Jacques Gasqui - Hubert Goldschmidt

Infinitesimal isospectral deformations of the Grassmannian of 3-planes in R6

Memoires de la SMF 108 (2007), vi+92 pages

Resume :

Deformations infinitesimales isospectrales de la grassmannienne des 3-plans dans R6
Ce memoire a pour cadre la grassmannienne des n-plans de , avec , et son espace reduit qui est l'espace symetrique irreductible, quotient de par l'involution envoyant un n-plan sur son orthogonal. Un de nos principaux resultats est la construction de deformations infinitesimales isospectrales non triviales sur obtenant ainsi le premier exemple d'espace symetrique irreductible reduit et non infinitesimalement rigide. Nous donnons aussi un critere d'exactitude pour les formes differentielles de degre 1 sur mettant en jeu la nullite d'une transformee de Radon.

Mots clefs : Espace symetrique, grassmannienne, transformee de Radon, deformation isospectrale, forme symetrique, condition de Guillemin

Abstract:

We study the real Grassmannian of n-planes in , with , and its reduced space. The latter is the irreducible symmetric space , which is the quotient of the space under the action of its isometry which sends a n-plane into its orthogonal complement. One of the main results of this monograph asserts that the irreducible symmetric space possesses non-trivial infinitesimal isospectral deformations; it provides us with the first example of an irreducible reduced symmetric space which admits such deformations. We also give a criterion for the exactness of a form of degree one on in terms of a Radon transform.

Key words: Symmetric space, Grassmannian, Radon transform, infinitesimal isospectral deformation, symmetric form, Guillemin condition

Joseph Ayoub

les six operations de Grothendieck et le formalisme des cycles evanescents dans le monde motivique (II)

Asterisque 315 (2007), vi+362 pages

Resume :

Ce deuxieme volume regroupe les chapitres 3 et 4 de notre etude de la fonctorialite des categories homotopiques stables des schemas. Dans le volume precedent, nous nous sommes concentres sur les six operations f*, f*, f!, f!, et et leurs proprietes de constructibilite et d'exactitude.

On commence ce volume par la construction des foncteurs motifs proches , analogues motiviques des foncteurs cycles proches bien connus en cohomologie etale. On etend ensuite le formalisme des cycles evanescents a ces foncteurs. En particulier, on calcule l'effet du foncteur dans le cas ou f est a reduction semi-stable. On montre aussi que les preservent les motifs constructibles, qu'ils commutent au produit tensoriel exterieur et aux foncteurs de dualite. On definit ensuite un operateur de monodromie et on montre qu'il est nilpotent.

Le dernier chapitre, de nature differente des trois autres, reprend en detail la construction de la categorie homotopique stable des S-schemas.

Mots clefs : Motifs, six operations de Grothendieck, dualite de Verdier, cycles evanescents, -homotopie des schemas, categories de modeles

Abstract:

The Grothendieck six operations and the vanishing cycles formalism in the motivic world (II)
This second volume contains chapter 3 and 4 of our study of the functoriality of the stable homotopy categories of schemes. In the previous volume, we concentrated on the six operations f*, f*, f!, f!, and , their constructibility and exactness.

This volume begins with the construction of the nearby motive functors which are the analogue of the nearby cycles functors, well-known in etale cohomology. We then extend the vanishing cycles formalism to these functors. In particular, we compute the effect of the functor in the case where f has semi-stable reduction. We show also that preserve constructible motives and commute with external tensor product and duality. We then define a monodromy operator and prove that this operator is nilpotent.

The last chapter, which is of different nature than the previous ones, recall in full details the construction of the stable homotopy category of S-schemes.

Key words: Motives, Grothendieck six operations, Verdier duality, vanishing cycles, -homotopy theory, model categories

IMPAN Lecture Notes is an irregular
book series based on selected lectures delivered at the Institute of Mathematics, Polish Academy of Sciences.
To appear:

Peter Imkeller

Malliavin's calculus and applications in stochastic control and finance

IM PAN Lecture Notes, Vol. 1,
Warsaw 2008,

From the cover:
Malliavin's calculus alias the stochastic calculus of variations nowadays finds numerous applications
in stochastic analysis and finance, ranging from enhancements of the speed of convergence of Monte-Carlo
algorithms for stochastic equations to the fine structure of solutions of stochastic control problems
in backward stochastic differential equations (BSDE).

We develop this calculus by starting with everybody's notion of differential calculus on
finite dimensional Euclidean space. We generalize it to infinite dimensional sequence space,
and use a natural isomorphism between sequence and path space to carry it over to canonical Wiener space.
In a generalized version of the Clark-Ocone representation formula it is seen to provide
the right framework to interpret solutions of BSDE.

Peter Imkeller is professor for Probability at Humboldt-Universitat zu Berlin since 1996.
His main areas of interest range from stochastic analysis and dynamics, with a
particular focus in the statistical and probabilistic analysis of metastability in paleoclimatic
time series and low-dimensional climate models, to stochastic finance, with a
focus on risk assessment in insurance and environment.