David Kazhdan

Integration in valued fields

We develop a theory of integration over valued fields of residue characteristic zero. In particular, we obtain new and base-field independent foundations for integration over local fields of large residue characteristic, extending results of Denef, Loeser, and Cluckers. The method depends on an analysis of definable sets up to definable bijections. We obtain a precise description of the Grothendieck semigroup of such sets in terms of related groups over the residue field and value group. This yields new invariants of all definable bijections, as well as invariants of measure-preserving bijections.

Isoperimetric Inequalities for Ramanujan Complexes and Topological Expanders

AbstractExpander graphs have been intensively studied in the last four decades (Hoory et al., Bull Am Math Soc, 43(4):439–562, 2006; Lubotzky, Bull Am Math Soc, 49:113–162, 2012). In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every d there are unbounded degree simplicial complexes of dimension d with these properties. However, a major open problem, formulated by Gromov (Geom Funct Anal 20(2):416–526, 2010), is whether bounded degree high dimensional expanders exist for

Iwahori–Hecke algebras for p-adic loop groups

{d \geq 2}

Affine Gindikin–Karpelevich Formula via Uhlenbeck Spaces

d≥2. We present an explicit construction of bounded degree complexes of dimension

On the depth r Bernstein projector

{d = 2}

Algebraic groups over a 2-dimensional local field: Some further constructions

d=2 which are topological expanders, thus answering Gromov’s question in the affirmative. Conditional on a conjecture of Serre on the congruence subgroup property, infinite sub-family of these give also a family of bounded degree coboundary expanders. The main technical tools are new isoperimetric inequalities for Ramanujan Complexes. We prove linear size bounds on