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Greg Parker: Lectures

May 14, 2024
TITLE: A Fredholm framework for singular deformation problems.

ABSTRACT: Recent work in several directions has led to singular elliptic operators with infinite-dimensional obstructions, for which the standard Fredholm theory does not apply. In particular, this is the case for singular Fueter sections. My previous work and the work of Donaldson and Takahashi has shown that for Fueter sections, the freedom to deform a singular set may be used to cancel the infinite-dimensional obstruction. The goal of this talk is to explain that this is not a trick for solving a particular problem, but rather the beginning of a unified approach where additional geometric freedom may be used to place these singular problems back within the standard Fredholm framework (almost). As a demonstration of this, I will talk about new gluing results for \mathbb Z_2 harmonic spinors and 1-forms based on joint work with Siqi He.

September 13, 2022
TITLE: Gluing Z2-Harmonic Spinors

ABSTRACT: Z2-harmonic spinors are singular generalizations of classical harmonic spinors and appear in two contexts in gauge-theory. First, they arise as limits of sequences of solutions to equations of Seiberg-Witten type on low-dimensional manifolds; second, they are the simplest type of (singular) Fueter section–objects which arise naturally on calibrated submanifolds in the study of gauge theory on manifolds with special holonomy. These two pictures are related by proposals for defining invariants of manifolds with special holonomy.

In this talk, after giving an introduction to these ideas, I will discuss a gluing result that shows a generic Z2-harmonic spinor on a 3-manifold necessarily arises as the limit of a family of two-spinor Seiberg-Witten monopoles. Due to the singularities of the Z2-harmonic spinor, the relevant operators in the gluing problem are only semi-Fredholm and possess an infinite-dimensional cokernel. To deal with this, the proof requires the analysis of families of elliptic operators degenerating to a singular limit, and the study of deformations of the singularities which are used to cancel the infinite-dimensional cokernel. At then end, I will discuss related problems in gauge theory and calibrated geometry.