An overview on Lie pseudgroups and geometric structures
Monday, 20.10.25, 16:15-17:45, Seminarraum 404
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.
Topological aspects of compact holonomy and closed G₂ manifolds
Monday, 24.11.25, 16:15-17:45, Seminarraum 404
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.
Tangentialer Sp(1)-Bordismus
Monday, 1.12.25, 16:15-17:45, Seminarraum 404
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Monday, 15.12.25, 16:15-17:45, Seminarraum 404
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Monday, 2.2.26, 16:15-17:45, Seminarraum 404