We will talk about relations between ergodic properties of group actions and the structure of group characters. The latter is equivalent to the classification of all finite-type factor representations. The outstanding conjecture (often attributed to Vershik) is that for a large class of groups their group characters must have the form \mu(Fix(g)), where $\mu$ is a G-invariant measure for some special group action on a measure space, Fix(g) is the set of all fixed points of group element $g$. We will then establish Vershik's conjecture for the family of Higman-Thompson groups. Since these groups have no non-trivial ergodic measures, we get that they have no non-trivial factor representations. Examples of other classes of groups known to satisfy Vershik's conjecture will be also discussed. The talk will be based on two recent preprints by Dudko and Medynets, "Finite factor representations of Higman-Thompson Groups" ArXiv 1212.1230 and "On characters of inductive limits of symmetric groups" Arxiv 1105.6325

For a computable algebraic field F, the splitting set S_F of F is the set of polynomials with coefficients in F which factor over F, and the root set R_F of F is the set of polynomials with coefficients in F which have a root in F. In the first part of this talk, on October 2, 2012, we showed that under the bounded Turing (bT) reducibility, determining whether a polynomial has a root in a field F is more difficult than determining whether it factors over F, i.e. S_F is always bT-reducible to R_F, but there are fields F where R_F is not bT-reducible to S_F. In the second part, we will define a Rabin embedding g of a field into its algebraic closure, and for a computable algebraic field F, we compare the relative complexities of R_F, S_F, and g(F) under m-reducibility and under bT-reducibility.

The characterization of all K_{2,3}-minor-free graphs is well-known: 2-connected K_{2,3}-minor-free graphs are either K_4 or outerplanar. In this talk, we provide a characterization of all K_{2,4}-minor-free graphs. For the 3-connected graphs, we have an infinite family which yields 2n-8 graphs on n vertices along with some small special examples on at most eight vertices. The 2-connected graphs are then formed by joining the 3-connected ones with outerplanar graphs subject to some restrictions. This is joint work with Mark Ellingham, Kenta Ozeki and Shoichi Tsuchiya.

In this talk on a more classical part of ergodic theory, that of Z-actions, I will explain the construction of rank-one transformations via cutting and stacking that goes back to von Neumann and Kakutani and has been used to create examples and counterexamples of various mixing-like properties. Following the explanation of the subject, I will present some of my work on when such transformations are mixing. Some of the results presented are joint work with Cesar Silva.

We give a complete description of gradings by abelian groups on certain classes of nilpotent Lie algebras, including Lie algebras of maximal class. This has consequences concerning symmetries on homogeneous spaces of nilpotent Lie groups. These results are joint with Elizabeth Remm and Michel Goze (Univesite de Haut Alsace, France).

Tea at 3:30 pm in SC 1425.

Tea at 3:30 pm in SC 1425.

Tea at 3:30 pm in SC 1425.

Tea at 3:30 pm in SC 1425.