# Seminars & Colloquia Calendar

Geometric Analysis Seminar

## Variations of the Z2-Dirac operator

#### Gregory Jacob Parker (MIT)

Location:  Zoom
Date & time: Tuesday, 26 April 2022 at 2:50PM - 3:50PM

Abstract: It is a classical result that on a compact 3-manifold $$Y$$, the set of metrics for which there exists a harmonic spinor of the spin Dirac operator is a codimension 1 subset in the space of all metrics. In this talk, I will discuss an extension of this result to the case of the $$mathbb Z_2$$-Dirac operator, which is defined as the Dirac operator on the complement of a codimension 2 submanifold $$mathcal Zsubseteq Y$$ twisted by a flat connection on $$Y-mathcal Z$$ whose holonomy lies in $$mathbb Z_2$$. The deformation problem for solutions of this operator carries an infinite-dimensional obstruction for a fixed $$mathcal Z$$. Coupling the operator to the geometry of $$mathcal Z$$  by considering the infinite-dimensional family of Dirac operators parameterized by embedded submanifolds gives rise to a Fredholm problem up to a loss of regularity phenomenon. The proof of the result then requires the use of the Nash-Moser Implicit Function Theorem or related techniques.