Fall-2021. Math 242 (class # 25980): Symplectic Geometry

Instructor: Alexander Givental
Lectures: TuTh 12:30-2 in 4 Evans
Text: Symplectic Geometry and its Applications, V. Arnold and S. Novikov (eds.) ( Encyclopaedia of Mathematical Science. Dynamical Systems IV ) freely available in the electronic PDF format for patrons of the UC Berkeley library
Recommended: Mathematical Methods of Classical Mechanics by Vladimir Arnold, Lectures on Symplectic Geometry by Ana Cannas da Silva, Lectures on Symplectic Manifolds and The local structure of Poisson manifolds by Alan Weinstein.
Prerequisites: Math 214 (Differential Manifolds) or equivalent. At various points in the course students would benefit from some familiarity with root systems and Morse theory. Homework: due weekly on Th before the class (in person or by email givental-at-math-dot-berkeley-dot-edu)
Office hours: TuTh 3:30-5:30 pm in 701 Evans, and on Zoom during weekends (time to be specified)
Grading policy: Here is one I tried successfully in some courses and intend to use this time. The starting point is: 50% homework + 50% take-home final (most likely during the RRR week). Next: each individual weekly hw score which is lower (percentage-wise) than your score on the final will be dropped - together with its weight. E.g.: if half of your hw is below your final score and half above, then your total score is composed 2/3 from the final score and 1/3 from the remaining hw. Thus, there are many reasons why you want to do hw (as well as many other exercises, not assigned as hw), and do it well, yet a particular hw score can only improve your overall performance, but can never hurt your ultimate result compared to the final exam. Besides, I don't have a preconceived distribution of As and Bs, and would be happy to give everyone an A, should every one demonstrate good knowledge of the subject.

Course outline: Unlike Riemannian geometry, whose purpose is to classify Riemannian metrics (according to Felix Klein's "Erlangen program" of characterizing various geometries), symplectic geometry should not be viewed as yet another branch. It is rather a language, or a common landscape for classical mechanics, quantization, PDEs, representations of Lie groups, and probably much else. As any language, it is elegant and powerful but not particularly deep. In the contrary, symplectic topology is a deep theory studying intricate rigidity properties of phase spaces of Hamiltonian mechanics. It is not our goal to sink into symplectic topology, but time permitting, we'll have an informal primer of it as well.

For the most of the course, we will loosely follow the article Symplectic Geometry by Vladimir Arnold and myself, found on pages 1--135 in the aforementioned volume Dynamical Systems IV. Officially it is a survey paper, and as such is not required to contain proofs. In reality the exposition of all the essential material was organized as if a detailed textbook was first written, and then all trivial proofs removed and left as exercises to the reader, while all non-trivial ones left as outlines. This makes the text suitable for a graduate-level course, and testing this conjecture is a part of the plan.

HW1, due Th, Sep. 2: Read 1.1-2.1, 2.5, 4.1-4.4 in Ch.1. Solve: 1 and 3 from the (ever growing) list of excercises.
HW2, due Th, Sep. 9: Read 4.5, 3.1 in Ch. 1, and 1.1, 1.3 in Ch. 2. Solve: 10 and 13.
HW3, due Th, Sep. 16: Read 1.5, Section 2, and 3.1 in Ch. 2. Solve: 20, 23.
HW4, due Th, Sep. 23: 3.2-3,4 from Ch. 2, and the excerpt from Weinstein's paper. Solve: 24, 26.
HW5, due Th, Sep. 30: Read 4.1-4.2 from Ch. 2. Solve: 30, 31.
HW6, due Th, Oct. 7: Read Ch. 3 Section 1 (Variational principles). Solve 33,34.
HW7, due Th, Oct. 14: Read 1.4-1.5, 2.1-2.4 from Ch. 3. Solve: 37, 39.
HW8, due Th, Oct. 21: Read: the same. Solve: 49, 50.
HW9, due Th, Oct. 28: Read: Read Section 3 (Hamiltonian systems with symmetries) from Chapter 3. Solve: 51, 52.
HW10, due Th, Nov. 4: Read: the same, especially 3.8. Solve: 53, 55.
HW11, due Th, Nov. 11: Read: Chapter 4, Section 1. Solve: 56, 57.
HW12, due Th, Nov. 18: Read the rest of Chapter 4. Solve: 58, 61.
HW13, due Tu, Nov. 23: Due to Thanksgiving, and cancelled Nov. 18 lecture, let's skip this week's homework; but I recommend reading pp. 148--154 (from Kirillov's paper on Geometric Quantization).
HW14, due Th, Dec. 2: Read pages 38-42 (as well as 2.4 of Chapter 4). Solve 64 and 66.

HW solutions
excerpt from A. Weinstein's The local structure of Poisson manifolds up to the splitting theorem.