Speaker: Xiaoyu Su
Title: Self- similar C*-algebras and Schur Complement.
I will introduce self-similar representations/C*-algebras associated with self-similar groups, in particular the Cuntz-Pimsner algebra, mentioning possible interesting questions behind it. Also, I will talk a little bit about spectral graph theory (understand how the spectra of
various operators defined on functions on the graph are related to the geometry of the graph), and how to use Schur complement renormalization to find the spectra of certain Markov operators with examples(eg. the Grigorchuk group).
Speaker: Alex Weygandt
Title: Smooth Subalgebras of Group C*-Algebras
A major task in noncommutative geometry is the computation of the K-theory of C*-algebras associated to (discrete) groups, as this is a natural receptacle for invariants of manifolds on which the groups act. One method by which this is achieved is by working with subalgebras which are (hopefully) easier to work with, and give the same K-theory. In this talk, I will discuss a few methods by which such subalgebras are produced, and (time permitting) discuss their applications to topology and geometry.
Speaker: Zheng Kuang
Title: Growth of Groups via Orbital Graphs
I will define the notion of growth of groups and introduce a technique of determining subexponential growth via orbital graphs. Then I will do some examples of growth of groups generated by preperiodic kneading sequences. In the end, I will introduce some open problems of this technique.
Speaker: Alex Mau
Title: The Behavior of the Norm of a Polynomial in a Nearly Algebraic Element of a Normed, Unital, Power Associative Algebra
The notion of a norm is extended in a conventional way to algebras on arbitrary fields with absolute value, and a criterion is presented for determining whether an object $T$ in a normed, unital, power-associative algebra has a minimal polynomial in the minimal extension of the algebra to a normed algebra on the metric completion of the underlying field with respect to its absolute value. A correspondence is stated and proven between the norms of objects expressed as polynomials in $T$ and the norms of related polynomials in the roots of the minimal polynomial of $T$.
Speaker: Zheng Kuang
Title: Groups Generated by Bounded Automata and their Schreier Graphs
I will introduce the notions of groups generated by bounded automata, contracting group actions, and Schreier graphs. Then I will describe a process of plotting Schreier graphs of groups generated by bounded automata. The process is called Inflation of Graphs. If time allows, I will introduce a Python package named kneading_orbital_graph for a visualization of Inflation of Graphs.