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.

Secondary invariants are important in geometry and topology. While primary invariants only depend on the topology of the underlining manifolds, secondary invariants also depend on certain auxiliary geometric data (e.g. metrics or connections etc. ) of the underlining manifolds. Some of the well-known secondary invariants are Chern-Simon invariants, eta invariant and rho invariant, where the latter two were introduced by Atiyah, Patodi and Singer. In this talk, I will discuss some of my recent work and joint work with others on these secondary invariants (and their higher versions). In particular, I shall talk about the higher eta invariant and the higher rho invariant, and their connections to the Baum-Connes conjecture and positive scalar curvature problems.

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 Four Color Theorem is easy to state:  If you have a (nice enough) map of countries, and you want to color them so that no two adjacent countries are the same color, then you never need more than four colors to do it!  This theorem was proved in the 70's, but its proof is controversial among mathematicians due to its heavy use of a computer.  We will discuss the history of this theorem, and then prove the Five Color Theorem, which states you never need more than five colors to color our map, which is much easier than four!

This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. Tea at 3:30 pm in 1425 Stevenson Center