Idisplays

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Im Rahmen des Alumni-Tages hält Prof. Dr. Ralf Schindler einen Vortrag. Abstract: Das erste der im Rahmen des ICM-Vortrags von David Hilbert im Jahre 1900 vorgestellten Probleme fragte, wieviele reelle Zahlen es gäbe. Ist Cantors Kontinuumshypothese wahr? Diese Frage war seither Ansporn und Triebkraft mathematischer Grundlagenforschung. Durch neuere Ergebnisse ist das Problem wieder in den Fokus der Forschung gerückt. Interessierte sind herzlich willkommen!

A central question in the theory of automatic continuity is whether the topology of a topological group is uniquely determined by its algebraic structure. My PhD project investigates this question in the setting of p-adic Lie groups. In this talk, I will discuss the group SL2(Qp) as a first example. We will see that SL2(Qp) admits a unique topology, and I will outline the strategy used to prove this result

Nonlocal energies, such as fractional Sobolev seminorms, arise naturally in mathematical models involving long-range interactions. In this talk, we consider minimizers of such energies vanishing on a set of obstacles with random positions and sizes. This leads to what is commonly called the (bilateral) fractional obstacle problem. Our goal is to obtain a simplified description of the problem in the limit of many obstacles of small average size. Homogenization results of this type are known in both deterministic and random settings, but existing results in the random case typically rely on strong structural assumptions. I will present a new homogenization result that works under substantially weaker hypotheses. The obstacles may be generated by a broad class of spatial random models, allowing both their locations and their sizes to vary in a quite general way. In particular, obstacles are allowed to overlap, giving rise to a complex microstructure. The analysis identifies the limit of the energies and highlights how this reflects the probability distribution of the obstacles. The talk will be kept at an elementary level, starting with the classical local setting based on Poisson?s equation and then transitioning to the nonlocal framework, where the Laplace operator is replaced by its fractional counterpart.

I will explain the result of applying Lurie?s tangent complex formalism to the infinity-category of E_1-groups, and how the resulting deformation-theoretic notions relate to classical concepts in algebraic topology.

Abstract: Within Berger?s classification of holonomy groups, G? is the distinguished case in dimension seven, and a G?-holonomy metric determines a parallel 3-form. 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. Subsequent results lead to the conjecture that special and exceptional holonomy manifolds should be formal. In this talk, we discuss the only counterexample known to date: a compact holonomy G? manifold that is not formal.

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