Geometry and Analysis Seminar

The Atiyah-Floer conjecture asserts the instanton Floer homology of a closed three-manifold (constructed via gauge theory) is isomorphic to the Lagrangian Floer homology of a pair of Lagrangian submanifolds associated to a splitting of the three manifold (constructed via symplectic geometry). This conjecture has remained open for more than three decades. In this talk I will explain two compactness results for the SO(3) case of the conjecture in the neck-stretching process. One result is related to the construction of a natural bounding chain in the Lagrangian Floer theory and a conjecture of Fukaya.

I'll be presenting my PhD work, in which I define two new algebraic structures on the equivariant symplectic cohomology of a convex symplectic manifold. The first is a collection of shift operators which generalise the shift operators on equivariant quantum cohomology in algebraic geometry. That is, given a Hamiltonian action of the torus T, we assign to a cocharacter of T an endomorphism of (S1 × T)-equivariant Floer cohomology based on the equivariant Floer Seidel map. The second is a connection which is a multivariate version of Seidel’s q-connection on S1 -equivariant Floer cohomology and generalises the Dubrovin connection on equivariant quantum cohomology.

Counting coherent sheaves on Calabi--Yau fourfolds is a subject in its infancy. An evidence of this is given by how little is known about perhaps the simplest case - counting ideal sheaves of length $n$. On the other hand, the parallel story for surfaces while with many open questions has seen many new results, especially in the direction of understanding virtual integrals over Quot-schemes. Motivated by the conjectures of Cao--Kool and Nekrasov, we study virtual integrals over Hilbert schemes of points of top Chern classes $c_n(L^{[n]})$ and their K-theoretic refinements. Unlike lower-dimensional sheaf-counting theories, one also needs to pay attention to orientations. In this, we rely on the conjectural wall-crossing framework of Joyce. The same methods can be used for Quot-schemes of surfaces and we obtain a generalization of the work of Arbesfeld--Johnson--Lim--Oprea--Pandharipande for a trivial curve class. As a result, there is a correspondence between invariants for surfaces and fourfolds in terms of a universal transformation.

Grothendieck's Quot schemes — moduli spaces of quotient sheaves — are fundamental objects in algebraic geometry, but we know very little about them. This talk will focus on a relatively simple special case: the Quot scheme Quotˡ(E) of length l quotients of a vector bundle E of rank r on a smooth surface S. The scheme Quotˡ(E) is a cross of the Hilbert scheme of points of S (E=O) and the projectivisation of E (l=1); it carries a virtual fundamental class, and if l and r are at least 2, then Quotˡ(E) is singular. I will explain how the ADHM description of Quotˡ(E) provides a conjectural description of the singularities, and show how they can be resolved in the l=2 case. Furthermore, I will describe the relation between Quotˡ(E) and Quotˡ of a quotient of E, prove a functoriality result for the virtual fundamental class, and use it to compute certain tautological integrals over Quotˡ(E).

The torus T of projective space also acts on the Hilbert
scheme of subschemes of projective space, and the T-graph of the
Hilbert scheme has vertices the fixed points of this action, and edges
the closures of one-dimensional orbits. In general this graph depends
on the underlying field. I will discuss joint work with Rob
Silversmith, in which we construct of a subgraph, which we call the
spine, of the T-graph of Hilb^N(A^2) that is independent of the choice
of field. The key technique is an understanding of the tropical ideal,
in the sense of tropical scheme theory, of the ideal of the universal
family of an edge in the spine.

We prove a homological mirror symmetry result for a one-parameter family of genus 2 curves (https://arxiv.org/abs/1908.04227), and then mention current joint work with H. Azam, H. Lee, and C.-C. M. Liu on generalizing this to the 6-parameter family of all genus 2 curves.

First we describe the B-model genus 2 curve in a 4-torus and the geometric construction of the generalized SYZ mirror. Then we set up the Fukaya-Seidel category on the mirror. Finally we will see the main algebraic HMS result on homogenous coordinate rings, which is at the level of cohomology. The method involves first considering mirror symmetry for the 4-torus, then restricting to the hypersurface genus 2 curve and extending to a mirror Landau-Ginzburg model with fiber the mirror 4-torus. 

Unlike for closed manifolds, the existence of positive scalar curvature (psc) metrics on connected manifolds with
nonempty boundary is unobstructed. We study and compare the spaces of psc metrics on such manifolds with various
conditions along the boundary: H ≥ 0, H = 0, H > 0, II = 0, doubling, product structure. Here H stands for the
mean curvature of the boundary and II for its second fundamental form. "Doubling" means that the doubled metric
on the doubled manifold (along the boundary) is smooth and "product structure" means that near the boundary the
metric has product form. We show that many, but not all of the obvious inclusions are weak homotopy equivalences.
In particular, we will see that if the manifold carries a psc metric with H ≥ 0, then it also carries one which is
doubling but not necessarily one which has product structure. This is joint work with Bernhard Hanke.

The idea of assigning weights to local coordinate functions is used in many areas of mathematics, such as singularity theory, microlocal analysis, sub-Riemannian geometry, or the theory of hypo-elliptic operators, under various terminologies. In this talk, I will describe some differential-geometric aspects of weightings along submanifolds. This includes a coordinate-free definition, and the construction of weighted normal bundles and weighted blow-ups. As an application, I will describe a canonical local model for isotropic embeddings in symplectic manifolds. (Based on joint work with Yiannis Loizides.)

In the talk I shall discuss an approach to the localisation technique, for spaces satisfying the curvature-dimension condition, by means of L1-optimal transport. Moreover, I shall present recent work on a generalisation of the technique to multiple constraints setting. Applications of the theory lie in functional and geometric inequalities, e.g. in the Lévy-Gromov isoperimetric inequality.