Project IV

Algebraic and operator-theoretic aspects of ergodic theory

Gabriel Fuhrmann

Description. Ergodic Theory is an area in mathematics in which one studies the evolution of deterministic systems through the lens of the measure theory. It has vast applications in various other branches of mathematics (e.g. geometry, number theory, topology) but also applies tools from various other branches to its own problems.

In this project, we will focus on such tools which originate from an algebraic and/or operator-theoretic background. On the more algebraic side, this may be the enveloping semigroup of a dynamical system or dimension groups (associated to certain homeomorphisms on the Cantor set); on the operator-theoretic side, this may be the notion of discrete spectrum and its application in the classification of so-called minimal equicontinuous systems. Don’t worry if none of this makes sense to you now, it is exactly the goal of this project that you get familiar with these notions in the first place.

There is a vast amount of literature available (and there is something for everyone’s taste) but for a start [1] and [2] may be good places to go.

Prerequisites. Depending on the exact direction you will be heading, Analysis 3 may be very useful. If you prefer to cover more topological topics, this would require Topology 3 instead.

Literature.

[1] Tanja Eisner, Bálint Farkas, Markus Haase, and Rainer Nagel. Operator theoretic aspects of ergodic theory. volume 272 of Graduate Texts in Mathematics. Springer, Cham, 2015.

[2] Harry Furstenberg. Recurrence in ergodic theory and combinatorial number theory. Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures.