Mini-Workshop on Symplectic Geometry, Utrecht University, March 15, 2019

Titles and abstracts for the talks

Álvaro del Pino Gómez:
Wrinkling of submanifolds of jet spaces

Darboux's theorem tells us that any contact manifold is locally modelled by the space of 1-jets of functions. This allows us to manipulate legendrians by looking at their front projection, i.e. their projection to the 0-jet. When we do this, we see that the theory of legendrians is essentially the theory of multiply-valued functions.

Higher jet spaces are endowed with canonical distributions too, but these are not contact anymore. Nonetheless, they are the simplest examples to be understood in order to develop a general topological theory of distributions. A particularly nice feature is that they are still very related to the contact case and, in particular, their integral submanifolds can still be understood as multiply-valued functions.

In this talk I will explain on-going work joint with L. Toussaint in which we use h-principle techniques to construct and manipulate integral submanifolds of jet spaces. Our approach is intimately related to the wrinkling theory developed by Eliashberg and Mishachev, and later used by Murphy and Álvarez-Gavela in the contact setting. I will try to make the connection between our result and theirs as explicit as possible.

Sonja Hohloch:
Direct limit approach to homoclinic Floer homology

Floer homology was originally devised for counting the number of intersection points of two `nice' Lagrangian submanifolds in a symplectic manifold.

The stable and unstable manifolds of a hyperbolic fixed point of a symplectomorphism are Lagrangians, but they are `highly noncompact’ as soon as they intersect transversely, meaning, they are only injectively immersed and display a wild accumulation and oscillation behaviour. These properties render the standard approach to Floer homology via pseudoholomorphic curves quite hopeless. Nevertheless, on 2-dimensional manifolds, analysis can be completely replaced by combinatorics --- without loosing any complexity concerning the abundance of intersection points of the stable and unstable manifold (the so-called `homoclinic points’).

In previous papers, we had constructed various Floer homologies generated by so-called `primary homoclinic points’ which form a (modulo iteration) finite subset of the set of homoclinic points.

Now we want to use for the construction of Floer homology all homoclinic points by passing to a (suitably constructed) direct limit.

In this talk, we will outline this new construction and (some of) its properties plus examples.

Tobias Ekholm:
Skeins on branes

We define open Gromov-Witten invariants of a Maslov zero Lagrangian L in a 3-dimensional Calabi-Yau manifold with values in the framed HOMFLY skein module of L. Combining the invariance properties of the resulting curve count with SFT-stretching in the case of knot conormals in the cotangent bundle of the 3-sphere, we establish the simplest form of Ooguri-Vafa large N duality that relates HOMFLY polynomials of knots to Gromov-Witten invariants in the resolved conifold. The talk reports on joint work with Vivek Shende.

Frances Kirwan (colloquium talk):
Implosion in symplectic and hyperkahler geometry

The hyperkahler quotient construction, which allows us to construct new hyperkahler spaces from suitable group actions on hyperkahler manifolds, is an analogue of symplectic reduction (introduced by Marsden and Weinstein in the 1970s), and both are closely related to the quotient construction for complex reductive group actions in algebraic geometry provided by Mumford's geometric invariant theory (GIT). Hyperkahler implosion is in turn an analogue of symplectic implosion (introduced in a 2002 paper of Guillemin, Jeffrey and Sjamaar), which is related to a generalised version of GIT providing quotients for non-reductive group actions in algebraic geometry.

Frances Kirwan:
Moment maps and non-reductive geometric invariant theory

When a complex reductive group acts linearly on a projective variety the GIT quotient can be identified with an appropriate symplectic quotient. The aim of this talk is to discuss an analogue of this description for GIT quotients by suitable non-reductive actions. In general GIT for non-reductive linear algebraic group actions is much less well behaved than for reductive actions. However when a linear algebraic group has internally graded unipotent radical U, in the sense that a Levi subgroup has a central one-parameter subgroup which acts by conjugation on U with all weights strictly positive, then GIT for a linear action of the group on a projective variety is almost as well behaved as in the reductive setting, provided that we are willing to multiply the linearisation by an appropriate rational character. In this situation we can ask for a moment map description of the quotient.