Differential Equations and Dynamical Systems (Part II) (with Jaaziel Lopez de la Luz and Clara Park)
This course is the next chapter in dynamical systems of differential equations: nonlinear dynamical systems. Content that we’ll cover includes the existence and uniqueness, linearization, the Hartman–Grobman theorem, and invariant and center manifold theory. Continuing and new students with some background in linear algebra, differential equations, and analysis are encouraged to (re)join!
Prerequisites: 3A, 3D, 13, 140A/B (can be concurrent)
An Introduction to Commutative Algebra and Algebraic Geometry (with Aurora Vogel)
Commutative algebra and algebraic geometry shows a beautiful link between algebra and geometry (in particular the geometry of curves that locally look like polynomials). In the course, we will cover the basics of rings, localization, and primary decomposition, among other topics. For each concept, the link between the algebra and geometry will be illuminated via examples and theory. I give two examples.
1. Solution sets to polynomial equations correspond to radical ideals (Hilbert's Nullstellensatz).
2. It turns out the maximal ideals of the polynomial ring C[x] correspond to points of the affine complex line. This roughly generalizes to other polynomial rings as well.
The course will follow Siegfried Bosch's "Algebraic Geometry and Commutative Algebra" freely available via Springer link. The goal is to cover the first 3 chapters of the text (omitting some sections), and if time permits to cover some of chapter 4.
Prerequisites: 121A, 120B
Set Theory without Choice (with Brian Ransom)
A good foundation for math needs to account for the way that mathematicians use infinite objects. This presents a challenge, in that we are limited to finite-length descriptions of these objects. One way to address this challenge is by using non-constructive assumptions like the Axiom of Choice (AC). By doing so, we gain the ability to describe more things in set theory at the cost of being (in a controlled way) more vague about the things we're describing.
In this DRP, we will quickly build intuition that the other axioms of set theory (the ZF axioms) shouldn't be able to prove AC as a theorem. The harder, and more interesting, question is to ask how we can formally prove this to be the case. To do so, the main focus of the DRP will be learning how to use symmetry to "confuse" worlds of set theory and cause failures of AC in these worlds. The primary technique we will learn relies heavily on the method of forcing in set theory.
Prerequisites: Prior experience with the method of forcing.