I am depositing various papers, mostly expository, on this page.
On the algebraic topology of finite spaces: My undergraduate thesis, on topological spaces with a finite number of points. They are the same as posets, and it turns out that they have the same weak homotopy type as the nerve of the poset.
$P(\omega)/fin$ and projections in the Calkin algebra (published in Proc. AMS (2008)): Some research I did as an undergraduate on set-theoretic pathology in operator algebras. A small erratum: the "lattice" of projections in the Calkin algebra is not actually a lattice at all, but this does not affect any of the results of the paper. The join mentioned in Theorem 3.2 can be replaced by any sufficiently small upper bound.
Set theory and operator algebras (with Ilijas Farah): Notes from a workshop by Ilijas Farah on set-theoretic problems in operator algebras.
The following are notes from classes I taught at Mathcamp. Beware that they have some minor errors.
The Category of Sets: introduction to category theory via the category of sets as a categorification of arithmetic; coproducts, products, exponential objects, and Yoneda's lemma.
Unique Factorization: basic theory of UFDs.
Multilinear Algebra: tensor products of vector spaces, exterior powers, determinants, duals and Hom-spaces, and traces.
Set Theory as a Foundation for Mathematics: construction of number systems and proving their basic properties in naive set theory.
Set Theory: ordinals, cardinals, axioms, and some elementary independence results.
Principal Ideal Domains: unique factorization, classification of modules and applications to linear algebra.
Boolean Algebras: basic theory, Stone's representation theorem, and some applications.
Pointless Topology: a quick, woefully incomplete introduction to locales.
Hyperreal Analysis: construction and sketch of the basic properties of hyperreals, without applications.
The Weierstrass Approximation Theorem: a proof with a view towards generalizing to the Stone-Weierstrass theorem.
Last updated 7/9/10.