Friday Fish seminar

In this fourth lecture on the Seiberg–Witten invariants we will finally introduce the Seiberg–Witten equations and set up
the basic framework we will need to study the moduli space of its solutions. We will start by studying basic properties of spinor
bundles and Dirac operators. Then we will introduce the equations and their gauge symmetries, and explain how they can be
extended to Sobolev spaces so that weak solutions are allowed. If time permits, we will briefly sketch how the Seiberg–Witten
equations can be regarded as the smallest topologically allowed minima of the Seiberg–Witten action functional.

The concept of a “differentiable stack” is a higher geometric notion similar to algebraic or topological stacks, aiming to transfer the study of manifolds to a more general class of “differential geometric spaces”. Examples include orbifolds, quotients of Lie group actions and many moduli spaces. While understanding differentiable stacks by unraveling their definition as categories is possible, there is also another way to approach their study: It can be seen that the 2-category of differentiable stacks is equivalent to a 2-category of Lie groupoids. Using this relation, it is possible to expand the mathematical toolbox applicable to either concept, for example, when computing the cohomology of a differentiable stack. It is possible to express it in terms of the associated Lie groupoid, which, in particular, frames differentiable stack cohomology as a generalization of equivariant cohomology. The existence of established models for equivariant cohomology has motivated research into finding generalised models for differentiable stack cohomology as well. In this talk, we will explore the problem and its key notions as well as its challenges. We will cover some approaches of generalizing preexisting models for equivariant cohomology including a recent advancement in the case of proper and regular Lie groupoids.

During the last talk, we were introduced to the spin and spin^c group and its
representation. In the 3rd talk of the Seiberg-Witten seminar, we will go from this linear algebra picture to the case of bundles. Specifically, we will focus on obstructions to the existence of spin and spin^c structure on manifolds. From this, we will see that all simply connected 4-manifold admit a spin^c structure. Finally, we will introduce spinors. Along the way, we will cover plenty of examples.

An exterior form is termed stable if its algebraic properties are preserved by all sufficiently small perturbations. Stable forms are fundamental to the study of G2/[split]G2 geometry and symplectic geometry, and additionally play a key role in the study of other geometries of interest, such as contact geometry.

In this talk, I shall introduce a new, general method for proving h-principles for stable forms, building on Gromov's technique of convex integration, and use this new method to prove 4 new h-principles related to [split]G2, symplectic and contact geometries. Applications to the non-constructability of compact [split]G2 and symplectic manifolds via geometric flows will then be discussed. Time permitting, I shall also explain (i) how this new method for proving h-principles subsumes all similar, previously established h-principles, and (ii) how this method may be extended to prove a fifth new h-principle in 6-dimensions related to para-complex geometry.

In this talk we will set the linear algebra preliminaries for the formulation of the Seiberg–Witten equations. These include Clifford algebras, Spin and Spin^c groups and spinor representations. Although the theory can be presented in great generality, for the sake of clarity we will often restrict to the case of interest for us: that of positive definite metrics on even-dimensional vector spaces.