The Chern—Simons functional is defined on the non-compact infinite-dimensional space of connections and so its Morse homology is not well-defined. However, its critical locus, the moduli space of flat connections, is finite-dimensional. I will explain how one can use shifted symplectic geometry of the moduli space of flat connections to define the relevant Morse homology groups and outline...
The symplectic version of the problem of packing K balls into a ball in the densest way possible (in 4 dimensions) can be extended to that of symplectically embedding an ellipsoid into a ball as small as possible. A classic result due to McDuff and Schlenk asserts that the function that encodes this problem has a remarkable structure: its graph has infinitely many corners, determined by...
In their landmark paper in 2016, Braverman-Finkelberg-Nakajima gave a mathematically rigour definition of the Coulomb branch of 3d N=4 supersymmetric gauge theories (of cotangent type). This broad class of spaces have many remarkable properties and are expected to play a key role in developing a geometric representation theory around "double affine Grassmannians". In particular, they...
Let M be a compact symplectic manifold, and D a normal-crossings symplectic divisor in M. We give a criterion (which can be naturally expressed in terms of the Kodaira dimension of M and log Kodaira dimension of M \ D, in the context where M and D come from the realm of algebraic geometry) under which the quantum cohomology of M is a deformation of the symplectic cohomology of M \ D. We will...
b-Structures and other generalizations (such as E-symplectic structures) are ubiquitous and sometimes hidden, unexpectedly, in a number of problems including the space of pseudo-Riemannian geodesics and regularization transformations of the three-body problem. E-symplectic manifolds include symplectic manifolds with boundary, manifolds with corners, compactified cotangent bundles and regular...
