In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik and Ryszard Szwarc, constructed and studied a certain Fredholm operator associated to a simplicial tree. The operator can be defined in at least two ways: from a combinatorial flow on the tree, similar to the flows in Forman's discrete Morse theory, or from the theory of unitary operator-valued cocycles. There are applications of the theory surrounding the operator to $C}^{*$-algebra $K$-theory, to the theory of completely bounded representations of groups that act on trees, and to the Selberg principle in the representation theory of $p$-adic groups.
This talk will present an extension of the constructions of Julg and Valette, and Pytlik and Szwarc, to CAT(0) cubical spaces which relies on the beautiful geometry of nonpositively curved spaces. The utility of this construction is demonstrated through a new proof of $K$-amenability for groups that act properly on bounded-geometry CAT(0)-cubical spaces. This talk is based on joint work with Nigel Higson and Erik Guentner.
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I will review recent results and open problems concerning constructions and ergodicity of dissipative dynamics for classical large interacting systems. In particular I will talk about dynamics defined by formally hypoelliptic generators and some generalised Dunkl type extensions, and how to control their long time behaviour (by use of coercive inequalities or generalised gradient type bounds).