Let A be a finite simple non-abelian Mal'cev algebra (e.g. a group, loop, ring). We identify the Fra{\"i}ss{\'e} limit D of the class of finite direct powers of A as a filtered Boolean power of A by the countable atomless Boolean algebra B and show that D also arises as congruence classes of the countable free algebra in the variety generated by A. Further we show that filtered Boolean powers of A by B are $\omega$-categorical, have the small index property, strong uncountable cofinality and the Bergman property. This is joint work with Nik Ruskuc (University of St Andrews).

Filtered Boolean powers and their automorphism groups

Tue, Feb. 6 3:30pm (MATH 3…

Topology

Juan Moreno

X

Last semester we saw the category of spectra introduced as the right place to study stable phenomena for spaces. This idea turns out to be incredibly useful largely because spectra behave very much like algebraic objects. We will spend a good part of the talk covering two instances of this: Duality - analogous to the notion of duality for vector spaces is something called Spanier-Whitehead duality for spectra. Localization - analogous to the notion of localizing abelian groups at a prime is something called Bousfield localization of spectra. With the remaining time we will see how these two notions can be used to capture important information about the stable homotopy groups of spheres. In particular we'll talk about a sort of "first approximation" to these groups: the stable image of J.

Duality, Localization, and stable homotopy groups of spheres