I will introduce another coarse invariant, closely related to the L2-isoperimetric profile, namely the return probability to the origin. I will introduce such an invariant and prove its links to the L2-isoperimetric profile and Nash inequalities on graphs.
Working group "Geometric group theory and its coarse neighbourhood"
Co-organized with Vincent Dumoncel.
Goal: Get a better understanding on topics interacting with geometric group theory
(dynamics, large-scale geometry, representations, operator algebras, logic, among others).
Format: "Mini-courses" intended to be elementary and accessible to a wide audience. They should last between one and two hours.
Room 1013 of bâtiment Sophie Germain, Tuesday (maybe Monday sometimes), around 4pm.
Online : BBB.
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- Tuesday 15-04-2025 (4pm): Third talk of Juan: Nash Inequalities and Upper bounds on the return probability to the origin.
ABSTRACT: In this last talk, we will see the links between upper bounds on the return probability and the L2-isoperimetric profile via Nash inequalities. - Week of 21-04-2025: Holidays!
- Monday 28-04-2025 or Tuesday 29-04-2025: Corentin Correia will talk about quantitative orbit equivalence for amenable groups.
- Monday 05-05-2025 or Tuesday 06-05-2025: second talk of Corentin.
- Talks of Juan Paucar:
- (2/3) 01-04-2025: Return probability to the origin and L2-isoperimetric profile
Abstract, notes and recording - (1/3) 18-03-2025: Isoperimetric profile
Abstract, notes and recordingI will start with a brief motivation by discussing about Sobolev inequalities. I will proceed to discuss the notion of isoperimetric profile for graphs as well as proving its invariance under quasi-isometries. I will finish by proving the relationship between the isoperimetric profile and the growth for Cayley graphs (Coulhon Saloff-Coste inequality).
- (2/3) 01-04-2025: Return probability to the origin and L2-isoperimetric profile
- Talks of Kostyantyn Krutoy:
- (3/3) 11-03-2025: Operator algebras: uniform Roe algebras of uniformly locally finite spaces
Abstract, notes and recordingIn this talk, we will introduce the uniform Roe algebras of uniform locally finite metric spaces. We will discuss some examples of these algebras and examine their basic properties. We conclude the series of talks with a brief introduction to the rigidity problem of uniform Roe algebras.
- (2/3) 04-03-2025: Coarse Geometry: Property A and Coarse Embeddability into a Hilbert Space
Abstract, notes and recordingIn this talk, we will examine two fundamental properties of coarse spaces: property A and coarse embeddability into a Hilbert space. These properties play a crucial role in the coarse Baum-Connes conjecture and the rigidity problem for Roe algebras. We will demonstrate that they are preserved under coarse equivalences and present illustrative examples. Their significance for the rigidity problem will be addressed in the third talk.
- (1/3) 18-02-2025: Coarse geometry: coarse spaces and coarse maps
Abstract, notes and recordingIn this talk, we will explore the fundamental concepts of coarse geometry and compare them with the framework of geometric group theory introduced by Vincent in the preceding lecture series. We will introduce coarse structures as an abstraction of metric spaces viewed through the lens of large-scale geometry and discuss coarse equivalences, which are maps preserving large-scale properties.
- (3/3) 11-03-2025: Operator algebras: uniform Roe algebras of uniformly locally finite spaces
- Talks of Vincent Dumoncel:
Quasi-isometries of finitely generated groups: invariants and rigidity
Abstractand notes of VincentA finitely generated group is naturally a metric space, that can be studied from a large-scale point of view. Quasi-isometries are the maps preserving this structure, and a natural problem is then to determine whether two finitely generated groups are quasi-isometric. We introduce two invariants that provide answers in some cases: the volume growth and the number of ends. The second part will be devoted to the quasi-isometric rigidity of ℤ.