Last week we introduced the approachability ideal and internally approachable structures, and made some basic observations. We continue this week using the trichotomy theorem to guarantee the existence of an exact upper bound for a certain sequence, and use this to prove the existence in ZFC of a scale of length \aleph_{\omega+1} in a reduced product \omega_k for k \in A, an infinite subset of \omega.

The interplay between structural  Ramsey theory and topological dynamics of automorphism groups has been extensively studied since their connection was established in a paper by Kechris-Pestov-Todorcevic, while earlier works of Pestov, and Glasned and Weiss exhibited the phenomena in special cases. This line of research was extended to metric structures and approximate Ramsey property by Melleray and Tsankov. We establish the approximate Ramsey property for the class of finite-dimensional normed vector spaces and deduce that the group of linear isometries of the universal approximately homogeneous Banach space, the Gurarij space, is extremely amenable, that is, every continuous action on a compact Hausdorff space has a fixed point. Dualizing our ideas, we show that the class of finite-dimensional simplexes with a distinguished extreme point and  affine surjections satisfies the approximate Ramsey property. As a consequence, we find that the universal minimal flow of the group of affine homeomorphisms of the Poulsen simplex is its natural action on the Poulsen simplex. This is a joint work (in progress) with Aleksandra Kwiatkowska (UCLA), Jordi Lopez Abad (ICMAT Madrid and USP) and Brice Mbombo (USP).

Abstract: Charles Darwin, in his On the Origin of Species, laid the foundation for all of modern biology. But he had doubts of his own theory: "I...confine myself to one special difficulty...actually fatal to my whole theory. I allude to the neuters or sterile females in insect communities...they cannot propagate their kind." Put another way, how can evolution explain altruistic behaviour? Evolution has proven the test of time, so was Darwin wrong in second guessing? As it turns out, some basic mathematics was all that was needed to solidify his theory.
 

Technology has become an integral part of everyday life and the classroom. With the availability of computer algebra systems online, on computers, and even on cellphones integration and differentiation have become trivial exercises. But, how does this affect how we approach teaching Calculus? I will discuss various ways that I have integrated technology and some of the pitfalls in software. I will also discuss how technology can be used to create visualizations to reach out to different learning styles of students.