Algebraically constructible functions

An algebraic version of Kashiwara and Schapira's calculus of constructible functions is used to describe local topological properties of real algebraic sets, including Akbulut and King's numerical conditions for a stratified set of dimension three to be algebraic. These properties, which include generalizations of the invariants modulo 4, 8, and 16 of Coste and Kurdyka, are defined using the link operator on the ring of constructible functions.Comment: AMS-TeX v2.1, 25 page

Real intersection homology II: A local duality obstruction

We show that the intersection pairing on our real intersection homology groups is not a dual pairing in general.Comment: 7 pages. Second author grant support adde

The knowledge illusion: who is doing what thinking?

Focusing on students’ attempts to explain the relative significance of different factors in Hitler’s rise to power, Catherine McCrory explores the vexed question of why students who seem able to express necessary historical knowledge on one occasion cannot effectively reproduce it on another. Drawing on a detailed analysis of what it actually means to ‘know’ something, she plans a series of accessible activities allowing as many students as possible to secure essential knowledge for themselves, rather than simply relying on the authority of the teacher who told them. She goes on to explain how careful diagnosis of the gaps between what students say and the reasoning that underpins their utterances can help teachers to decide where they can usefully ‘give’ students particular insights and where the students need to ‘arrive at’ those insights through their own cognitive labou

Virtual Betti numbers of real algebraic varieties

The weak factorization theorem for birational maps is used to prove that for all nonnegative i the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a "virtual Betti number" beta_i defined for all real algebraic varieties, such that if Y is a closed subvariety of X, then beta_i(X) = beta_i(X\Y) + beta_i(Y).Comment: 13 pages. Talk by first author, AMS meeting, Northeastern University, October 5-6, 200