Vorträge des SFB 1442

Recent years have seen substantial progress in understanding algebraic K-theory of singular varieties. Still, computations of (higher) K-theory of singular varieties are quite rare, especially in dimension >1. In this talk, I will explain some techniques that allow one to perform computations in higher dimensions, with a particular focus on cubic surfaces and threefolds, where complete computations can be obtained in many cases.

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.

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.

The Bernstein decomposition makes precise how the category of smooth complex representations of a p-adic reductive group G is built from supercuspidals. The center of the category factors accordingly and is largely well-understood. If one replaces the complex numbers by a coefficient field of characteristic p, the analogous category has strikingly different features. For one, the (Bernstein) center is rather small and often determined by the center of G. This was shown by Ardakov and Schneider for a general locally pro-p group G (under a mild hypothesis). In this talk, I will discuss the corresponding question in the derived setting. I will present various results related to the graded center of the derived category of smooth mod p representations. The talk is based on joint work (in progress) with Peter Schneider.

For any prime $p$, a finite group has as many irreducible complex characters of degree prime to $p$ as the normalizer of a Sylow $p$-subgroup. This equality, conjectured by John McKay in 1971, was reduced in 2007 by Isaacs--Malle--Navarro to a conjecture on representations of finite simple groups. Thanks to their classification we know that the latter are essentially finite groups of Lie type. Deligne--Lusztig theory helps to prove the McKay conjecture by this approach. For groups of characteristic different from $p$, the normalizers of Sylow $p$-subgroups belong to a larger class of subgroups related to parabolic subgroups of the ambient algebraic group for which Deligne-Lusztig varieties and induction functors have been used in the 1990s to provide a substitute to parabolic induction. This works well for unipotent characters. An important step is the construction of a Jordan decomposition of characters which is equivariant with respect to automorphisms of the simple group.