Focused weeks 2011
A serie of focused weeks will be held at the Laboratoire Jean Leray.
The informations on the lectures can be found below.
Summary of participants:
- Vincent Borrelli: 29 March - 01 April
- Frédéric Bourgeois 23 - 27 May
- Baptiste Chantraine 23 - 27 May
- Marc Chaperon 23 - 27 May
- Emmanuel Ferrand 23 - 27 May
- David Gay 20 - 24 June
- Brad Henry 23 - 27 May
- Ko Honda: 16 - 27 May
- Yael Karshon: 2 - 6 May
- Sikimeti Ma'u 20 - 24 June
- Tim Perutz 20 - 24 June (TBC)
- Fran Presas: 14 - 18 March
- Dan Rutherford 23 - 27 Ma
- Josh Sabloff 23 - 27 May
- David Théret 23 - 27 May
- Lisa Traynor 23 - 27 May
- Thomas Vogel: 14 - 18 March
- Chris Wendl: 9 - 13 May
- Katrin Wehrheim 20 - 24 June
Hengel structures (March 14th - March 18th)
Participants :
- Fran Presas: 14 - 18 March
- Thomas Vogel: 14 - 18 March
Convex integration (March 29th - April 1st)
- Vincent Borrelli : minicourse on convex integration
-
mardi 29 mars 17h, salle Hypatia -- Séance 1 : Qu'est-ce que le h-principe ? (notes I)
Résumé : On présente le h-principe ainsi que quelques théorèmes célèbres
qui en relèvent. On passe en revue trois méthodes générales permettant de
demontrer l'existence d'un h-principe en pointant pour chacune leurs
qualités et leurs défauts.
mercredi 30 mars 17h, salle Hypatia -- Séance 2 : L'intégration convexe 1-dimensionnelle (notes II)
Résumé : L'intégration convexe est l'une des méthodes permettant d'établir
l'existence d'un h-principe. Elle a été mise au point par Gromov et elle
s'est avérée à la fois générale et féconde. Le cas 1-dimensionnel en
concentre toutes les idées.
jeudi 31 mars 17h, salle hypatia -- Séance 3 : Le h-principe pour les relations amples (notes III)
Résumé : On passe de l'intégration convexe 1-dimensionnelle à
l'intégration convexe multi-dimensionnelle. On établit un théorème
d'existence d'un h-principe pour les relations différentielle dites
amples. De ce théorème se déduisent un certain nombre de résultats
célèbres dont le fameux retournement de la sphère de Smale. Une version
plus élaborée
permet également de retrouver le théorème des plongements isométriques de
Nash-Kuiper. Une des conséquences de ce théorème est que l'on peut
retourner la sphère parmi les immersions C1-isométriques.
vendredi 1er avril, séminaire de géométrie 10h15
Intégration convexe, tores plats et visualisation.
En 1954, F. Nash énonce un théorème déconcertant : il n'y a pas
d'obstruction à l'existence de plongements isométriques en petite
codimension. Complété par N. Kuiper, son résultat implique qu'il existe
des plongements isométriques de tores plats dans l'espace euclidien de
dimension trois. Bien sûr, la courbure de Gauss interdit à de tels tores
d'être de classe $, mais ils sont tout de même de classe $ et
possèdent en tout point un espace tangent. Plus tard, en revisitant les
travaux de nombreux géomètres, M. Gromov invente une technique qui
généralise et éclaire de facon extraordinaire la manière dont F. Nash et
N. Kuiper ont construit leurs plongements isométriques : c'est la
technique de l'intégration convexe. A l'aide de cette méthode, une
implémentation est possible et la visualisation des tores plats plats
devient envisageable...
Hamiltonian group actions (May 2nd - May 6th)
- Yael Karshon : minicourse on Hamiltonian group actions
-
mardi 3 mai 14h, salle Hypatia : Introduction to symplectic toric manifolds
Abstract: "what every symplectic geometer needs to know about
symplectic toric manifolds". If there's time, I will include
the proof of Delzant's theorem, that a symplectic toric manifold
is determined up to isomorphism by its momentum polytope.
mercredi 4 mai 10h, salle Hypatia : Introduction to coadjoint orbits
Abstract: I will give basic definitions and facts about coadjoint orbits,
Kostant's theorem that a transitive Hamiltonian group action is a covering
of a coadjoint orbit, and new results and conjectures about the Gromov
width of coadjoint orbits.
vendredi 6 mai 10h, salle hypatia : Counting toric actions on symplectic four-manifolds.
Abstract: I will explain recent work with Liat Kessler and Martin
Pinsonnault: we use holomorphic curves to reduce the question of
counting toric actions to a combinatorial question.
- Yael Karshon : Hamiltonian group actions (Colloquium)
Jeudi 5 mai, 17h, salle de séminaires
Abstract: This talk is about actions of compact Lie groups
on symplectic manifolds, specifically those that are generated
by momentum maps. Such group actions model symmetries in classical
mechanics and also arise in purely mathematical contexts.
The momentum map encodes manifold information into polytopes and graphs.
The purpose of the talk is to give a taste of the field
through a sample of examples and results, old and new.
Holomorphic Curves in Symplectic Fillings (May 9th - May 13th)
Participants :
- Chris Wendl : minicourse on
Holomorphic Curves in Symplectic Fillings
- ABSTRACT:
I will explain some joint work in progress with Jeremy Van Horn-Morris
and Sam Lisi (including also some recent work with Janko Latschev and
Klaus Niederkrüger) on classifying symplectic fillings of contact
3-manifolds. The central concept is a generalization of the notion of
a supporting open book decomposition, called a "spinal" open book,
which gives rise to a foliation of the symplectization by
J-holomorphic curves. Whenever the decomposition contains a planar
page, this makes possible various computations in Embedded Contact
Homology and Symplectic Field Theory, and it implies that the
deformation classes of symplectic fillings can be identified with
diffeomorphism classes of bounded Lefschetz fibrations. As an
example, this permits a complete classification of the strong fillings
of S^1-invariant contact structures on circle bundles. Spinal open
books also naturally lead to a new type of contact surgery, related to
Eliashberg's symplectic capping construction, which yields non-exact
symplectic cobordisms that one can use to give "low-tech" proofs of a
wide range of non-fillability results.
-
Tentatively, the presentation will be divided into the following five lectures:
- overview of spinal open books, definitions and results (1 hour)
- finite energy foliations, ECH, SFT and computations (2 hours)
- spinal open books and bounded Lefschetz fibrations (1 hour)
- strong vs. weak fillings (1 hour)
- spine removal surgery and symplectic cobordisms (1 hour)
Generating functions (May 23rd - May 27th)
Participants :
- Frédéric Bourgeois 23 - 27 May
- Baptiste Chantraine 23 - 27 May
- Marc Chaperon 23 - 27 May
- Emmanuel Ferrand 23 - 27 May
- Brad Henry 23 - 27 May
- Marco Mazzucchelli 23 -27 May
- Petya Pushkar 23 - 27 May
- Dan Rutherford 23 - 27 Ma
- Josh Sabloff 23 - 27 May
- Jean-Claude Sikorav 26 - 27 May
- David Théret 23 - 27 May
- Lisa Traynor 23 - 27 May
- Lisa Traynor and Josh Sabloff : minicourse on
Applications of Generating Families to Legendrian and Lagrangian
Submanifolds
- ABSTRACT:
The theory of generating families is classical. Since the time
of Hamilton and Jacobi, it has been known that Lagrangian submanifolds of
symplectic manifolds and Legendrian submanifolds of contact manifolds can
be locally described, or “generated”, by functions. In the early 1990s,
Viterbo showed that generating families could be used to study global
phenomena. In this series of lectures, we will explore how generating
families -- and homological invariants derived from the difference between
two generating families -- can be employed to study a variety of problems
in the theory of Legendrian and Lagrangian submanifolds. Topics will
include:
- Invariants for links and knots from generating families;
- Restrictions to the shapes of hyperplane slices of Lagrangian
submanifolds;
- Existence of and obstructions to Lagrangian cobordisms between
Legendrian submanifolds.
-
- Lecture 1 (May 24, 9.30 - 10.30, Salle Eole): An Introduction to
Generating Families and an Overview to the Topics of the Mini-Course
- Lecture 2 (May 24, 10.45 - 11.45, Salle Eole): Invariants for Knots and Links
- Lecture 3 (May 25, 10.45 - 11.45, Salle Eole): Slices of
Lagrangian Submanifolds
- Lecture 4 (May 26, 9.30 - 10.30, Salle Hypathia): Obstructions to the
Existence of Lagrangian Caps of Legendrian Submanifolds
- Lecture 5 (May 26, 10.45 - 11.45, Salle Hypathia): A Generating Family
TQFT and Constructions of Lagrangian Cobordisms
- Lecture 6 (May 27, 10.45 - 11.45, Salle Hypathia): Open Problems and New
Directions for Generating Families
-
Talk Marc Chaperon (May 23, 10.30 - 11.30, Salle Eole):
Title: Weak solutions of Hamilton-Jacobi equations, following Wei Qiaoling
Abstract: The "minimax" weak solutions of Hamilton-Jacobi equations
introduced by the speaker in 1990 (and called "variational" by Claude
Viterbo) coincide with the Crandall-Lions viscosity solutions in the case
of Hamiltonians that are convex with respect to the vertical variable, but
differ from them in general. In her thesis,Qiaoling Wei shows that the
solutions of a Cauchy problem obtained by iterating the minimax procedure
tend to the viscosity solution when the step of the subdivision of the
time interval tends to zero.
- Talk Dan Rutherford (May 25, 9.30 - 10.30, Salle Eole):
Title: A combinatorial Legendrian knot DGA from generating families
Abstract : This is joint work with Brad Henry. A generating family for a
Legendrian knot $ in standard contact $\mathbb{R}^3$ is a family of
functions $ whose critical values coincide with the front projection
of $. Pushkar introduced combinatorial analogs of generating families
which have become known as Morse complex sequences. In this talk, I will
describe how to associate a differential graded algebra (DGA) to a
Legendrian knot with chosen Morse complex sequence. In addition, I will
discuss the geometric motivation from generating families and the
relationship with the Chekanov-Eliashberg invariant.
-
Talk Brad Henry (May 27, 9.30 - 10.30, Salle Hypathia):
Title: Connections between the Chekanov-Eliashberg DGA, Morse complex
sequences, and the generating family DGA of Legendrian knots
Abstract: This talk compliments and, in part, continues the discussion
from Dan Rutherford's talk on the generating family differential
graded algebra. A Morse complex sequence, abbreviated MCS, is a finite
sequence of chain complexes associated to the front projection L of a
Legendrian knot. Its definition is geometrically motivated by the
fiber-wise Morse–Smale chain complexes coming from a suitably generic
generating family and metric for the knot. An MCS can be encoded
graphically using certain vertical markers on L.
After placing an equivalence relation on the set of MCSs on L we
describe a surjective map from the equivalence classes to the set of chain
homotopy classes of augmentations of L_N, where L_N is the Ng resolution
of L. In the case of Legendrian knot classes admitting
representatives with two-bridge front projections, this map is
bijective. We also exhibit two standard forms for MCSs and describe their
usefulness. If an MCS is in what we call A-form, then the
generating family DGA of this Morse complex sequence is stable tame
isomorphic to the CE-DGA of L_N. This work is partly joint with Dan
Rutherford (Duke). The definition of an MCS and the equivalence
relation originate from work of Petya Pushkar.
Holomorphic quilts (June 20th - June 24th)
Participants :
- David Gay 20 - 24 June
- Sikimeti Ma'u 20 - 24 June
- Tim Perutz 20 - 24 June (TBC)
- Katrin Wehrheim 20 - 24 June
Program
- Lundi (Salle Éole) 9h30 -- 12h00 Katrin Wehrheim : Quilted Floer homology
- Mardi (Salle Hypatia) 9h30 -- 12h00 David Gay : Morse 2-functions
- Mercredi (Salle Hypatia) 9h30 -- 12h00 Tim Perutz : Lagrangian Matching invariants
- Jeudi (Salle Hypatia) 9:30 -- 12h00 Sikimeti Ma'u : Functors and bimodules
- Vendredi (Salle Hypatia) 9:30 -- 12h00 : À décider selon les préférences du publique.
