Termine

(Joint work with Y. Barnea, M. Ershov, A. Le Boudec, M. Vannacci and Th. Weigel.) The commensurator of a group encapsulates (up to a suitable equivalence) all isomorphisms between finite index subgroups of the group. We study the commensurator of a free group F and of a free pro-p group, and also the p-commensurator of F (which is the subgroup of the commensurator that respects the pro-p topology on F), with a focus on normal subgroup structure. As well as 'global' results about the commensurator as a whole, we obtain some new constructions of simple groups: finitely generated simple groups with a free commensurated subgroup, and nondiscrete compactly generated simple locally compact groups that possibly have a free pro-p open subgroup.

To every ring one can associate its algebraic K-groups, which form a rich invariant and play a role in several areas of mathematics. However, these groups are often difficult to compute. When a ring is constructed from simpler pieces, a descent result provides a way to recover its K-groups from the K-groups of these building blocks. In recent years, techniques from higher algebra and derived algebraic geometry have led to new descent results, with applications extending beyond algebraic K-theory itself. In this talk, I will give an introduction to this circle of ideas and discuss some applications.

1st talk:
K-theoretic classification works surprisingly well for C*-algebras which - are separable and simple - have approximations (in terms of finite-dimensional algebras) which also are topologically finite-dimensional in a suitable sense - have some underlying (amenable) dynamical structure. A long-term project within the CRC is it to understand and perhaps even to classify such underlying structures for a given classifiable C*-algebra. I will explain how the principles above can be used to write the well-known CAR algebra (an infinite tensor product of 2x2 matrices) in new ways, with substantially different underlying dynamical structures, and how the latter can be distinguished in terms of dimension-type invariants. The examples are based on the so-called paper-folding subshift from symbolic dynamics. (This is joint work with Grigoris Kopsacheilis.)

2nd talk:
I will talk about stable rank one, that is, the density of invertible elements in a C*-algebra. For commutative C*--algebras, this property occurs precisely in topological dimension zero or one, and can therefore be interpreted as a notion of noncommutative low dimensionality. Stable rank one guarantees more accessible computations of K-theory and other C*-algebraic invariants. The space Act(G,X) of actions of a countable group G on the Cantor set X by homeomorphisms carries a natural Polish topology, making it amenable to methods from Baire category theory. By restricting to suitable Polish subspaces of Act(G,X), we investigate situations in which stable rank one occurs generically, that is, on a dense G_\delta subset, for the associated crossed product C*-algebras. This is joint work with Jamie Bell and David Kerr.

Further information

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The analysis of Dirac-type equations introduces structural challenges, which are absent in wave or Klein?Gordon equations, due to their first-order, and spinorial nature. In particular, the natural notion of energy for Dirac fields is derived from conserved currents rather than stress?energy tensors. The main focus of this talk is a spinor-adapted geometric, current-based energy framework which treats Dirac equations directly, without reduction to wave equations. I explain why this shift becomes necessary in the presence of gauge coupling and discuss how Dirac currents, gauge-covariant vector field methods, and divergence identities naturally interact with the Maxwell field, which supplies decay through its own energy hierarchy. I conclude with an overview of the current status of the Maxwell?Dirac system near Minkowski spacetime