Metric spaces Coarse spaces Growth and amenability Translation algebras Coarse algebraic topology Coarse negative curvature Limits of metric spaces Rigidity Asymptotic dimension Groupoids and coarse geometry Coarse embeddability Bibliography.

Lectures on coarse geometry

http://www.personal.psu.edu/ndh2/math/Papers_files/Higson,%20Lafforgue,%20Skandalis%20-%202002%20-%20Counterexamples%20to%20the%20Baum-Connes%20conjecture.pdf

Counterexamples to the baum-connes conjecture

We find lower bounds on the topology of the fibers $${F^{-1}(y)\subset X}$$
 of continuous maps F : X → Y in terms of combinatorial invariants of certain polyhedra and/or of the cohomology algebras H*(X). Our exposition is conceptually related to but essentially independent of Part 1 of the paper.

Singularities, Expanders and Topology of Maps. Part 2: from Combinatorics to Topology Via Algebraic Isoperimetry

The main result of this paper is the existence of Galois representations associated with the mod p (or mod p m ) cohomology of the locally symmetric spaces for GLn over a totally real or CM eld, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for GLn as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of Shimura varieties. Thus, we prove new results on the p-adic geometry of Shimura varieties (of Hodge type). Namely, the Shimura varieties become perfectoid when passing to the inverse limit over all levels at p, and a new period map towards the ag variety exists on them, called the Hodge-Tate period map. It is roughly analogous to the embedding of the hermitian symmetric domain (which is roughly the inverse limit over all levels of the complex points of the Shimura variety) into its compact dual. The Hodge-Tate period map has several favorable properties, the most important being that it commutes with the Hecke operators away from p (for the trivial action of these Hecke operators on the ag variety), and that automorphic vector bundles come via pullback from the ag variety.

/pdf/on-torsion-in-the-cohomology-of-locally-symmetric-varieties-tx5m3qdazx.pdf

On torsion in the cohomology of locally symmetric varieties

The coarse Baum–Connes conjecture and groupoids

Je suis parvenu dans ma these a construire une k-theorie bivariante pour les algebres de banach. Cela m'a permis de demontrer la conjecture de baum-connes pour les groupes de lie semi-simples et les groupes reductifs p-adiques, ainsi que pour les sous-groupes discrets de type fini de sl#3(f), avec f une extension finie de q#p et pour les sous-groupes discrets cocompacts de sp(n, 1), sl#3(r) et sl#3(c). J'ai aussi demontre une conjecture analogue a la conjecture de baum-connes pour les algebres l#1 (et plus generalement toutes les bonnes completions) des sous-groupes fermes des groupes de lie semi-simples et des groupes reductifs p-adiques.

K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes

For any reductive group G over a global function field, we use the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence for G in the direction \"from automorphic to Galois\". Moreover we obtain a canonical decomposition of the spaces of cuspidal automorphic forms indexed by global Langlands parameters. The proof does not rely at all on the Arthur-Selberg trace formula.

Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale

We show a strong form of property (T) for some real or p-adic semisimple groups. This strong form of property (T) means that the trivial representation is isolated among representations with a small exponential growth in Hilbert spaces or even in uniformly convex Banach spaces. We show that hyperbolic groups do not have strong property (T). We construct families of expanders which do not imbed uniformly in any uniformly convex Banach space

Un renforcement de la propriété (T)

The overlap number of a finite (d + 1)-uniform hypergraph H is the largest constant c(H) is an element of (0, 1] such that no matter how we map the vertices of H into R-d, there is a point covered by at least a c(H)-fraction of the simplices induced by the images of its hyperedges. Motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, we address the question whether or not there exists a sequence {H-n}(n=1)(infinity) of arbitrarily large (d + 1)-uniform hypergraphs with bounded degree for which inf(n >= 1) c(H-n) > 0. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of (d + 1)-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant c = c(d). We also show that, for every d, the best value of the constant c = c(d) that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete (d + 1)-uniform hypergraphs with n vertices, as n -> infinity. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any h, s and any epsilon > 0, there exists K = K(epsilon, h, s) satisfying the following condition. For any k >= K and for any semi-algebraic relation R on h-tuples of points in a Euclidean space R-d with description complexity at most s, every finite set P subset of R-d has a partition P = P-1 boolean OR ... boolean OR P-k into k parts of sizes as equal as possible such that all but at most an epsilon-fraction of the h-tuples (P-i1, ... , P-ih) have the property that either all h-tuples of points with one element in each P-ij are related with respect to R or none of them are.

/pdf/overlap-properties-of-geometric-expanders-47z7jm8sb0.pdf

Vincent Lafforgue

Papers

Counterexamples to the baum-connes conjecture

K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes

Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale

Un renforcement de la propriété (T)

Overlap properties of geometric expanders