Termine

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The configuration spaces Conf(X,n) of a topological space X play an important role in geometric topology. Recent developments have shown that, for fixed spaces X and Y, it can be fruitful to study compatible families of maps from Conf(X,n) to Conf(Y,n) as n varies. There are several ways of assembling together configuration spaces and maps thereof as n varies, including embedding calculus, configuration categories, and Ran spaces. In this talk we show that the mapping spaces arising from the latter two approaches agree. In particular, we prove that the natural map Aut_{/Fin}(Conf(R^d))-->Aut_{/Fin}(Entr(Ran(R^d))) is an equivalence. Here Entr(Ran(R^d)) is the opposite of the exit-path ?-category of the Ran space of R^d equipped with its usual stratification, and Conf(R^d) is the configuration category of R^d. As an application, we prove a corrected version of a conjecture of Ayala, Francis and Tanaka concerning the relationship between Top(d) and Aut_{Fin}(Entr(Ran(R^d))). The main ingredient is a stratified nerve theorem, establishing a presentation of the enter path ?-category of a conically stratified space X-->P with locally weakly contractible strata as a localization of the poset associated with a sufficiently good open cover of X. As a by-product of these methods, we also obtain a much simpler proof of the exodromy equivalence for conically stratified spaces with weakly contractible strata. This is a joint work with Alexander Kupers.

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.

Vaughan Jones and I showed in the late 1990's that whenever a subfactor has an intermediate subfactor, the fundamental subfactor symmetries are captured by what we called the ``Fuss-Catalan algebras''. If the subfactor has two intermediate subfactors in general position, i.e. we are given a quadrilateral of subfactors, describing the quantum symmetries is more complicated and not much progress had been made since the work of Grossman, Izumi and Jones some 20 years ago. I will discuss recent work with Junhwi Lim in which we determine the quantum symmetries of a quadrilateral of subfactors when the four algebras form a co-commuting square and satisfy some additional conditions. The subfactor symmetries we find are related to partition algebras and Bell numbers.