Past Geometry and Analysis Seminar

Forthcoming events in this series
16 March 2021
14:15
The Quot scheme Quotˡ(E)
Abstract

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).

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8 March 2021
14:15
The spine of the T-graph of the Hilbert scheme
Diane MacLagan
Abstract

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.

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1 March 2021
14:15
Homological mirror symmetry for genus two curves
Catherine Cannizzo
Abstract

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. 

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22 February 2021
14:15
Spaces of metrics of positive scalar curvature on manifolds with boundary
Christian Bär
Abstract

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.

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15 February 2021
14:15
Weightings and normal forms
Eckhard Meinrenken
Abstract

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.)

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8 February 2021
14:15
Punctured invariants and gluing
Dan Abramovich
Abstract
Associativity in quantum cohomology is proven using a gluing formula for Gromov-Witten invariants. The gluing formula underlying orbifold quantum cohomology has additional interesting features. The Gross-Siebert program requires an analogue of quantum cohomology in logarithmic geometry, with underlying gluing formula for punctured logarithmic invariants. I'll attempt to explain how this works and what new subtle features arise. This is based on joint work with Q. Chen, M. Gross and B. Siebert (https://arxiv.org/pdf/2009.07720.pdf).

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1 February 2021
14:15
Leaf decompositions in Euclidean spaces
Krzysztof Ciosmak
Abstract

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.

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25 January 2021
14:15
Equivariant Lagrangian Floer homology and Extended Field theory
Guillem Cazassus
Abstract
Given a compact Lie group G and a Hamiltonian G-manifold endowed with a pair of G-Lagrangians, we provide a construction for their equivariant Floer homology. Such groups have been defined previously by Hendricks, Lipshitz and Sarkar, and also by Daemi and Fukaya. A similar construction appeared independently in the work of Kim, Lau and Zheng. We will discuss an attempt to use such groups to construct topological field theories: these should be seen as 3-morphism spaces in the Hamiltonian 3-category, which should serve as a target for a field theory corresponding to Donaldson polynomials.

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18 January 2021
14:15
Representation theory in geometric complexity theory
Christian Ikenmeyer
Abstract

Geometric complexity theory is an approach towards solving computational complexity lower bounds questions using algebraic geometry and representation theory. This talk contains an introduction to geometric complexity theory and a presentation of some recent results. Along the way connections to the study of secant varieties and to classical combinatorial and representation theoretic conjectures will be pointed out.

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7 December 2020
11:00
Two perspectives on the stack of principal bundles on an elliptic curve and its slices
Dougal Davis
Abstract

Let G be a reductive group, E an elliptic curve, and Bun_G the moduli stack of principal G-bundles on E. In this talk, I will attempt to explain why Bun_G is a very interesting object from the perspectives of both singularity theory on the one hand, and shifted symplectic geometry and representation theory on the other. In the first part of the talk, I will explain how to construct slices of Bun_G through points corresponding to unstable bundles, and how these are linked to certain singular algebraic surfaces and their deformations in the case of a "subregular" bundle. In the second (probably much shorter) part, I will discuss the shifted symplectic geometry of Bun_G and its slices. If time permits, I will sketch how (conjectural) quantisations of these structures should be related to some well known algebras of an "elliptic" flavour, such as Sklyanin and Feigin-Odesskii algebras, and elliptic quantum groups.

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