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.

Frames provide a mathematical framework for stably representing signals as linear combinations of basic building blocks that constitute an overcomplete collection. Finite frames are frames for finite dimensional spaces, and are especially suited for many applications in signal processing. The inherent redundancy of frames can be exploited to build compression and transmission algorithms that are resilient not only to lost of information but also to noise. For instance, tight frames constitute a particular class of frames that play important roles in many applications. After giving an overview of finite frame theory, I will consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors. A frame that positively answers this question will be called scalable. I will give various characterizations of the set of scalable frames, and present some topological descriptions of this set. (This talk is based on joint work with G. Kutyniok, F. Philipp and E. Tuley). Tea at 3:30 pm in SC 1425.

I will present the construction of the Poisson Boundary of a group, originally defined by Furstenberg, and explain its various properties and applications. The Poisson Boundary can be thought of as the exit boundary of a random walk on the group and can be identified with the space of harmonic functions on the group. The first talk will focus on the construction of the Poisson Boundary and various results due primarily to Furstenberg and Zimmer about boundaries. The second talk will focus on the dynamical behavior of the boundary and its applications to ergodic theory.