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Yuanqi Wang: Lectures

May 14, 2024
TITLE: Spectral theory of singular G2-instantons

ABSTRACT: G2-instantons on 7-dimensional manifolds generalize
both flat connections in dimension 3, and anti self-dual connections in dimension 4. Donaldson-Segal program expects a certain count of G2-instantons and other objects could yield a topological invariant for 7–manifolds, called the prospective G2–Casson invariant. Related to the compactification/boundary of the moduli space, Walpuski proposed to construct singular G2–instantons via gluing. The analytic part of this singular perturbation problem is expected to encounter indicial roots, that are essentially related to the spectrum of a certain Dirac operator on the standard 5-dimensional unit sphere.

In this talk, we report some work on the spectral theory, the consequent obstruction theory, and some expansions of harmonic sections related to these G2-instantons with 1-dimensional singularities. This is the preliminary of a joint project with Thomas Walpuski and Henrique Sá Earp.

Slides of Lecture