Tag - Symplectic geometry

Sebastian Hurtado-Salazar: On Zimmer’s conjecture

6th April, 2017

The group SLn(ℤ) (when n > 2) is very rigid, for example, Margulis proved all its linear representations come from representations of SLn(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.

For example, conjecturally the only action on SLn(ℤ) on an (n−1) dimensional manifold (up to some trivialities) is the one on the (n−1) sphere coming projectivizing natural action of SLn(ℝ) on ℝn. I'll describe some recent progress on these questions due to A. Brown, D. Fisher and myself.

Chris Wendl: Rigid holomorphic curves are generically super-rigid

31st March, 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. By the same techniques, we can also show that unbranched covers of simple J-holomorphic curves are generically regular. These results are based on a decomposition of the space of branched covers into smooth strata on which certain twisted Cauchy-Riemann operators have kernel and cokernel of constant dimension.

Dusa McDuff: The stabilized symplectic embedding problem

30th March, 2017

I will discuss some recent work (mostly joint with Dan Cristofaro-Gardiner and Richard Hind) on the stabilized symplectic embedding problem for ellipsoids into balls. The main tools come from embedded contact homology.

Felix Schlenk: The many forms of rigidity for symplectic embeddings

30th March, 2017

We look at the following chain of symplectic embedding problems in dimension four.

E(1,a)→Z4(A), E(1,a)→C4(A), E(1,a)→P(A,ba)(b ∈ ℕ≥2), E(1,a)→T4(A).

Here E(1,a) is a symplectic ellipsoid, Z4(A) is the symplectic cylinder D2(A)×R2, C4(A)=D2(A)×D2(A) is the cube and P(A,bA)=D2(A)×D2(bA) the polydisc, and T4(A)=T2(A)×T2(A), where T2(A) is the 2-torus of area A. In each problem we ask for the smallest A for which E(1,a) symplectically embeds. The answer is very different in each case: total rigidity, total flexibility with a hidden rigidity, and a two-fold subtle transition between them.

François Lalonde: Continuous covers on symplectic manifolds

24th March, 2017

In this talk, we first introduce the notion of a continuous cover of a manifold parametrised by any compact manifold endowed with a mass 1 volume-form. We prove that any such cover admits a partition of unity where the usual sum is replaced by integrals. We then generalize Polterovich's notion of Poisson non-commutativity to such a context in order to get a richer definition of non-commutativity and to be in a position where one can compare various invariants of symplectic manifolds, for instance the relation between critical values of phase transitions of symplectic balls and eventual critical values of the Poisson non-commutativity. Our first main theorem states that our generalisation of Poisson non-commutativity depends only on real one-parameter spaces since intuitively the Hilbert curve in any high dimensional parameter space fills out the entire manifold and preserves the measure. Our second main theorem states that the Poisson non-commutativity is a (not necessarily strictly) decreasing function of the size of the symplectic balls used to cover continuously any given symplectic manifold. This function has other nice properties as well that do not prevent it from undergoing singularities similar to phase transitions.

Daniel Alvarez-Gavela: The simplification of caustics

23rd March, 2017

We present a full h-principle (relative, parametric, C0-close) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we prove that if there is no homotopy-theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of an ambient Hamiltonian isotopy. The main ingredients in the proof are a refinement of the holonomic approximation lemma and the construction of a local wrinkling model for Lagrangian and Legendrian submanifolds. We give sample applications of our h-principle, including an Igusa-type theorem which states that higher singularities are unnecessary for the homotopy-theoretic study of the space of Legendrian knots in the standard contact Euclidean 3-space. This last result can be understood as a generalization of the Reidemeister theorem for families of Legendrian knots parametrized by a space of arbitrarily high dimension.

Alexandru Oancea: Symplectic homology for cobordisms

23rd February, 2017

Symplectic homology for a Liouville cobordism (possibly filled at the negative end) generalizes simultaneously the symplectic homology of Liouville domains and the Rabinowitz-Floer homology of their boundaries. I intend to explain a conceptual framework within which one can understand it, and give a sample application which shows how it can be used in order to obstruct cobordisms between contact manifolds.

Kei Irie: C∞ closing lemma for three-dimensional Reeb flows via embedded contact homology

16th February, 2017

Cr closing lemma is an important statement in the theory of dynamical systems, which implies that for a Cr generic system the union of periodic orbits is dense in the nonwondering domain. C1 closing lemma is proved in many classes of dynamical systems, however Cr closing lemma with r > 1 is proved only for few cases. In this talk, I'll prove C∞ closing lemma for Reeb flows on closed contact three-manifolds. The proof uses recent developments in quantitative aspects of embedded contact homology (ECH). In particular, the key ingredient of the proof is a result by Cristofaro-Gardiner, Hutchings and Ramos, which claims that the asymptotics of ECH spectral invariants recover the volume of a contact manifold. Applications to closed geodesics on Riemannian two-manifolds and Hamiltonian diffeomorphisms of symplectic two-manifolds (joint work with M. Asaoka) will be also presented.

Sara Tukachinsky: Relative quantum product and open WDVV equations

2nd February, 2017

The standard WDVV equations are PDEs in the potential function that generates Gromov-Witten invariants. These equations imply relations on the invariants, and sometimes allow computations thereof, as demonstrated by Kontsevich-Manin (1994). We prove analogous equations for open Gromov-Witten invariants that we defined in a previous work. For (ℂPn,ℝPn), the resulting relations allow the computation of all invariants. The formulation of the open WDVV requires a lift of the big quantum product to relative cohomology. Surprisingly, this brings us to use moduli spaces of disks with geodesic conditions. No prior knowledge of the subjects mentioned above will be assumed.