Stanford Symplectic Geometry Seminar 2004-2005

Schedule of talks:

Monday Oct. 11, 4:00 p.m., 380-383N.

Dieter Kotschick (U. of Munich)

Homology of symplectomorphism groups and extended flux homomorphisms

Abstract: We shall discuss an existence theorem for foliated bundles with symplectic total holonomy, and its relationship to the homology of symplectomorphism groups. It turns out that the classical flux homomorphism on the identity component of the group of symplectomorphisms can sometimes be extended to the full symplectomorphism group not as a homomorphism, but as a crossed homomorphism that is related to the geometry of foliated bundles.

Monday Oct. 18, 4:00 p.m., 380-383N.

Yasha Eliashberg (Stanford)

Quantum jumps and contact non-squeezing

Abstract: I will talk about a joint work with Sang-Seon Kim and Leonid Polterovich about non-aqueezing phenomena in contact geometry analogous to the famous Gromov non-squeezing theorem in symplectic geometry. It turns out that the corresponding results in contact geometry exhibit a quantum behavior: there are non-squeezing results for domain of size larger than $\hbar$ but not for small ones. This problem is tightly related to the problem of positive loops in the group of contact transformations.

Monday Oct. 25, 4:00 p.m., 380-383N

Brett Parker (Stanford)

Perturbed Holomorphic Graphs in Lagrangian Torus Fibrations

Abstract: I will describe a method for constructing the moduli space of holomorphic curves in some Lagrangian torus fibrations using a moduli space of graphs.

See NCSGS announcement.
Some people will be driving down from Stanford. If you need a ride, e-mail Robert Lipshitz for information on who is going.

Monday Nov. 1, NCSGS in 187 Dwinelle Hall at U.C. Berkeley

Grigory Mikhalkin (U. of Utah)

Complex, real and tropical curves

Abstract: Tropical varieties are limit objects of the so-called amoebas of complex algebraic varieties under a certain deformation of the ambient complex structure. This talk will focus on tropical curves and their applications for a range of classical problems in complex and real algebraic geometry.

and

Thomas Vogel (U. of Munich)

Existence of Engel structures

Abstract: We develop a construction of Engel structures based on the decomposition of manifolds into round handles and some results from contact topology. We prove that every parallelizable 4-manifold admits an Engel structure.

Monday Nov. 8, 4:00 p.m., 380-383N

Boris Kruglikov (U. of Tromsø)

A criterion of formal integrability for systems of PDEs (part 1)

Abstract: We will describe obstructions to formal integrability for a PDEs system of generalized complete intersection type. For this we introduce certain multi-brackets of functions on the space of infinite jets (non-linear differential operators). These multi-brackets generalize the Jacobi-Mayer brackets, known from the contact geometry of scalar first order equations. The obstructions will be expressed via these brackets.

Monday Nov. 15, 4:00 p.m., 380-383N

Boris Kruglikov (U. of Tromsø)

A criterion of formal integrability for systems of PDEs (part 2)

Abstract: We prove the criterion for a system of linear PDEs. Then we explain what changes in the non-linear case. The criterion will be applied to solve the following problems: 1) Determine locally when a surface metric is of Liouville type, i.e. its geodesic flow is qudratically integrable. 2) Find a criterion when two functions K, H can be Gaussian and mean curvatures of a surface in the Euclidean space.

Monday Nov. 22, 4:00 p.m., 380-383N

Robert Lipshitz (Stanford)

A 4-dimensional view of Heegaard Floer homology (area exam talk)

Abstract: Given a Heegaard splitting for a 3-manifold Y, we will construct the Heegaard Floer homology groups of Y (introduced by Ozsváth-Szabó) by counting certain holomorphic curves in the product of the Heegaard surface and a strip. (The original construction counts certain holomorphic curves in a symmetric product of the Heegaard surface.) We will emphasize the basic constructions and examples, most of which are substantially similar to those used by Ozsváth-Szabó, rather than technical aspects of our construction.

Monday, Dec. 6, 2:30 p.m., 380-383N

Aleksey Zinger (Stanford)

On the Genus-One Gromov-Witten Invariants of Complete Intersection Threefolds

Abstract: I will describe a formula relating the genus-one Gromov-Witten invariants of a projective complete intersection threefold to the GW-invariants of the ambient projective space. Along with a separate desingularization result, this formula allows one to compute the genus-one GW-invariants of such threefolds. It might be possible to use this formula to verify the genus-one mirror symmetry prediction for curves in Calabi-Yau threefolds.

Monday, Dec. 6, 4:00 p.m., 380-383N

Frederic Bourgeois (U. Libre de Bruxelles)

Contact homology and action functional

Abstract: Contact homology is a powerful invariant for contact structures, and can be thought of as an infinite dimensional Morse theory for the action functional. I'll explain how to extract additional information from the action functional to enhance or help compute contact homology, in two different settings. These emerged in joint works with Vincent Colin and Mei-Lin Yau respectively.

Monday, Jan. 10, 4:00 p.m., 380-383N

Jian He (Stanford)

Introduction to the Gromov-Witten / Hurwitz correspondence

Abstract: I will begin to describe the Gromov-Witten / Hurwitz correspondence discovered by A. Okounkov and R. Pandharipande. This is the first in a series of talks on the subject. In it, I hope to discuss most of Section 0 of the paper "Gromov-Witten theory, Hurwitz theory, and completed cycles" by Okounkov-Pandharipande.

Monday, Jan. 17, 4:00 p.m., 380-383N

Jian He (Stanford)

GW/H: Psi classes and ramification conditions

Abstract: I will discuss the correspondence between integrals of psi classes and ramification conditions; relative Gromov-Witten theory; and hopefully a proof of the relative Gromov-Witten / Hurwitz correspondence, assuming the absolute case. This material comprises sections 1.1, 1.2, and 1.3 of the paper "Gromov-Witten theory, Hurwitz theory, and completed cycles" by Okounkov-Pandharipande.

Monday, Jan. 24, 4:00 p.m., 380-383N

Ciprian Manolescu (Princeton U.)

Symplectic Floer theories, Hilbert schemes, and the Jones polynomial

Abstract:We will explain how the Jones polynomial of links appears in symplectic geometry. Seidel and Smith have used Lagrangian Floer homology to define a new link invariant. We describe a set of generators for the Seidel-Smith chain complex which can also be used to compute the Jones polynomial, or as a set of generators for the Heegaard Floer chain complex of the double branched cover.

Monday, Jan. 21, 2:30 p.m., 891 Evans Hall at U.C. Berkeley

Xiang Tang (U.C. Davis)

Foliations, modular forms, and deformation quantization

Abstract: Inspired by the Rankin-Cohen brackets on modular forms, Connes and Moscovici constructed a universal deformation formula for the Hopf algebra associated to a codimension one foliation. In this talk, we will explain how to use symplectic geometry to understand their deformation and various related structures. In particular, we will show that the Rankin-Cohen deformation is equivalent to the Weyl-Moyal product.

Monday, Jan. 21, 4:10 p.m., 60 Evans Hall at U.C. Berkeley

Bertram Kostant (M.I.T.)

Gelfand--Zeitlin theory from the perspective of classical --> --mechanics

Abstract: The space $M(n)$ of $n\times n$ matrices is a Poisson manifold. Gelfand-Zeitlin theory gives rise a maximal Poisson commutative algebra of functions on $M(n)$. We show that the corresponding Poisson vector fields are globally integrable and give to a new commutative group $A$ of Poisson automorphisms on $M(n)$. The orbits of $A$ are explicitly given and give rise to new decompositions of $M(n)$.

The group $A$ leads to a solution of a classical analogue of the Gelfand-Kirillov conjecture

Monday, Feb. 7, 4:00 p.m., 380-383N

Yasha Eliashberg (Stanford)

Differential equations arising in SFT

Note: This is also the Gromov-Witten seminar of the week.

Friday, Feb. 11, 2:00 p.m., 380-383N

John Etnyre (U. of Pennsylvania)

Contact geometry and the genus of open book decompositions

Abstract: After discussing a result of Giroux relating open book decompositions and contact structures on 3-manifolds I will discuss how the genus of an open book relates to geometric properties of the contact structure. In particular I will discuss relations with tightness/fillability, the Weinstein conjecture and the definition of invariants of contact structures and Legendrian knots.

Monday, Feb. 14, 4:00 p.m., 380-383N

Dieter Kotschick (U. of Munich)

Some symplectic topology of complex surfaces

Abstract: Certain standard results about compact Kahler surfaces, which are quite hard to prove in the traditional approach via algebraic geometry, can be proved more efficiently using the basics of symplectic topology. I will explain in detail a particular instance of this phenomenon, revolving around the notion of minimality. Time permitting, I shall also discuss further topological consequences using Seiberg--Witten gauge theory.

Monday, Feb. 28, 2:30 p.m., 380-383N

Cheol-Hyun Cho, (Northwestern U.)

A-infinity structure of open-closed maps

Abstract: I will first review the construction of A-infinity algebra of Lagrangian submanifold by Fukaya, Oh, Ohta and Ono, and explain computations of product structures for toric Fano manifolds. Then,I will explain A-infinity algebra for quantum cohomology, and A-infinity morphism between quantum cohomology and Hochschild cohomology of Fukaya category.

Tobias Ekholm (U. of Southern California)

Pseudo-holomorphic disks with boundary on a Legendrian submanifold of a $1$-jet space and gradient flow trees with cusps.

Abstract: The projection of a generic Legendrian submanifold $L\subset J1(M)\approx T^\ast M\times\R$, where $M$ is a Riemannian manifold, to $J0(M)\approx M\times\R$ consits of graphs of functions over the complement of a stratified subset of codimension one in $M$. A cusped gradient flow tree is a certian map of a tree into $M$, with the property that its edges coincide with gradient flow lines of differences of the functions mentioned above. We show that if $\dim(L)\le 2$, or if $\dim(L)>2$ and the projection of $L$ to $J0(M)$ has only cusp-edge singularities, then there is a $1-1$-correspondence between rigid pseudo-holomorphic disks in $T^\ast M$ with boundary on (the projection of) $L$ and rigid cusped gradient flow trees. We will also discuss some ramifications of this result.

Monday, March 7, 4:00 p.m., 380-383N

Amnon Neeman (Australian National U., Canberra)

Classical invariant theory meets statistics

Abstract: We will explain how a very classical problem on mixture models can be translated to a question in classical invariant theory, and then briefly outline how we solved the problem. This is a report on joint work with two statisticians, Peter Hall and Ryan Elmore.

Monday, April 4, 2:30 p.m., 939 Evans Hall at U.C. Berkeley

Henrique Bursztyn (U. of Toronto)

Quasi-Poisson manifolds and Dirac geometry

Abstract: Quasi-Poisson manifolds (introduced by Alekseev, Kosmann-Schwarzbach and Meinrenken) arise as semi-classical limits of quasi-Hopf algebras and provide an atractive framework for the study of the symplectic/Poisson geometry of certain moduli spaces in gauge theory (via the theory of group-valued momentum maps). In this talk, I will show how quasi-Poisson manifolds can be naturally described in terms of Dirac structures, and how this point of view sheds light on many of their properties (e.g. their decomposition on leaves analogous to the symplectic foliation of a Poisson manifold). If time permits I will discuss the close connection between Dirac geometry and the hamiltonian theory of quasi-Poisson actions.

Monday, April 4, 4:10 p.m., 939 Evans Hall at U.C. Berkeley

Matt West (Stanford)

Asynchronous variational integrators

Abstract: Variational Integrators are discretizations of Lagrangian ordinary or partial differential equations based on direct discretizations of Hamilton's principle. They automatically preserve symplectic structures for ODEs and satisfy integral multi-symplectic principles for PDEs, and can readily be designed to preserve momentum maps. Many standard existing methods are in fact variational, including the popular solid-mechanics Newmark code.

In this talk we present a class of variational integrators for hyperbolic Lagrangian PDEs that use fully asynchronous spacetime meshes. These Asynchronous Variational Integrators permit the use of locally adapted timesteps that conform to spatially adapted meshes, such as are common in 3D finite element codes. They have a discrete multi-symplectic structure, which results in excellent energy and momentum conservation. Numerical results are presented for some solid mechanics problems.

Monday, April 11, 4:00 p.m., 380-383N

Olga Plamenevskaya (M.I.T.)

Khovanov homology, Heegaard Floer invariants, and contact structures.

Abstract: Khovanov homology is a combinatorially defined invariant of a link in S^3. A recent theorem of Ozsvath and Szabo relates Khovanov homology to Heegaard Floer homology of 3-manifolds. After reviewing this result, I will discuss how the correspondence between the two theories can be extended to include invariants of contact 3-manifolds and transverse knots in S^3.

Monday, April 18, 4:00 p.m., 380-383N

Peter Ozsváth (U.C. Berkeley / Columbia U.)

Heegaard Floer homology and rational surgeries.

Monday, May 2, 2:30 p.m., 380-383N

Alexander Karabegov (Abilene Christian U.)

Formal symplectic groupoids and contravariant connections.

Abstract: We give a Fedosov-type geometric construction of a formal symplectic groupoid over a Poisson manifold endowed with a Poisson torsion-free contravariant connection. We show that there are Poisson manifolds that do not admit such connections.

Wei-Dong Ruan (U. of Illinois at Chicago)

The homological mirror symmetry conjecture for weigted projective spaces and their non-commutitive deformations.

Abstract: In 1994, Kontsevich proposed the homological mirror symmetry conjecture for Fano varieties and Calabi-Yau manifolds that predicts the equivalence of the derived category of coherent sheaves on the manifold and the Fukaya category for the mirror. In this talk, we will consider the case of weighted projective space for all dimension. We will prove the homological mirror symmetry in this case through the category of constructible sheaves on the complex side and the Fukaya-Oh Morse category on the symplectic side.

Monday, May 16, 4:00 p.m., 380-383N

Nadya Shirokova (Max Planck Institute / Stanford)

On the local system of Ozsvath-Szabo homologies.

Abstract: We consider the local system of OS homologies on the configurational space of 3-manifolds and extend it to the singular locus.

This web page is maintained by Robert Lipshitz, who can be e-mailed for more information.