This essay will present a brief outline of the theory of Clifford algebras, together with a small amount of material about quadratic forms. I follow loosely the well known book Geometric algebra by Emil Artin, but with elegant modifications that I saw originally in some lecture notes by Raoul Bott, (dating from 1962), subsequently included in [Atiyah-Bott-Shapiro:1964]. In the first section, I’...

We describe Bott towers as sequences of toric manifolds M, and identify the omniorientations which correspond to their original construction as toric varieties. We show that the suspension of M is homotopy equivalent to a wedge of Thom complexes, and display its complex K-theory as an algebra over the coefficient ring. We extend the results to KO-theory for several families of examples, and com...

one can quantize an invertible symbol σ ∈ C∞(S∗X; hom(π∗E, π∗F )) as an elliptic pseudodifferential operator in the Φ-b or edge calculus, ΨΦ-b(X;E,F ), introduced in [7] or alternately as an elliptic operator in the Φ-c or φ calculus, ΨΦ-c(X;E,F ), introduced in [8]. As on a closed manifold, either of these operators will induce a bounded operator acting between natural L-spaces of sections but...

This paper considers left/right action-preserving morphisms between Bott-Samelson bimodules from a combinatorial perspective; specifically, using a diagrammatic representation. We examine the maps formed by a basis known as Libedinsky’s light leaves, which constructs a map based on a string r, composed of the letters s and t, and a binary string b. The eventual goal is to find a formula for the...

Locally conformally Kähler (LCK) manifolds with potential are those which admit a Kähler covering with a proper, automorphic, global potential. Existence of a potential can be characterized cohomologically as vanishing of a certain cohomology class, called the Bott-Chern class. Compact LCK manifolds with potential are stable at small deformations and admit holomorphic embeddings into Hopf manif...