Zeit und Ort

Montag, 2.2.26, 16:15–17:45, Seminarraum 404

Zeit und Ort

Montag, 26.1.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

In this talk, I will give an overview of my current research on Dirac operators on conical domains. I will outline the main ideas behind the proof that the Dirac operator is regular and self-adjoint for a large class of local boundary conditions. Previously, such results were only available for two-dimensional corner domains under very restrictive boundary conditions, or for rotationally symmetric convex three-dimensional cones equipped with MIT bag boundary conditions.

Recently, Pankrashkin proved self-adjointness of the Dirac operator with MIT bag boundary conditions on all (possibly non-smooth) convex domains. Combining this result with my own, the class of admissible domains extends to a huge family of domains with convex edges and arbitrary (non-convex, non-Lipschitz, non-connected, ...) "smooth" cones. That said, non-convex polyhedra are not yet understood.

Zeit und Ort

Montag, 19.1.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

We discuss smooth local boundary conditions for Dirac-type operators, giving existence and non-existence results for local self-adjoint boundary conditions and discuss concrete parametrizations of the space of all those conditions in low dimensions. We also give concrete conditions when these boundary conditions are elliptic/regular/Shapiro-Lopatinski (i.e. in particular giving rise to self-adjoint Dirac operators with domain in H^1). This is joint work with Hanne van den Bosch (Universidad de Chile) and Alejandro Uribe (University of Michigan).

Zeit und Ort

Montag, 15.12.25, 16:15–17:45, Seminarraum 404

Zeit und Ort

Montag, 1.12.25, 16:15–17:45, Seminarraum 404

Zeit und Ort

Montag, 24.11.25, 16:15–17:45, Seminarraum 404

Zusammenfassung

Within Berger’s classification of holonomy groups, G₂ is the distinguished case in dimension seven, and a G₂-holonomy metric determines a parallel 3-form ϕ. As in other special geometries, the existence of such metrics imposes topological constraints on compact manifolds; analogues in Kähler geometry include the hard Lefschetz property, the Hodge decomposition, and formality. Formality, first discovered as a property of compact Kähler manifolds by Deligne, Griffiths, Morgan, and Sullivan in 1975, depends on the rational homotopy type of a manifold.

We review recent developments in the topology of compact holonomy G₂ manifolds by focusing on two results: one showing that compact holonomy G₂ manifolds need not be formal (arXiv:2409.04362), and another presenting examples of compact closed G₂ manifolds (dϕ=0) that satisfy all known topological obstructions to admitting holonomy G₂ metrics, for which the existence of such metrics cannot be confirmed or excluded with current techniques.

Zeit und Ort

Montag, 20.10.25, 16:15–17:45, Seminarraum 404

Zusammenfassung

The space of (local) symmetries of a given geometric structure has the natural structure of a Lie (pseudo)group. Conversely, geometric structures admitting a local model can be described via the pseudogroup of symmetries of such local model.

The main goal of this talk is to provide several examples and give an intuitive understanding of the slogan above, which can be made precise at various levels of generality (depending on the definition of "geometric structure") and using different tools/methods.

Moreover, I will sketch a new framework, which include previous formalisms (e.g. G-structures or Cartan geometries) and allows us to prove integrability theorems. In particular, I will provide intuition on the relevant objects which make this approach work, namely Lie groupoids endowed with a multiplicative "PDE-structure" and their principal actions. Poisson geometry will give us the guiding principles to understand those objects, which are directly inspired from, respectively, symplectic groupoids and principal Hamiltonian bundles.

This is based on a forthcoming book written jointly with Luca Accornero, Marius Crainic and María Amelia Salazar.