Harvard Mathematics Spring 2020 Tutorial
Lectures: Mondays 4:30-5:45 pm SC 309A, Tuesdays 7:30-8:45 pm SC 310;
Problem sessions/office hours: Thursdays 7:30-8:30 pm SC 116


Caustics in dimension 3

Symplectic geometry is a modern and rapidly-developing field of mathematics that began with the study of the geometric ideas that underlie classical mechanics. This tutorial will begin similarly by introducing the Lagrangian and Hamiltonian formulations of classical mechanics and their resulting dynamical properties, before re-expressing them in the language of differential geometry, that is, in terms of symplectic manifolds. This new setting reveals that classical mechanics is about symmetries in geometry; this will be illustrated by the detailed study of symmetries of mechanical systems, with many concrete examples drawn from physics such as orbital motion and rigid bodies. We will finish with an informal introduction to the problems and techniques considered in modern symplectic topology, such as non-squeezing theorems and the use of pseudoholomorphic curves. There will be ample scope for students to follow their own interests, exploring connections to topics such as quantum mechanics, algebraic geometry, and dynamical systems.


Multivariable calculus, and familiarity with open/closed sets and convergence as in 112 or 131 will be assumed, but no prior knowledge of physics or geometry will be required. The theory of manifolds and differential forms will be integrated throughout the tutorial and motivated from a physical perspective, from which it becomes most transparent and intuitive. If you're not sure you meet the prerequisites, take a look at the notes for the first few lectures below.


Each week there will be two 75 minute lectures, as well as a 1 hour problem session/office hours set aside for students to work together on problems (which will not be handed in). Each student will give two presentations in the semester: one short 15-minute presentation early in the semester, and a 30-minute presentation near the end of the teaching period, for which topics will be provided (subject to change based on class size). Finally, a short expository paper will be due at the end of exam period: a list of interesting suggested topics can be found below. Students who are not required to write a paper may instead choose half of the problems to complete and submit at the end of the exam period.



Here are all the lecture notes combined into a single file, and solutions to the problems kindly provided by Abi. There are almost certainly mistakes and typos in the notes: please let me know of any you find!

General References

Suggested Paper Topics