Friday Fish seminar

Holonomic approximation is the main tool used in proving that the h-principle holds on open manifolds for a large class of partial differential relations. In the first part of this talk I discuss a generalization of holonomic approximation which also works on closed manifolds, using mild singularities known as wrinkles. In the second part, we use this result to carry out the first step of a strategy proposed by Laudenbach and Meigniez to study h-principles via a certain type of singular foliations known as Haefliger structures. We end with an application to the classifying space of principal groupoid bundles which encode Haefliger structures with a transverse geometry.

First, I will cover the material alluded to in the title. In the remaining two hours, I will define gerbes, try to give an idea of when they appear, talk about connections, loop spaces, Courant algebroids, vector bundle twists and topological T-duality. I will run out of time at some point in the middle.

In this talk we will continue to explore the world of topological strings, specifically the A-twisted nonlinear sigma model (NLSM) as introduced by Jesse Straat on May 16th. The aim will be to actually compute the observables of a NLSM with a (smooth) toric variety as target. The trick will be to look at a so-called gauged linear sigma model (GLSM) whose low-energy limit is the NLSM we want. When studying the observables, we encounter the quantum cohomology ring, an extension of the ordinary cohomology, as a way of combining this information into a ring with product. It turns out that these observables are closely related to the Gromov–Witten invariants of classes corresponding to the operators we consider.

If time permits, we will look at a different formulation of quantum cohomology using the Gromov–Witten invariants directly.

Note that the talk will be accessible for both physicists and mathematicians that want to see such an approach to quantum cohomology.

Minimal geodesic flows on closed hyperbolic manifolds (manifolds which support a negative curvature metric) exhibit a strong rigidity: their flows are topologically conjugate. Slightly more generally, for any geodesic flow on a hyperbolic manifold, it has a non-empty set of globally minimal geodesics, which always shadow geodesics of a given negative curvature metric.

For Reeb flows on star-shaped contact surfaces, it is impossible to be minimal when the surface is strictly not convex fiberwise. However, for a slight generalization of minimality, called coarse minimality, we can produce examples of non-convex hypersurfaces with coarsely minimal Reeb flows. Additionally, there is a semi-conjugacy coarsely minimal flows and the geodesic flow a given negative curvature metric.

A common question for geometric structures is that of rigidity, i.e. given two geometric structures sufficiently close, are they equivalent? In this talk we discuss this question for regular foliations on a closed manifold.

In the first part we give an overview of the current state of the art and highlight its relation with the rigidity for group actions. We will see that several rigidity results for foliations require the leaves to be compact.

In an attempt to understand whether such results can be extended to Riemannian foliations, we seek examples of such foliations which are infinitesimally rigid with non-compact leaves. In the second part, we use the relation to group actions to construct infinitesimally rigid Lie foliation with non-compact, dense leaves. This is based on joint work in progress with Stephane Geudens.