Math 191 Winter 2019

Section 1

We will discuss basic aspects of smooth 4-manifolds. The topics we may include are: intersection forms, Kirby calculus, symplectic 4-manifolds, Seiberg-Witten invariants, trisections, surfaces in 4-manifolds. Prerequisites: Ma 109, Ma 151. It is suitable for undergrads who have taken Ma 151.

Section 2

We plan to discuss some of the recent and ongoing developments in the arithmetic of elliptic curves over the rationals, especially in regards to the Birch and Swinnerton-Dyer conjecture. The developments rely on an intriguing mix of techniques arising from Iwasawa theory, congruences between modular forms, the theory of Beilinson--Kato elements and Heegner points, along with related Euler systems. We hope to survey some of the underlying ideas.    Prerequisites are some background in algebraic geometry (Ma130) and algebraic number theory (Ma160). 

Section 3

The course will give an introduction to algebraic K-theory. We start with the basics on K_0,K_1,K_2 and Quillen’s definition and fundamental theorems about higher algebraic K-theory. Then we’ll discuss in some details the algebraic K-theory of number rings, in particular finite generation. At the end I’ll hope to discuss some more recent developments , notably the work of Tabuada et al. on a characterization of higher algebraic K-theory by a universal property.

 Prerequisites are some background in algebraic geometry (Ma130), algebraic number theory (Ma160) and algebraic topology (Ma151). While one can probably follow with a minimal background in the first two subjects, we will rely on algebraic topology more seriously. In particular, some knowlege about category theory, H-spaces and the description of homotopy types by simplicial sets will be necessary. However, I might also develop some of the background material in class.