Morse homology of the complex Chern—Simons functional from derived symplectic geometry

• Pavel Safronov (University of Edinburgh)

Room: 5323 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 connections to other objects, such as skein modules of 3-manifolds and Donaldson-Thomas invariants. This talk is based on work in progress joint with Sam Gunningham.

Infinite staircases in the symplectic ball packing problem

• Ana Rita Pires (University of Edinburgh)

Room: 5323 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 Fibonacci numbers, that fit together to form an infinite staircase. This ellipsoid embedding function can be equally defined for other targets, and in this talk I discuss for which other targets the function also has an infinite staircase. In the case when these targets can be represented by lattice (moment) polygons, the targets seem to be exactly those whose polygon is reflexive (i.e., has one interior lattice point). In a specific family of irrational polygons, the answer involves self-similar behavior akin to the Cantor set. This talk is based on various projects, joint with Dan Cristofaro-Gardiner, Tara Holm, Alessia Mandini, Maria Bertozzi, Tara Holm, Emily Maw, Dusa McDuff, Grace Mwakyoma, Morgan Weiler, and Nicki Magill.

Symplectic geometry of Coulomb branches

• Gwyn Bellamy (University of Glasgow)

Room: 5323 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 conjectured that these spaces should be examples of symplectic singularities. In this talk I'll describe some of the basic properties of these spaces and outline a simple proof of the fact that they have do indeed have symplectic singularities.

Quantum cohomology as a deformation of symplectic cohomology

• Nick Sheridan (University of Edinburgh)

Room: 5323 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 give an idea of what quantum cohomology and symplectic cohomology are. We will also show that the `skeleton' of M \ D has strong symplectic rigidity properties in this context, and explain some conjectures about what happens when our criterion is not satisfied. If time permits we will explain the relationship with mirror symmetry. This is joint work with Strom Borman and Umut Varolgunes.

From symplectic to Poisson manifolds and back

• Eva Miranda (Universitat Politècnica de Catalunya + CRM-Barcelona)

Room: Yew Lecture Theatre 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 symplectic foliations. Their deformation quantization was studied à la Fedosov by Nest and Tsygan. How general can such structures be? In this talk, I first explain how to associate an E-symplectic structure to a Poisson structure with transverse structure of semisimple type (joint work with Ryszard Nest) and I will connect this to a result by Cahen, Gutt and Rawnsley on tangential star products. This result illustrates how E-symplectic manifolds serve as a trampoline to the investigation of the geometry of Poisson manifolds and the different facets of their quantization. This should let us address a number of open questions in Poisson Geometry and the study of its quantization from a brand-new perspective.