Joint IAS-PU Symplectic Geometry Seminar

Wrapped Floer theory and Homological mirror symmetry for toric Calabi-Yau manifolds

Yoel Groman
Columbia University
October 9, 2017
Consider a Lagrangian torus fibration a la SYZ over a non compact base. Using techniques from arXiv:1510.04265, I will discuss the construction of wrapped Floer theory in this setting. Note that this setting is generally not exact even near infinity. The construction allows the formulation of a version of the homological mirror symmetry conjecture for open manifolds which are not exact near infinity.

Floer theory in spaces of stable pairs over Riemann surfaces

Timothy Perutz
University of Texas, Austin; von Neumann Fellow, School of Mathematics
May 4, 2017
I will report on joint work with Andrew Lee, which explores the notion that spaces of stable pairs over Riemann surfaces (in the sense of Bradlow and Thaddeus) could form a natural home for a “non-abelian” analog of Heegaard Floer homology for 3-manifolds - just as the g-fold symmetric product is the home of Heegaard Floer homology - thereby circumventing the problems with singularities that beset instanton-type theories. In an initial foray into this area, we set up a theory not for Heegaard splittings but for fibered 3-manifolds, based on fixed-point Floer homology.

Lagrangian Floer theory in symplectic fibrations

Douglas Schultz
Rutgers University
April 27, 2017
Given a fibration of compact symplectic manifolds and an induced fibration of Lagrangians, one can ask if we can compute the Floer cohomology of the total Lagrangian from information about the base and fiber Lagrangians. The primary example that we have in mind is the manifold of full flags in ${\mathbb C}^3$ which fibers as ${\mathbb P}^1 \to {\rm Flag}({\mathbb C}^3) \to {\mathbb P}^2$, and a Lagrangian $T^3$ that fibers over the Clifford torus in ${\mathbb P}^2$.

Symplectic field theory and codimension-2 stable Hamiltonian submanifolds

Richard Siefring
Ruhr-Universität Bochum
April 20, 2017
Motivated by the goal of establishing a "symplectic sum formula" in symplectic field theory, we will discuss the intersection behavior between punctured pseudoholomorphic curves and symplectic hypersurfaces in a symplectization. In particular we will show that the count of such intersections is always bounded from above by a finite, topologically-determined quantity even though the curve, the target manifold, and the symplectic hypersurface in question are all noncompact.

On Zimmer's conjecture

Sebastian Hurtado-Salazar
University of Chicago
April 6, 2017

The group $\mathrm{SL}_n(\mathbb Z)$ (when $n > 2$) is very rigid, for example, Margulis proved all its linear representations come from representations of $\mathrm{SL}_n(\mathbb R)$ and are as simple as one can imagine. Zimmer's conjecture states that certain "non-linear" representations ( group actions by diffeomorphisms on a closed manifold) come also from simple algebraic constructions.

Rigid holomorphic curves are generically super-rigid

Chris Wendl
Humboldt-Universität zu Berlin
March 31, 2017
I will explain the main ideas of a proof that for generic compatible almost complex structures in symplectic manifolds of dimension at least 6, closed embedded J-holomorphic curves of index 0 are always "super-rigid", implying that their multiple covers are never limits of sequences of curves with distinct images. This condition is especially interesting in Calabi-Yau 3-folds, where it follows that the Gromov-Witten invariants can be "localized" and computed in terms of Euler classes of obstruction bundles for a finite set of disjoint embedded curves.

The many forms of rigidity for symplectic embeddings

Felix Schlenk
University of Neuchâtel
March 30, 2017
We look at the following chain of symplectic embedding problems in dimension four. \[E(1, a) \to Z_4(A),\ E(1, a) \to C_4(A),\ E(1, a) \to P(A, ba) (b \in {\mathbb N}_{\geq 2}),\ E(1, a) \to T_4(A).\] Here $E(1, a)$ is a symplectic ellipsoid, $Z_4(A)$ is the symplectic cylinder $D_2(A) \times R_2$, $C_4(A) = D_2(A) \times D_2(A)$ is the cube and $P(A, bA) = D_2(A) \times D_2(bA)$ the polydisc, and $T_4(A) = T_2(A) \times T_2(A)$, where $T_2(A)$ is the 2-torus of area $A$. In each problem we ask for the smallest $A$ for which $E(1, a)$ symplectically embeds.