Vorträge des SFB 1442

Abstract: For an étale topological groupoid, we consider a dualisable category of sheaves associated to it and study its K-theory. In the case of groupoids arising from proper group actions, we show that K-theory is controlled by a sheaf which is constructible with respect to the stratification by orbit types. This is based on ongoing work joint with D. Mukherjee and T. Nikolaus.

Main objects in function field arithmetic are Anderson A-modules and their A-motives and A-comotives (dual A-motives). Of particular interest are those which are "abelian", since their A- motives are finitely generated, and those which are "coabelian", since their A-comotives are finitely generated. U. Hartl showed that for an Anderson A-module that is abelian and coabelian, there is a perfect pairing between its motive and its comotive, and hence an isomorphism between the dual of the motive, and the comotive. The pairing, however, is not given explicitly, and Hartl asks for such an explicit description. It also remained open whether abelian A-modules are the same as coabelian A-modules. After introducing the main objects, we will explain the main tasks. We will show that indeed abelian and coabelian are equivalent notions, and provide an explicit pairing between the A-motive and the A-comotive when the Anderson A-module is abelian/coabelian. Due to the nature of the A-motive and A-comotive, we will deal a lot with non-commutative polynomial rings and non-commutative Laurent series fields.

Abstract: Motivated by Dieudonné theory, V. Drinfeld and E. Lau introduced a "decompletion" of the ring of Witt vectors W(R) of a derived p-complete ring R such that (R/p)_{red} is perfect, extending a construction of T. Zink. I will explain various characterizations of this decompletion (called the sheared Witt vectors) and some examples. I will also explain two applications: a conjectural description of p-divisible groups over such rings (due to Drinfeld and Lau), and a decompletion of the stack of prismatization. (Joint work in progress with Bhargav Bhatt, Vadim Vologodsky, and Mingjia Zhang, and partially also with Artem Kanaev.)

In broad terms, classification in mathematics means finding a clear and useful way to describe the isomorphism classes of a given category, such as those of finite simple groups or division algebras. In the case of C*-algebras and groupoids, the main goal is to classify objects up to Morita equivalence. Morita equivalence can be described by the existence of a pair of mutually inverse imprimitivity bimodules. This suggests that the right setting is the category whose objects are C*-algebras and whose arrows are isomorphism classes of Hilbert bimodules. However, once we identify bimodules up to isomorphism, the resulting category becomes non-concrete and difficult to handle. A better approach is to include bimodule homomorphisms as a second layer of arrows, which gives a bicategory (a weak 2-category). Many well-known C*-algebraic constructions, such as Cuntz?Pimsner algebras, can then be described by universal properties in this bicategory. I will explain how this framework leads to a gauge-equivariant homotopy classification of graph C*-algebras of finite regular graphs. The talk is based on joint work with Adam Dor-On and Efren Ruiz.

A first-order sentence in the language of group theory is a mathematical statement whose variables refer only to elements of a group, and two groups are said to be elementarily equivalent if they satisfy the same first-order sentences. I will introduce these concepts using simple examples, then turn to a question posed by Tarski in the 1940s, known as the Tarski problem: are non-abelian free groups elementarily equivalent? Despite the apparent simplicity of its formulation, this problem remained open until the early 2000s, when it was solved by Sela and by Kharlampovich and Myasnikov.

I will give an introduction to Lue Pan?s approach to the study of p-adic completed cohomology of Shimura varieties, focusing on the example of quaternionic Shimura curves. I will then discuss applications to locally analytic representations of the multiplicative group of the non-split quaternion algebra over Qp. This talk is based on joint work with Zhenghui Li and Benchao Su.