Math 132: Smooth Manifolds, Spring 2017

Euler characteristic, Poincaré-Hopf and vector fields

The culmination of the class will be the use of oriented intersection theory to prove the celebrated Poincaré-Hopf theorem: if v: X--> TX is a smooth vector field, with X compact, oriented, and v admits only finitely many zeros, then the global sum of the indices of v equals the Euler characteristic of X.

This result is astounding because it relates differential geometric (i.e. analytic) information (the zeros of a vector field) with topological information (the Euler characteristic of X). An immediate consequnce is the Hairy Ball theorem: every vector field on the 2-sphere must admit a zero. Physically this means that there is always some point on the surface of the earth where the wind is not blowing or, more comically, it's impossible to comb a cocunut without at least one cowlick.

For a more down-to-earth introduction to this circle ideas see the following Chapters from the book A Mathematical Gift I by Ueno, Shiga and Morita.

• Euler characteristic and the Poincaré-Hopf theorem.

Learning mathematics

Learning a new skill, in general, can be a tricky obstacle to overcome and we all proceed in our own peculiar ways; some of us have natural talent, and some of us need to persevere to gain mastery. Learning mathematics is a challenge and taking upper-level college mathematics courses (such as Math 132) is extremely difficult: a student in mathematics needs excellent problem-solving skills, the ability to understand and manipulate abstract concepts, and exceptional clarity of thought (to the point of pedantry).

Here are some interesting articles that focus on how we learn, and how you can be more successful in learning mathematics. If you come across an article or text that you find useful then let me know and I can add it to the list.

• Mathematics Professors and Mathematics Majors’ Expectations of Lectures in Advanced Mathematics: An article discussing some of the possible disconnect between student and teacher expectations in college-level proof-based courses, written by Prof. K. Weber.
• Reading Mathematics: Learning to read mathematics can, for some of us, be a task. These notes are from a fantastic little book called 'How to study for a mathematics major', by Lara Alcock. While the examples given in these notes may not be the most challenging, the ideas that Alcock is attempting the reader to engage with are worth thinking about.
• Writing Mathematics: Writing college-level mathematics effectively takes time and effort, and is unlike most of what we've had to write before. These notes are taken from Alcock's book.

The concept of a manifold

As mentioned in class, it was Georg Friedrich Bernhard Riemann who first introduced the concept of a manifold (albeit in a manner completely unrecognisable to the modern mathematician) in his 1854 Habilitationsschrift. Moreover, he introduced the notion of 'distance' on a manifold: this is the precursor (believe it or not!) to Riemannian geometry. As if being the founder of modern differential geometry wasn't cool enough (and providing the mathematical tools for Einstein to formulate his theory of general relativity), Riemann also:

• laid the foundations for a rigorous theory integration, applying his methods to Fourier series, thereby inspiring Georg Cantor to investigate and develop set theory;
• revolutionised the study of complex analysis by introducing Riemann surfaces to study multivalued complex differentiable functions;
• wrote a ludicrously short paper on number theory (his paper on the subject), and formulated the famous hypothesis which now bears his name.

John Milnor

John Milnor is one of the most celebrated mathematicians of the twentieth century, being awarded the Fields Medal in 1962 and the National Medal of Science in 1967 (among other prestigious awards). At the age of twenty-five he showed the existence of manifolds that were homeomorphic, but not diffeomorphic, to the 7-sphere: such spheres were labeled exotic spheres. In addition to being an awesome mathematician, Milnor is renowned for his clarity of exposition and has authoured many fundamental textbooks in geometry and topology.

• Wikipedia article.
• Video lectures on differential topology from 1956.
• Recollections by Milnor on the discovery of exotic spheres. It's advanced but provides great insight.
• Section 2.1 in these notes by L. Nicolaescu (Notre Dame) provide a reasonably straightforward description of an exotic 7-sphere.