Yann Ollivier

Page Web professionnelle/Professional Web page

The reasonable man adapts himself to the world; the unreasonable man persists in trying to adapt the world to himself. Therefore, all progress depends on the unreasonable man.

G. B. Shaw

I am currently a CNRS "Chargé de recherche" (research scientist) at UMPA, École normale supérieure de Lyon. If this page is up-to-date, I'm 31.

This academic year I am visiting the TAO team (machine learning, optimization, evolutionary computing) at the computer science department in Orsay.

Enseignants-chercheurs : vrai ou faux ? Texte sur ma page Web personnelle, qui n'engage pas mes employeurs.

My mathematical interests are in probability and geometry in the broad sense (though I'm a priori interested in any field of mathematics). More specific research topics include Markov chains, Ricci curvature, concentration of measure, random groups, hyperbolic groups, and general relativity.

You may also visit my personal Web page. In particular, it contains non-professional, wide-audience mathematical texts and mathematical programs.

Scientific publications arranged by topic:

Years given in parentheses denote redaction time. For published texts, the year given without parentheses is the official publication year (i.e. the year of actual printing on paper).

Markov chains, concentration, Ricci curvature

(2009)

Curvature, concentration, and error estimates for Markov chain Monte Carlo

With Aldéric Joulin.

pdf (277K)
ps.gz

Ann. Probab., to appear.

Under a discrete positive curvature assumption, we get explicit finite-time bounds for convergence of empirical means in the Markov chain Monte Carlo method. This allows to improve known bounds on several examples such as the Ornstein-Uhlenbeck process, waiting queues, spin systems at high temperature or Brownian motion on positively curved manifolds.

(2008)

A survey of Ricci curvature for metric spaces and Markov chains

pdf (410K)
ps.gz (230K)

Expository manuscript, to appear in Proceedings of the 1st Math. Soc. Japan Seasonal Institute, Kyoto, 2008.

This is a gentle introduction to the context and results of my preprint Ricci curvature of Markov chains on metric spaces. It begins with a description of classical Ricci curvature in Riemannian geometry, as well as a reminder for discrete Markov chains. A number of open problems are mentioned.

2009 (2007)

Ricci curvature of Markov chains on metric spaces

pdf (564K)
ps.gz (291K)

J. Funct. Anal. 256 (2009), n°3, 810--864.
Some of the results are announced in the note Ricci curvature of metric spaces, C.R. Math. Acad. Sci Paris 345 (2007), n°11, 643--646.

We define the Ricci curvature of metric measure spaces as the amount by which small balls are closer (in transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back the value of Ricci curvature of a tangent vector. For example, the discrete cube is positively curved, as well as processes with positive Ricci curvature in the Bakry-Émery sense. Positive Ricci curvature is shown to imply a spectral gap, a Lévy-Gromov Gaussian concentration theorem and a kind of logarithmic Sobolev inequality.
(Erratum: In theorem 49 (and only there), we need to assume that X is locally compact. This is omitted in the published version.)

2007 (2006)

With Pierre Senellart.

pdf (242K)
ps.gz

Proc. AAAI-07, Vancouver, Canada, July 2007, 1427--1433.

The goal is to find "related nodes" to a given node in a graph/Markov chain (e.g. a graph of Web pages). We propose the use of discrete Green functions, a standard tool from Markov chains. We test this method versus more classical ones on the graph of Wikipedia. Accompanying Web site.

(2004)

Groupe de travail "Concentration de la mesure"

html

Notes (in French) of a working seminar I organized in 2004 about the various approaches to concentration of measure (geometrical, analytical, probabilistic).

2003 (2000)

Rate of convergence of crossover operators

pdf (220K)
ps.gz

Random Struct. Algor. 23 (2003), n° 1, 58--72.

We study the genetical dynamics of mating (crossover) operators in finite populations. We prove that the convergence to equilibrium is exponential but that there is a non-eventually vanishing bias depending on population size.

(1999)

Diamètre observable des sous-variétés de $S^n$

et $\mathbb{C}\! P^n$

pdf (268K)
ps.gz

Master's dissertation, advised by P. Pansu.

I explain some of the results of Gromov's "Chapter 3 1/2", about the observable diameter and concentration of measure on submanifolds of the sphere and on some projective complex algebraic varieties.

Random groups, geometric group theory

2005

A January 2005 invitation to random groups

pdf (1658K)
ps.gz (1130K)

Ensaios Matemáticos [Mathematical Surveys], 10. Sociedade Brasileira de Matemática, Rio de Janeiro, 2005.

Small book reviewing currently known facts and numerous open problems about random groups. This text is aimed at those having some basic knowledge of geometric group theory and wanting to discover the precise meaning of "random groups" and hopefully provides a roadmap to working on the subject.
January 2010 random groups updates

How to order a printed copy.

(2005)

Cubulating random groups at density less than $1/6$

With Daniel T. Wise.

pdf (498K)
ps.gz (324K)

Trans. Amer. Math. Soc., to appear.

Random groups at density $d<1/5$

have a codimension-1 subgroup and thus do not have property $(T)$

. Moreover at density $d<1/6$

they act freely and cocompactly on a CAT(0) cube complex and thus have the Haagerup property.

2007 (2004)

Some small cancellation properties of random groups

pdf (223K)
ps.gz

Internat. J. Algebra Comput. 17 (2007), n°1, 37--51.

We prove that random groups at density d satisfy an isoperimetric inequality with sharp constant $1-2d$

. Also when $d<1/5$

the random presentation satisfies the Dehn algorithm, whereas it does not for $d>1/5$

. We use a somewhat improved local-global criterion.

2007 (2004)

Kazhdan groups with infinite outer automorphism group

With Daniel T. Wise.

pdf (227K)
ps.gz (250K)

Trans. Amer. Math. Soc. 359 (2007), n°5, 1959--1976.

We prove that any countable group embeds in $Out(G)$

for some group G with property $(T)$

(this answers a question of Paulin). We also get Kazhdan groups which are not Hopfian, or not coHopfian. For this we use the graphical small cancellation technique of Gromov.

2005 (2004)

Collapsing of random quotients of hyperbolic groups with torsion

pdf
ps.gz

C.R. Math. Acad. Sci. Paris 341 (2005), n°3, 137--140.

Random quotients of hyperbolic groups with "harmful" torsion collapse at densities smaller than expected.

(2003)

Spectral interpretations of Property $(T)$

pdf
ps.gz

Expository manuscript.

Simple proof that property $(T)$

is equivalent to a uniform spectral gap for the random walk operator with values in unitary representations, and of the $\lambda_1>1/2$ criterion.

(2003)

Cayley graphs containing expanders, after Gromov.

pdf
ps.gz

Expository manuscript.

Description of the steps in Gromov's construction of a group whose Cayley graph contains (more or less isometrically) a family of expanders, with some technical points.

2006 (2003)

Growth exponent of generic groups

pdf (327K)
ps.gz

Comment. Math. Helv. 81 (2006), n° 3, 569--593.

The growth exponent of a generic group (in the density model) is arbitrarily close to that of the free group. This answers a question of Grigorchuk and de la Harpe.

2005 (2003)

Cogrowth and spectral gap of generic groups

pdf (309K)
ps.gz

Ann. Inst. Fourier (Grenoble) 55 (2005), n° 1, 289--317.

We show that the spectral gap of the Laplacian (or random walk operator) on a generic group is very probably almost as large as in a free group. Moreover this spectral gap is robust under random quotients of hyperbolic groups (in the density model).

2006 (2003)

On a small cancellation theorem of Gromov

pdf (228K)
ps.gz

Bull. Belg. Math. Soc. 13 (2006), n° 1, 75--89.

If a group presentation has for relators the words read on the cycles of a labelled graph, and if the labelling satisfies a generalized small cancellation condition, then the group is hyperbolic.

2004 (2002)

Sharp phase transition theorems for hyperbolicity of random groups

pdf (838K)

GAFA, Geom. Funct. Anal. 14 (2004), n° 3, 595--679.

Generalisation of Gromov's result that a random group is infinite hyperbolic if the number of relators is less than some critical value and trivial above this value: this is still true when taking a random quotient of a hyperbolic group. The critical value can be computed and depends on the properties of the random walk on the group.

2003 (2002)

Critical densities for random quotients of hyperbolic groups

C.R. Math. Acad. Sci. Paris 336 (2003), n° 5, 391--394.

Short paper announcing the results of Sharp phase transition theorems for hyperbolicity of random groups.

General relativity and statistical physics

(2008)

On the kinetic Fokker-Planck equation in Riemannian geometry

With Fabrice Debbasch and Cédric Villani.

In preparation.

2009 (2008)

Large-scale non-linear effects of fluctuations in relativistic gravitation

With Claire Chevalier and Fabrice Debbasch.

pdf (327K)

Nonlinear Analysis: Theory, Methods & Applications 71 (2009), n°12, e199--e202.

This is a survey of our results on the large-scale effects of fluctuations in cosmology. It contains a synthesis of results from the texts below.

2009 (2008)

Multiscale cosmological dynamics

With Claire Chevalier and Fabrice Debbasch.

pdf
ps.gz

Physica A 388 (2009), 5029--5035.

Since general relativity is non-linear, fluctuations (e.g. gravitational waves or irregularities in matter density) around a given mean produce non-zero average effects. For example, we show that gravitational waves of currently undetectable amplitude and frequency could influence expansion of the universe roughly as much as the total matter content of the universe. This should be taken into account when considering dark matter/dark energy problems.

2005 (2004)

Observing a Schwarzschild black hole with finite precision

With Fabrice Debbasch.

pdf
ps.gz

Astron. Astrophys. 433 (2005), n°2, 397--404.

Observing a fluctuating black hole yields the impression that it is surrounded with ``apparent matter'' of negative energy.

Miscellaneous

(2009)

Habilitation à diriger des recherches

pdf (515K)

The introduction (in French) of my Habilitation manuscript, which contains a presentation of my research for a general mathematical audience.

(2003)

Thèse de doctorat

pdf (1210K)
ps.gz (574K)

Advisors: M. Gromov and P. Pansu

My PhD dissertation. In addition to an overall presentation, it mainly contains some of the texts above.

Other expository texts and lecture notes

Various texts on my personal Web page, eg: the various meanings of entropy in mathematics; introduction to concentration of measure; presentation of different cohomology theories in various settings; introductions to geometric group theory; and more. (Mostly in French.)

Any comment? Write to yann.ollivier (domain:) normalesup.org
Last modified: February 21, 2010.