Geometry and Analysis Seminar

Work by Donaldson, Yang Li et al. introduced into $G_2$ geometry the concept of adiabatic limits and adiabatic associative submanifolds. The underlying $G_2$ manifolds are assumed to admit a very strong structure: a coassociative K3 fibration. 

This seminar will concern the more general situation of G2 manifolds endowed with a calibrated distribution. We will discuss the following issues: 

(i) How to extend the notions of adiabatic limits and adiabatic associative submanifolds to this situation.

(ii) How adiabatic associative submanifolds are related to calibrated geometry, to Fueter maps and to Mirror Symmetry.

(iii) The deformation theory of adiabatic associative submanifolds. In particular, we will show that this deformation theory is similar to that of (classical) associative submanifolds, but it relies on perturbations of the calibrated distribution rather than of the background $G_2$ structure.

This is joint work with K. Kawai (Saitama University - Japan).

Z/2 harmonic spinors arise naturally in gauge theory and calibrated geometry, notably as limiting objects in compactness problems. They are harmonic sections of Dirac operators twisted by Euclidean line bundles ramified along codimension-two subsets. Understanding the analysis of these operators is an important step towards developing a deformation theory of Z/2 harmonic spinors. In this talk, I will discuss the closed extensions of such Dirac operators and their characterization in terms of the Gelfand–Robbin quotient. When the branching locus is a smooth codimension-two submanifold, we construct a geometric realization of this quotient, allowing us to describe Fredholm extensions through residue conditions along the branching locus. I will also discuss elliptic regularity in adapted Sobolev spaces. This is joint work with Thomas Walpuski (arXiv:2503.01392).