Project A: Algebra and logic

In research area A two fundamental principles are used. On the one hand, group representation theory is employed for the investigation of objects coming from number theory and algebraic geometry. On the other hand, the logic based projects A4 and A5 use non-linear group actions for further investigations of algebraic groups. Using methods from model theory and set theory, these projects also reach into topological and geometric questions.

In Project A1 the primary goal is to understand the category of semistable vector bundles on algebraic varieties, in particular curves, over p-adic fields. Vector bundles contain a lot of interesting geometric information about the variety. A natural group which is associated with a variety is its fundamental group. In the analogous classical situation over the complex numbers the Narasimhan-Seshadri correspondence describes stable vector bundles in terms of unitary representations of the topological fundamental group. Over p-adic fields this principle of setting vector bundles into correspondence with representations of fundamental groups is only partially realized so far. One is still far from understanding the precise domain of this correspondence as well as from giving a characterization of its range. These are two of the central questions to be addressed in A1.
Project A2 aims at expanding a program devised by Langlands to understand nonabelian extensions of number fields by means of the representation theory of reductive algebraic groups. The underlying principle again is a correspondence between representations of Galois groups of local or global number fields and representations of reductive groups. The classical Langlands program only deals, in its local form, with l-adic Galois representations of a p-adic field when the prime numbers l and p are different. The category of p-adic Galois representations of a p-adic field is considerably more complicated. The main challenge in A2 is to find an extension of the local Langlands correspondence to p-adic Galois representations.
One of the most important algebraic methods to investigate group representations is to transform them into modules over a group algebra. In the p-adic context the arising p-adic group algebras are still rather mysterious objects which need to be further investigated in their own right. This all the more so as they also turn up in other contexts, for example in global number theory where they act on Selmer groups and are used to construct p-adic L-functions. For the latter purpose it is crucial to compute algebraic K-groups of p-adic group rings. This is the second goal in A2. It builds very much on methods from algebraic topology.
One of the important advantages which one expects from a p-adic extension of the Langlands program is that objects on both sides of the correspondence may vary in families. This leads naturally to the consideration of various kinds of moduli spaces and period domains. This phenomenon also plays a role in A1. Semistable vector bundles on an algebraic curve, for example, form a moduli space which itself is an algebraic variety. One may even pursue the idea that correspondences like the ones described above derive from actual maps between moduli spaces.
Project A3 will be devoted to a systematic study of certain moduli spaces over local and global function fields in positive characteristics which are analogs of Shimura varieties over characteristic zero number fields. The construction of such a moduli space involves a specific reductive group. A main goal of the project is to describe the bad reduction of these moduli spaces in group theoretical terms. Tools are the affine buildings also studied in project B4. It is expected that the picture is somewhat similar to the bad reduction of Shimura varieties. Over function fields however, group theoretic methods are much more powerful. A thorough group theoretic understanding of the situation will most likely also refertilize the theory of Shimura varieties. For certain groups the local moduli spaces also have analogs in characteristic zero, which are moduli spaces of Barsotti-Tate groups. In cooperation between A2 and A3 it will be investigated whether these latter spaces make it possible to understand the p-adic local Langlands correspondence as a map between spaces.
In contrast to projects A1, A2 and A3, project A4 investigates nonlinear actions of algebraic and other, abstractly given groups. The questions addressed here concern characterizations of groups acting on certain natural geometries, like Tits buildings, associated to algebraic groups. Model theory has a natural interest in these questions as from the model theoretic point of view one wants to find characterizations which imply that an abstractly given group is (isomorphic to) an algebraic group. This yields information about the first-order structures involved which can then be used for questions concerning algebraic groups and geometry, like the questions about the definable subsets in valued fields.
Project A5, while still investigating group actions will also use methods from descriptive set theory and set theory. We will investigate actions of Polish groups which resemble algebraic groups in the way they can act and plan to construct new examples. This leads to questions about Borel actions and orbit equivalence relations also considered in Project B7. Our emphasis will be more on the model and set theoretic aspects of these questions than Project A4.


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Publications and preprints of the SFB

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