Past Geometry and Analysis Seminar

23 November 2020
14:15
Vidit Nanda
Abstract

Given a nested pair X and Y of complex projective varieties, there is a single positive integer e which measures the singularity type of X inside Y. This is called the Hilbert-Samuel multiplicity of Y along X, and it appears in the formulations of several standard intersection-theoretic constructions including Segre classes, Euler obstructions, and various other multiplicities. The standard method for computing e requires knowledge of the equations which define X and Y, followed by a (super-exponential) Grobner basis computation. In this talk we will connect the HS multiplicity to complex links, which are fundamental invariants of (complex analytic) Whitney stratified spaces. Thanks to this connection, the enormous computational burden of extracting e from polynomial equations reduces to a simple exercise in clustering point clouds. In fact, one doesn't even need the polynomials which define X and Y: it suffices to work with dense point samples. This is joint work with Martin Helmer.

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  • Geometry and Analysis Seminar
16 November 2020
14:15
Andrea Mondino
Abstract

In the talk I will survey the fast growing field of metric measure spaces satisfying a lower bound on Ricci Curvature, in a synthetic sense via optimal transport. Particular emphasis will be given to discuss how such (possibly non-smooth) spaces naturally (and usefully) extend the class of smooth Riemannian manifolds with Ricci curvature bounded below.

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  • Geometry and Analysis Seminar
9 November 2020
14:15
Davesh Maulik
Abstract

In this talk, I will discuss some results on the structure of the cohomology of the moduli space of stable SL_n Higgs bundles on a curve. 

One consequence is a new proof of the Hausel-Thaddeus conjecture proven previously by Groechenig-Wyss-Ziegler via p-adic integration.

We will also discuss connections to the P=W conjecture if time permits. Based on joint work with Junliang Shen.

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  • Geometry and Analysis Seminar
2 November 2020
14:15
Egor Shelukhin
Abstract

We describe how Smith theory applies in the setting of Hamiltonian Floer homology filtered by the action functional, and provide applications to questions regarding Hamiltonian diffeomorphisms, including the Hofer-Zehnder conjecture on the existence of infinitely many periodic points and a question of McDuff-Salamon on Hamiltonian diffeomorphisms of finite order.

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  • Geometry and Analysis Seminar
26 October 2020
14:15
Ben Davison
Abstract
Given a suitably friendly category D we can take the 3-Calabi Yau completion of D and obtain a 3-Calabi-Yau category E. The archetypal example has D as the category of coherent sheaves on a smooth quasiprojective surface, then E is the category of coherent sheaves on the total space of the canonical bundle - a quasiprojective 3CY variety. The moduli stack of semistable objects in the 3CY completion E supports a vanishing cycle-type sheaf, the hypercohomology of which is the basic object in the study of the DT theory of E. Something extra happens when our input category is itself 2CY: examples include the category of local systems on a Riemann surface, the category of coherent sheaves on a K3/Abelian surface, the category of Higgs bundles on a smooth complete curve, or the category of representations of a preprojective algebra. In these cases, the DT cohomology of E carries a cocommutative coproduct. I'll also explain how this interacts with older algebraic structures in cohomological DT theory to provide a geometric construction of both well-known and new quantum groups.

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  • Geometry and Analysis Seminar
19 October 2020
14:15
Hector Papoulias
Abstract

The Spin(7) and SU(4) structures on a Calabi-Yau 4-fold give rise to certain first order PDEs defining special Yang-Mills connections: the Spin(7) instanton equations and the Hermitian Yang-Mills (HYM) equations respectively. The latter are stronger than the former. In 1998 C. Lewis proved that -over a compact base space- the existence of an HYM connection implies the converse. In this talk we demonstrate that the equivalence of the two gauge-theoretic problems fails to hold in generality. We do this by studying the invariant solutions on a highly symmetric noncompact Calabi-Yau 4-fold: the Stenzel manifold. We give a complete description of the moduli space of irreducible invariant Spin(7) instantons with structure group SO(3) on this space and find that the HYM connections are properly embedded in it. This moduli space reveals an explicit example of a sequence of Spin(7) instantons bubbling off near a Cayley submanifold. The missing limit is an HYM connection, revealing a potential relationship between the two equation systems.

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  • Geometry and Analysis Seminar
12 October 2020
14:15
Lothar Gottsche
Abstract

This is a report on joint work with Martijn Kool. 

Recently, Marian-Oprea-Pandharipande established a generalization of Lehn’s conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of higher rank. 

Using Mochizuki’s formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg- Witten invariants and intersection numbers on products of Hilbert schemes of points. We use this to  verify our conjectures in examples. 

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  • Geometry and Analysis Seminar
22 June 2020
14:15
Fatemeh Rezaee
Abstract

In this talk, I will explain a new wall-crossing phenomenon on P^3 that induces non-Q-factorial singularities and thus cannot be understood as an operation in the MMP of the moduli space, unlike the case for many surfaces.  If time permits, I will explain how the wall-crossing could help to understand the geometry of the associated Hilbert scheme and PT moduli space.

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  • Geometry and Analysis Seminar
15 June 2020
14:15
Tom Bridgeland
Abstract

I will describe an ongoing research project which aims to encode the DT invariants of a CY3 triangulated category in a geometric structure on its space of stability conditions. More specifically we expect to find a complex hyperkahler structure on the total space of the tangent bundle. These ideas are closely related to the work of Gaiotto, Moore and Neitzke from a decade ago. The main analytic input is a class of Riemann-Hilbert problems involving maps from the complex plane to an algebraic torus with prescribed discontinuities along a collection of rays.

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  • Geometry and Analysis Seminar
8 June 2020
14:15
Tommaso Pacini
Abstract

Calibrated geometry, more specifically Calabi-Yau geometry, occupies a modern, rather sophisticated, cross-roads between Riemannian, symplectic and complex geometry. We will show how, stripping this theory down to its fundamental holomorphic backbone and applying ideas from classical complex analysis, one can generate a family of purely holomorphic invariants on any complex manifold. We will then show how to compute them, and describe various situations in which these invariants encode, in an intrinsic fashion, properties not only of the given manifold but also of moduli spaces.

Interest in these topics, if initially lacking, will arise spontaneously during this informal presentation.

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  • Geometry and Analysis Seminar

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