewicz formula; open questions.

- separation and Poincaré profiles: definitions and coarse invariance; expander graphs; calculations for trees, Euclidean and real hyperbolic spaces; upper bounds using asymptotic dimension; conformal dimension; Diestel-Leader graphs and the thick-thin "dichotomy"; open questions.

- (Co)homological dimension: definition and coarse invariance.

January 21st. Lecture 1: Motivation for coarse geometry. Motivating Questions.
Lecture 2: The growth function, definition and coarse invariance. Groups with polynomial growth part 1.

January 28th. Lecture 1: Brief recap. Groups with polynomial growth, part 2.
Lecture 2: Assouad's embedding theorem. Groups with exponential growth, part 1.

February 4th. Lecture 1: Milnor's Theorem. Groups with exponential growth.
Lecture 2: Intermediate Growth. Grigorchuk's First Group.

February 11th. Lecture 1: Asymptotic dimension. Product formula.
Lecture 2: Hurewicz formula. Calculations of asymptotic dimension.

February 18th. Lecture 1: Sperner's Lemma. Asymptotic dimension of Euclidean spaces.
Lecture 2: Finite asymptotic dimension for hyperbolic spaces.

February 25th. Lecture 1: Topological Couplings. Coarse invariance of (co)homological dimension.
Lecture 2: Measure couplings and measure subgroups. Coarse invariance of isoperimetric profiles.

March 4th. Lecture 1: Separation profile. Definition and coarse invariance.
Lecture 2: Caculations of separation profiles: trees and Euclidean spaces

March 11th. Lecture 1: Separation profiles and expanders.
Lecture 2: Overview of Poincaré profiles.

Lecture 2: More examples and applications. Unannotated Slides. Annotated Slides.

January 17th (Definition of tdlc groups, Van Dantzig's theorem, classification of tdlc groups up to homeomorphism. N.B. The lemma on page 13 should include second countability as an assumption, separability is not sufficient.)

January 24th (Examples: Aut(Γ) continued, direct, semi-direct, restricted and wreath products.)

January 31st (Cayley-Abels graphs, definition and existence for compactly generated groups)

February 7th (Cayley-Abels representations, uniqueness of Cayley-Abels graphs up to quasi-isometry)

February 14th (Locally elliptic groups, Caprace-Monod Minimal Normal Subgroups Theorem)

February 21st (Caprace-Cornulier Non-Embedding Theorem, Topological chief factors)

February 28th (Preliminaries of essentially chief series)

March 7th (Existence and uniqueness of essentially chief series)

Clarifications up to and including Mar 7th Second countability versus separability. More detail in the proof of the Minimal Normal Subgroup Theorem.

A very recent survey of important open problems in the area has just been posted by Pierre-Emmanuel Caprace and Nicolas Monod, it can be downloaded from here.