Zurich Lectures in Advanced Mathematics
ISBN 978-3-03719-063-0
June 2008, 334 pages, softcover, 17 x 24 cm.
The central objects in the book are Lagrangian submanifolds and their invariants, such as Floer homology and its multiplicative structures, which together constitute the Fukaya category. The relevant aspects of pseudo-holomorphic curve theory are covered in some detail, and there is also a self-contained account of the necessary homological algebra.
Generally, the emphasis is on simplicity rather than generality. The last part discusses applications to Lefschetz fibrations, and contains many previously unpublished results. The book will be of interest to graduate students and researchers in symplectic geometry and mirror symmetry.
EMS Series of Lectures in Mathematics
ISBN 978-3-03719-018-0
June 2008, 119 pages, softcover, 17 x 24 cm.
So what is all the fuss about wavelets?
You can find out by reading these notes. They will introduce you to the central concepts surrounding wavelets and their applications. By focusing on the essential ideas and arguments, they enable you to get to the heart of the matter as quickly as possible. They then point you to the appropriate places in the literature for detailed proofs and real applications, so you can continue your study.
They begin with the notion of time-frequency analysis, present the multiresolution analysis and basic wavelet construction, introduce you to the many friends, relatives and mutations of wavelets, and finally give a selection of applications.
They are suitable for beginning graduate students and above. A preliminary chapter containing some of the prerequisite concepts and definitions is included for reference.
Zurich Lectures in Advanced Mathematics
ISBN 978-3-03719-065-4
July 2008, 396 pages, softcover, 17 x 24 cm.
The book starts with an introduction to Geometric Invariant Theory (GIT). The fundamental results of Hilbert and Mumford are exposed as well as more recent topics such as the instability flag, the finiteness of the number of quotients, and the variation of quotients.
In the second part, GIT is applied to solve the classification problem of decorated principal bundles on a compact Riemann surface. The solution is a quasi-projective moduli scheme which parameterizes those objects that satisfy a semistability condition originating from gauge theory. The moduli space is equipped with a generalized Hitchin map.
Via the universal Kobayashi?Hitchin correspondence, these moduli spaces are related to moduli spaces of solutions of certain vortex type equations. Potential applications include the study of representation spaces of the fundamental group of compact Riemann surfaces.
The book concludes with a brief discussion of generalizations of these findings to higher dimensional base varieties, positive characteristic, and parabolic bundles.
The text is fairly self-contained (e.g., the necessary background from the theory of principal bundles is included) and features numerous examples and exercises. It addresses students and researchers with a working knowledge of elementary algebraic geometry.
Asterisque 313 (2007), 122 pages
Dans la preuve de Drinfeld et Lafforgue de la correspondance de Langlands pour GLr sur les corps de fonctions, l'etape la plus difficile consiste a construire des compactifications des espaces de module (ou plutot des champs) de chtoucas de Drinfeld. Pour verifier la proprete, Lafforgue a utilise la reduction semistable a la Langton et une analyse detaillee des proprietes modulaires qui definissent les compactifications. Si l'on espere demontrer la correspondance de Langlands sur les corps de fonctions pour d'autres groupes reductifs, une des questions naturelles est de generaliser les compactifications de Lafforgue dans le contexte d'un groupe reductif arbitraire. Dans ce cas, l'approche de Lafforgue semble difficile a mettre en ?uvre.
Ce texte presente une facon de construire des compactifications des champs de chtoucas a modifications multiples qui generalisent celle des champs de chtoucas de Drinfeld. Notre approche repose sur une methode plus generale : la theorie geometrique des invariants. Dans le cas des champs de chtoucas de Drinfeld, nous retrouvons les compactifications de Lafforgue et decouvrons de nouvelles compactifications, entre autres des compactifications qui sont duales de celles de Lafforgue. De plus, notre methode est susceptible de produire des compactifications des champs de G-chtoucas pour un groupe reductif quelconque G.
Mots clefs : Chtoucas - Varietes modulaires de Drinfeld - Modules des fibres sur les courbes - Corps de fonctions - Theorie geometrique des invariants
Compactification of the stacks of shtukas and geometric invariant theory
In the proof of Drinfeld and Lafforgue of the Langlands correspondence for GLr over function fields, the most difficult part is to construct compactifications of moduli spaces (or stacks) classifying Drinfeld's shtukas. If one hopes to prove the Langlands correspondence over function fields for other reductive groups G, it is natural to generalize the above constructions for the stacks of G-shtukas. However, the approach of Lafforgue based on the semistable reduction due to Langton seems difficult to carry out.
In this article, we use the geometric invariant theory to give a new method to construct compactifications of moduli spaces of Drinfeld's shtukas. This rediscovers not only the compactications constructed by Drinfeld and Lafforgue, but also gives rise to new families of compactications.
ISBN : 978-2-85629-243-3
Asterisque 314 (2007), x+464 pages
Resume :
D'apres les travaux de Morel, Voevodsky et d'autres mathematiciens, on dispose de la notion du type d'homotopie motivique stable d'un S-schema lisse. Cet objet vit dans la categorie homotopique stable des S-schemas .
Ce travail est divise en deux volumes et chaque volume en deux chapitres. Dans le premier chapitre, on montre que du point de vue de la fonctorialite, les categories se comportent comme les categories derivees des faisceaux -adiques. En effet, le formalisme des operations de Grothendieck f*,f*, f! et f! s'etend sans changement au monde motivique. Dans le second chapitre, on etudie les proprietes de constructibilite des motifs et on developpe la dualite de Verdier. Le troisieme chapitre est consacre a la theorie des motifs proches et motifs evanescents. Dans le dernier chapitre, on reprend la construction des categories .
Mots clefs : Motifs, six operations de Grothendieck, dualite de Verdier, cycles evanescents, -homotopie des schemas, categories de modeles
The Grothendieck six operations and the vanishing cycles formalism in the motivic world
By the work of Morel, Voevodsky and other mathematicians, one has the notion of the stable motivic homotopy type of a smooth S-scheme. This object lives in the stable homotopy category of S-schemes .
This work consists of two volumes and each of them is divided into two chapters. In the first chapter, we show that from the view point of functoriality, the categories behave like the derived categories of -adic sheaves. Indeed, the formalism of Grothendieck operations f*, f*, f! and f! extends to the motivic world. In the second chapter, we study the constructibility of motives and develop Verdier duality. The third chapter deals with the theory of nearby motives and vanishing motives. In the last chapter, we give a self-contained treatment of the construction of the categories .
ISBN : 978-2-85629-244-0
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Memoires de la SMF 107 (2006), vi+116 pages
Etude mathematique du modele betaplan: ondes equatoriales et resultats de convergence
On s'interesse a un modele de fluides en rotation rapide, decrivant le mouvement de l'ocean dans la zone equatoriale. Ce modele est connu sous le nom de Saint-Venant, ou systeme ≪ shallow water ≫ , auquel on ajoute un terme de rotation dont l'amplitude est lineaire en la latitude ; en particulier il s'annule a l'equateur. Apres une introduction physique au modele, on decrit les differentes ondes en jeu et l'on etudie en detail les resonances associees a ces ondes. On exhibe ensuite un systeme limite formel (dans la limite d'une forte rotation), obtenu comme d'habitude en filtrant les ondes, et l'on demontre qu'il est bien pose. Enfin on demontre trois types de resultats de convergence : un theoreme de convergence faible vers un systeme geostrophique lineaire, un theoreme de convergence forte des solutions filtrees vers la solution unique du systeme limite, et enfin un resultat ≪ hybride ≫ de convergence forte des solutions filtrees vers une solution faible du systeme limite. En particulier on demontre l'absence d'ondes equatoriales confinees dans le mouvement moyen, quand la rotation augmente.
Mots clefs : Fluides tournants, Betaplan, ordre de Kelvin, ordre de Poincare, ordre de Rossby, confinement equatorial, compacite faible, filtrage, analyse harmonique
We are interested in a model of rotating fluids, describing the motion of the ocean in the equatorial zone. This model is known as the Saint-Venant, or shallow-water type system, to which a rotation term is added whose amplitude is linear with respect to the latitude; in particular it vanishes at the equator. After a physical introduction to the model, we describe the various waves involved and study in detail the resonances associated to those waves. We then exhibit the formal limit system (as the rotation becomes large), obtained as usual by filtering out the waves, and prove its wellposedness. Finally we prove three types of convergence results: a weak convergence result towards a linear, geostrophic equation, a strong convergence result of the filtered solutions towards the unique strong solution to the limit system, and finally a ``hybrid'' strong convergence result of the filtered solutions towards a weak solution to the limit system. In particular we obtain that there are no confined equatorial waves in the mean motion as the rotation becomes large.
ISBN : 978-2-85629-228-0