This talk will consist of four points:

1. The basic definition of K-theory
2. A brief history of K-theory
3. Algebraic versus topological K-theory
4. The unity of K-theory

Using a splitting of the algebraic Brown-Peterson spectrum (at the prime 2), I describe the E_2-term of the motivic Adams-Novikov spectral sequence over a p-adic field (p > 2) and identify an analogue of the alpha family within it. Inspired by classical computations from topology and previous work over algebraically closed fields, I determine the behavior of this family, discovering new phenomena (like the existence of nontrivial d_2-differentials) along the way. This produces an ``infinite result" in the stable motivic homotopy groups of the 2-complete sphere spectrum over p-adic fields.

K-homology is the dual theory to K-theory. This talk will give the basic definition (following Atiyah, Brown-Douglas-Fillmore, and Kasparov) of K-homology as abstract elliptic operators. A different approach ( due to Baum-Douglas) will also be indicated. This second definition of K-homology is closely connected to the D-branes of string theory. K-homology will then be used to state the BC (Baum-Connes) conjecture.

After a brief review on the Schwarz Reflection Principle in one variable, I will discuss the general situation in higher dimensions using the language of the so-called Segre varieties associated with real analytic hypersurfaces in $\mathbb C^n$. I will then explain how to use it for proving boundary regularity results for holomorphic mappings.

The three conjectures are : Local Langlands, Baum-Connes, Aubert-Baum-Plymen. All three conjectures will be carefully stated. The point of view will then be developed that Aubert-Baum-Plymen provides a link between Local Langlands and Baum-Connes. This talk will include an introduction to p-adic numbers and to the representation theory of reductive p-adic groups.

Mike Hill, Mike Hopkins and I recently solved one of the oldest problems in algebraic topology. The theorem we proved is the opposite of what many people tried to prove about it in the 1970s. This talk will give some history and background of the problem and say a little about our method of proof.

Note: The talk starts at 2:30pm, and coffee will be served at 2pm.

This talk will continue to develop the three conjectures (Local Langlands, Baum-Connes, Aubert-Baum-Plymen) and the interactions among them.

We will review the concept of a stack as a category where descent works. Starting with a stack as a sheaf of groupoids satisfying effective descent, we will review characterizations of stacks in terms of sieves, and in terms of fibrant models.