Konstantinos Tyros (Université d'Athènes) : A combinatorial approach to nonlinear spectral gaps

Lieu : salle Kampé de Fériet, 1er étage, Bâtiment M2

Date : vendredi 28 février 2025

Heure : 14h

Orateur : Konstantinos Tyros

Affiliation : Université nationale et capodistrienne d'Athènes

Titre : A combinatorial approach to nonlinear spectral gaps

--- Résumé ---

A major open problem due to Pisier and Mendel/Naor asks whether every regular expander graph G satisfies a discrete Poincaré inequality for functions taking values in a Banach space (X, ∥ · ∥_X) with finite cotype q ⩾ 2, that is, there exists a constant γ > 0 such that for every f : V (G) → X we have

E_x,y∈V(G) ∥f(x) − f(y)∥_X ⩽ γE_{x,y}∈E(G) ∥f(x) − f(y)∥_X.

Works of Odell–Schlumprecht (1994), Ozawa (2004) and Naor (2014) yield a positive answer for spaces having an unconditional basis, in addition to finite cotype. However, little is known with respect to quantitative estimates: the aforementioned results provide an estimate for the Poincaré constant that depends super-exponentially on q.

In this talk, we shall present a novel combinatorial method for obtaining quantitative nonlinear spectral gaps that relies on a property of regular graphs that we call long-range expansion. In particular we shall discuss the following results.

(1) Every regular graph with the long-range expansion property satisfies a discrete Poincaré inequality for functions taking values in a Banach space with an unconditional basis and cotype q, with a Poincaré constant proportional to q¹⁰. This estimate is nearly optimal.

(2) For any integer d ⩾ 10, a uniformly random d-regular graph satisfies the long-range expansion property with high probability.

This is joint work with Dylan Altschuler, Pandelis Dodos and Konstantin Tikhomirov.