Termine

The eigenvalues of the Laplacian are quantities that underlie many physical phenomena including, for instance, heat conduction or overtones of a drum. In 1974, Kac asked (and in a way answered) the question of how the eigenvalues of the Laplacian change when the underlying domain is perforated by a large number of tiny holes. Soon thereafter, Rauch and Taylor completed Kac? analysis while linking the problem to a cocktail-party-level question of how fine one should crush the ice cubes to maximize its cooling effect on a drink (read: ambient liquid). Disregarding the analogies, they concluded that the correct quantity to look at is the capacity density of the perforations; scaling the number of perforations by inverse capacity then makes the eigenvalues tend to those of an effective (deterministic) Schrödinger operator as the perforation diameters tend to zero. In the talk, I will review these findings relying at first on the classical connection to Wiener sausage which earned this problem much independent attention by probabilists over several decades. I will then proceed to discuss how one can capture the fluctuations of the eigenvalues and prove a Central Limit Theorem for the eigenvalues that are simple in the aforementioned limit. The method of proof in this part is quite different, relying largely on martingale representation and rank-one type of perturbations. For centering by expected eigenvalues the CLT holds in all dimensions 2 and above. For centering by limiting eigenvalues one has to restrict to dimensions less than 6 as non-trivial corrections arise in other cases. The talk is based on an upcoming joint work with Ryoki Fukushima (University of Tsukuba).

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Shrinking gradient Kähler-Ricci solitons model finite-time singularities of the Kähler-Ricci flow on compact Kähler manifolds. I will discuss the existence problem for shrinking gradient Kähler-Ricci solitons in the non-compact toric setting. This talk is based on joint work with Ivin Babu and Alix Deruelle.

Let $G$ be an even orthogonal group over a $p$-adic local field. Given a parahoric subgroup $K_p$ of $G$ and a minuscule geometric cocharacter $\mu$, one can associate a canonical local model, which is a projective scheme over the ring of $p$-adic integers whose generic fiber is the flag variety attached to $\mu$. When $p>2$ and the triple $(G,K_p,\mu)$ arises from a global Shimura datum, it is known that the local model is étale locally isomorphic to the canonical integral model of the corresponding Shimura variety. In this talk, we focus on the case of PEL type D. We prove that the spin local models introduced by Pappas-Rapoport are isomorphic to the corresponding canonical local models. This confirms a conjecture of Pappas-Rapoport. As a corollary, we obtain an explicit moduli interpretation for flat integral moduli spaces of PEL type D. This talk is partly based on joint work with Ioannis Zachos and Zhihao Zhao.