Venue: MCM110
Title: Lecture Series on \mathbb{Z}_{2}-harmonic 1-forms and their geometric applications
Abstract: The $\mathbb{Z}_2$-harmonic 1-form arises in various compactification problems in gauge theory, special holonomy and calibrated geometry, including those involving flat $PSL(2,\mathbb{C})$ connections, Fueter sections, branched deformation of special Lagrangians and Donaldson's branched maximal submanifolds in indefinite spaces. In this series of talks, we will focus on the special holonomy aspect of the $\mathbb{Z}_{2}$-harmonic 1-form, with an eye towards to Donaldson’s proposal of adiabatic limit of coassociative Kovalev-Lefschetz fibration.
In talk I, we will describe a recent construction of non-degenerate $\mathbb{Z}_2$ harmonic 1-forms on $\mathbb{R}^n$ for $n \geq 3$, and explore their relation to Lawlor's necks-a family of special Lagrangian submanifolds in $\mathbb{C}^n$. We will also discuss a gluing construction in which these examples are glued to a regular zero of a harmonic 1-form on a compact manifold and explain its geometric idea.
In talk II, we will explain the technical aspects of the gluing construction presented in talk I. Specifically, we adapt Donaldson's framework for deforming multivalued harmonic functions to our gluing setting and establish a weighted version of the Hamilton–Nash–Moser–Zehnder implicit function theorem to prove the gluing result.
In talk III-IV, we will give a brief introduction to background metrical to Donaldson's proposal, such as G_2 manifolds, coassociative fibrations and K3 surfaces.
In talk V, we will go though Donaldson's proposal and discuss heuristically how our gluing technique and examples can be apply to resolve some singularities in the conjectural G_2 manifolds.
