Advances in Symplectic Geometry and Topology

A special session in the

Mathematical Congress of the Americas 2013

Guanajuato Mexico, 5-9 August 2013

List of speakers.

Organisers: Octav Cornea, University of Montréal; Eduardo González, UMASS Boston; Leonardo Macarini, Federal University of Rio de Janeiro; Andrés Pedroza, Universida de Colima.

Objective: The session's objective is to highlight recent discoveries in a broad range of fields where hard and soft symplectic tools are used.
Sponsors: CIMAT, CONACYT, the City of Guanajuato and grant DMS-1306543 of the National Science Foundation.

NSF Support for US based participants: We are anticipating support from the National Science Foundation grant DMS-1306543 to fund junior scholars. To request support, please send an email to: eduardo AT-SIGN-HERE, with a brief CV-like paragraph with a description of research, including PhD graduation year, dissertation title and advisor. The deadline is 30 May 2013. If it applies, please include a list of publications. We strongly encourage the participation of under-represented groups.

Preliminary list of participants To all participants supported by the NSF. You need to make your own travel arrangements. We will reimburse you after the conference is over. Travel to Guanajuato For more information regarding traveling to the city of Guanajuato, please visit traveling to Guanajuato.

Abstracts and Schedule.

The session will be held at the "Centro de convenciones" on Monday 5 August and Tuesday 6 August. Octav Cornea will deliver an invited lecture on Thursday 8 August.


Mohammed Abouzaid
Lagrangian immersions and the Floer homotopy type
A conjecture of Arnold would imply that every exact Lagrangian in a cotangent bundle is isotopic to the zero section through Lagrangian embeddings. We now know that every such Lagrangian is homotopy equivalent to the zero section. I willexplain how, combining the h-principle with the spectrum-valued invariants introduced by T. Kragh, one can hope to show that such Lagrangians are in fact isotopic to the zero section through Lagrangian immersions. I will discuss partial results obtained with Kragh, constraining the Lagrangian isotopy class of Lagrangians embeddings. PDF

Jim Bryan
The Crepant Resolution Conjecture in Gromov-Witten and Donaldson-Thomas theory and the orbifold topological vertex
Gromov-Witten theory provides fundamental deformation invariants of symplectic manifolds and orbifolds. The crepant resolution conjecture provides a conjectural equivalence between the Gromov-Witten theory of an orbifold and its resolution. In the case when the symplectic manifold is a Calabi-Yau threefold, Gromov-Witten theory is also conjectured to be equivalent to Donaldson-Thomas theory. We discuss the relationship between the crepant resolution conjectures in Gromov-Witten theory and in Donaldson-Thomas theory. In the case where the threefold is toric, these theories are encoded by the so-called topological vertex. On the Gromov-Witten side, the topological vertex encodes Hodge integrals, on the Donaldson-Thomas theory side, the topological vertex encodes the combinatorics of 3D partitions. We discuss the relationship among the various topological vertexes dictated by the Gromov-Witten/Donaldson-Thomas correspondence and by the crepant resolution conjectures. PDF

Henrique Bursztyn
Symplectic geometry in degree 2
This talk will discuss symplectic geometry in the context of graded manifolds, with focus on manifolds of degree 2". Our main goal is showing how graded symplectic geometry provides a natural framework, as well as tools, for the study of geometrical objects such as Poisson and Dirac structures, Courant algebroids, and generalized complex structures. Particular attention will be given to the description of symplectic reduction in this more general setting. PDF

Kenji Fukaya
Algebraic topological foundation of Floer theory of arbitrary genus
In this talk I would like to summarize the on going research how we use bordered Riemann surface of arbitrary genus to associate certain algebraic structure to a Lagrangian submanifold. The algebraic formulation is a joint work with Cielibak and Latschev and a structure which we call involutive bi-Lie-infinity structure. We need to resolve certain transversality issue to realize it from the moduli space. I will also explain the main idea how to do so.

Yael Karshon
Distinguishing symplectic blowups of CP^2
A symplectic manifold that is obtained from CP^2 by k blowups is encoded by k+1 numbers: the size of the initial CP^2 and the sizes of the blowups. We determine which different sizes yield symplectomorphic manifolds. This is joint work with Liat Kessler and Martin Pinsonnault. PDF

Michael Hutchings
Extensions of the Weinstein conjecture in three dimensions.
The Weinstein conjecture in three dimensions, proved by Taubes, asserts that every contact form on a closed three-manifold has a Reeb orbit. We discuss some recent theorems and conjectures that extend this result, for example by providing more than one Reeb orbit, or Reeb orbits with particular properties. PDF

Francois Lalonde
Moduli space of symplectic surfaces in rational ruled symplectic 4-manifolds, exotic structures and a theorem of Guest and al.
We explore the moduli space of symplectic surfaces in rational ruled symplectic 4-manifolds in many homology classes. We show that their computations are related to the classification of exotic structures on C2 2 and, in one of the simplest cases, to a theorem of Guest et. al. on the complex case.

Ernesto Lupercio
New Families of Topological Quantum Field Theories on Orbifolds
In this talk I present new families of orbifold Topological Quantum Field theories closely related to Chen-Ruan cohomology and to Orbifold String Topology. PDF

Dusa McDuff
Smooth Kuranishi atlases
Many invariants in symplectic geometry arise from counting geometric objects that arise as solutions to differential equations. In order to get consistent counts, one often must perturb the equations and then develop a suitable framework in which to interpret the results. There are many possible approaches to this problem, some using traditional analytic techniques such as finite dimensional reduction and others using new more powerful analytic tools such as polyfolds. This talk describes recent joint work with Katrin Wehrheim in which we revisit a construction originally due to Fukaya-Ono we use traditional analysis but develop a new algebro-topological framework in which to understand the invariants. PDF

Tim Perutz
From seen to unseen Lagrangians via algebraic geometry.
I will describe a general method for proving generation of the Fukaya category $F(X; D)$ of a Calabi-Yau manifold $X$, relative to an ample divisor $D$, by a subcategory of for which homological mirror symmetry (HMS) has already been proved. The argument, which was discovered independently and concurrently by the author and by Nick Sheridan, leverages homological algebra and Hodge theory for the mirror family. Besides its significance for proving HMS theorems, the argument is also a scheme for using algebraic geometry to obtain information in symplectic topology. PDF

Bernardo Uribe
Equivariant extensions of closed differential forms for non-compact Lie groups.
Consider a manifold endowed with the action of a Lie group. In this talk I will explain a relation between the cohomology of the Cartan complex and the equivariant cohomology by using the equivariant De Rham complex developed by Getzler, and I will show that the cohomology of the Cartan complex lies on the 0-th row of the second page of a spectral sequence converging to the equivariant cohomology. I will use this result to generalize a result of Witten on the equivalence of absence of anomalies in gauge WZW actions on compact Lie groups to the existence of equivariant extension of the WZW term, to the case on which the gauge group is the special linear group with real coefficients. The results described in this talk are joint work with Hugo Garcia-Compean and Pablo Paniagua. PDF

Yuri Vorobiev
Transversally-Hamiltonian group actions around symplectic leaves.
We study a class of (noncanonical) actions of compact Lie groups on a Poisson manifold in the semilocal context, around an embedded symplectic leaf of nonzero dimension. Our approach is based on the method of coupling Poisson structures and the averaging technique for Ehresmann-Poisson connections. In the case when a compact Lie group acts in a Hamiltonian way relative to a transverse Poisson structure of the leaf, we show that the "averaged" Poisson structure is well-defined and isomorphic to the original one. This gives rise to an equivariant linearized Poisson model of the symplectic leaf. PDF

Jonathan Weitsman
The Morse Index Theorem for Hamiltonian Loop Group Spaces and Poincare Series of Moduli Spaces of Semistable Vector bundles.
In previous work (Invent. Math. 155, 225-251 (2004)) we used a version of Morse theory to prove an analog of the Kirwan surjectivity theorem for Hamiltonian loop group spaces. In this paper we continue the study of Morse theory for these spaces by giving an effective formula for the index of our Morse function at its critical points. We apply this formula to the computation of the Poincare series of moduli spaces of semi-stable vector bundles. (Jointly with Bott and Tolman) PDF

Chris Woodward
More on quantum cohomology and minimal models.
We analyze the quantum cohomology of toric manifolds and orbifolds using the minimal model program and quantum Kirwan map. We explain the connection between eigenvalues of quantum multiplication by the first Chern class and transition times in the minimal model program, and also explain that the quantum cohomology is invariant (in a certain sense) under flops. PDF

Notes from the session will be published here.

Last edited: Oct 2013.