Symplectic Geometry — 2nd Semester 2012/2013

Announcements

General information about the course (in english, in portuguese).

This will be a reading course. I will post here every week what you should read for the following week's class as well as the homework assignments.

For the first class (February 21st) you should read Part I in Ana Cannas's book and solve the problems 6,8,9 in Homework 1 and problem 2 in Homework 2 in the more recent edition of the same book (available here).

For the second class (March 1st) you should read Part V (Section 12) and solve problems 2 and 3 in Homework 8 and read Part II (Section 3) and solve problems 1 and 2 in Homework 3.

For the third class (March 8th) you should read Part III (Sections 6 to 9) and solve problems 1,2 and 3 in Homework 6.

For the fourth class (March 15th) you should read Part V (Sections 13 and 14) and solve Homework 9 and problem 1 in Homework 10.

For the fifth class (March 22th) you should read Part VI (Sections 15 and 16) and solve problems 1 and 2 in Homework 11 and problems 1, 2 and 3 in Homework 12.

For the sixth class (April 5th) you should read Part VI (Section 17) and solve problems 4, 5, 6 and 7 in Homework 12.

For the seventh class (April 12th) you should read Part VIII (Sections 21 and 22) and solve problems 1 to 5 in Homework 17.

For the eigth class (note the different day and time: April 18th at 2pm in room 4.35) you should read Part IX (Sections 23 and 24) and solve problems 1 to 4 in Homework 19.

For the ninth class (April 26th) you should read Part X (Sections 25 and 26) and solve problems 1 to 3 in Homework 20.

For the tenth class (May 17th) you should read Part X (Section 27) and Part XI (Sections 28, 29 and 30). You should solve all the problems in this Homework assignment and problems 2 to 4 in Homework 22.

Syllabus

Introduction to symplectic manifolds: Symplectic linear algebra and symplectic manifolds. Symplectomorphisms, symplectic and Hamiltonian vector fields. Cotangent bundle.

Darboux-Moser-Weinstein Theory: Isotopies. Classical Darboux Theorem. Moser Local Theorem. Weinstein Lagrangian Neighborhood Theorem.

Almost complex structures: Compatible almost complex structures. Integrability. Kahler manifolds.

Hamiltonian actions: Moment maps. Symplectic reduction: Marsden-Meyer-Weinstein Theorem. Toric manifolds. Convexity of the moment map: Atiyah-Guillemin-Sternberg Theorem.

J-holomorphic curves: Gromov Compactness Theorem. Gromov Nonsqueezing Theorem. Applications to symplectic topology: properties in dimension 4.

Bibliography

A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics, 1764, Springer-Verlag (2001).

K. Cieliebak, Introduction to Symplectic Geometry, Part A and Part B.

D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs, Oxford University Press, New York (1995).

D. McDuff and D. Salamon, J-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, 52. American Mathematical Society, Providence, RI (2012).

Evaluation

Homework.