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Abstract. This talk concerns the local topology of Riemannian man- ifolds with lower bounds on intermediate Ricci curvature. We obtain local topological restrictions for non-collapsed sequences of complete Riemannian manifolds satisfying Rick ? K. The proof is based on the construction of partially concave functions, using the cone-like structure of these spaces. This work is joint with Philipp Reiser.

The study of p-adic deformations of automorphic forms was initiated by Hida in the 1980s, following his discovery of systematic congruences between the Fourier coefficients of modular forms. The eigencurve, introduced by Coleman and Mazur, offers a geometric framework for understanding these congruences, and has since become a central tool in tackling deep number-theoretic conjectures, such as the Birch and Swinnerton-Dyer conjecture. At non-critical classical points of integer weight k>1, the eigencurve is known to be smooth, thanks to the classicality theorems of Hida and Coleman. In contrast, the structure of the eigencurve at weight k=1 is significantly more subtle and intricate. In this talk, after reviewing the seminal work of Bellaïche and Dimitrov in the so-called p-regular case (i.e. when crystalline Frobenius does not act by a scalar), I will present joint work with Adel Betina and Alice Pozzi giving a complete description of the local geometry in the more delicate p-irregular case. Time permitting, I will also discuss ongoing work comparing the completed local rings of the eigencurve at p-irregular weight one points with suitable Galois deformation rings.

I will speak on the classification problem of compact homogeneous Einstein spaces G/H. Recall that a G-invariant Einstein metric can be characterized as a critical point of the scalar curvature function restricted to the space of G-invariant metrics of volume one. To this end, I will introduce a new simplicial complex (defined by certain intermediate subgroups H

On an oriented surface M, the dynamics of a charged particle in a magnetic field is governed by a pair (g,\lambda) that consists of a Riemannian metric g together with a smooth function \lambda modelling the magnetic field. If every unit speed particle moves on a closed orbit (and the minimal period depends continuously on the orbit), we call (g,\lambda) a Zoll magnetic system. We show that there is a plethora of such systems on every closed oriented surface, essentially one for every closed 1-form on M. In negative Euler characteristic these are the first examples beyond the trivial case (constant curvature and large constant magnetic field). The construction of Zoll magnetic systems is based on so-called transport twistor spaces and holomorphic blow-down maps into ruled surfaces. Based on joint work with Gabriel P. Paternain.

Kasparov's bivariant K-theory (or KK-theory) is a powerful invariant for both C*-algebras and C*-dynamical systems, which was originally motivated as a tool to solve classical problems coming from topology and geometry. Its paramount importance for classification theory was discovered soon after, impressively demonstrated within the Kirchberg-Phillips theorem to classify simple nuclear and purely infinite C*-algebras. Since then, it can be said that every methodological novelty about extracting information from KK-theory brought along some new breakthrough in classification theory. Perhaps the most important example of this is the Lin-Dadarlat-Eilers stable uniqueness theorem (with a recent elegant proof by Cuntz), which forms the technical basis behind many of the most important articles written over the past decade. In the landmark paper of Carrion et al, it was demonstrated how the stable uniqueness theorem can be upgraded to a uniqueness theorem of sorts under extra assumptions. It was then posed as an open problem whether the statement of a desired "KK-uniqueness theorem" always holds. The recent work of Hua-White concerning nuclear embeddings of C*-algebras into II1-factors is based on a partial solution to this problem. In this talk I want to present the general affirmative answer: If A and B are separable C*-algebras and (f,g) is a Cuntz pair of absorbing representations whose induced class in KK(A,B) vanishes, then f and g are strongly asymptotically unitarily equivalent. The talk shall focus on the main conceptual ideas towards this theorem, and I plan to discuss variants of the theorem if time permits. It turns out that the analogous KK-uniqueness theorem is true in a much more general context, which covers equivariant and/or ideal-related and/or nuclear KK-theory.

Kolloquium über Geschichte und Didaktik der Mathematik