Differential Topology

Differential Topology (Spring 2023) is the study of differentiable manifolds, which are topological spaces that can locally be gently deformed to open subsets of Euclidean space. We’ll begin with a rigorous look at multivariable calculus, where the derivative is considered as a linear map on tangent spaces, something that isn’t usually covered in multivariable calculus classes. We’ll then look at intersections of manifolds. In particular, given two manifolds, what does it mean to see that they intersect “normally” (or, more formally, transversely) as opposed to tangentially?

We will also study vector fields on manifolds. This leads to one of many equivalent definitions of perhaps the most fundamental invariant of compact manifolds: the Euler characteristic. The zeros of a vector field on a manifold encode the Euler characteristic very neatly, thanks to the Poincaré-Hopf Theorem.

If time permits, we will also look at integration on manifolds, leading to a beautiful generalization of the Euler characteristic, namely de Rham cohomology. This is a way of determining global topological information about a manifold based on which functions have antiderivatives. For example, on \mathbb{C}\setminus{0}, the function \frac{1}{z} is a well-defined function, but it does not have a global antiderivative, because there is no logarithm function defined on all of $late \mathbb{C}\setminus{0}&bg=ffffff$. Remarkably, this failure exactly encodes the fact that \mathbb{C}\setminus{0} has a hole at the origin!

Knowledge of calculus and of analysis and topology at the level of the second and third quarters of the Fundamentals of Higher Mathematics sequence is required for this class.

This class will meet April 3–June 7. Everyone will attend lectures online on Mondays from 6:30–8:30 PM Pacific time. Online students will attend problem sessions on Tuesdays from 5:00–7:00 PM Pacific time. In-person students will attend problem sessions on Wednesdays from 6:30–8:30 PM Pacific time.