Math 242: Symplectic geometry
UC Berkeley, Spring 2009
Instructor
Michael Hutchings. [My last name with the last
letter removed]@math.berkeley.edu Tentative office hours: Tuesday
2:004:00, 923 Evans.
Textbooks
The main text for this course is Introduction to symplectic
topology, 2nd edition, by McDuff and Salamon. Much of what we
will discuss is covered in this book, although I will not follow it
closely and will give additional references as we go along. For the
last part of the course, An introduction to contact topology by
Geiges is a good reference.
Course outline
The rough plan for the course is to discuss the following topics (not
necessarily in linear order):
 Basic definitions; motivation from physics, geometry, and
lowdimensional topology. (cf McDuffSalamon, Ch. 1)
 Basic
facts about symplectic manifolds and symplectic linear algebra (MS
Ch. 2, 3, 4)
 Symplectic group actions (MS Ch. 5)

Symplectomorphisms and fixed points (MS Ch. 8, 9, 10, 11)
 The
HoferZehnder capacity, Gromov nonsqueezing, and the Weinstein
conjecture in R^2n (MS Ch. 12)
 Contact geometry, mainly in 3 dimensions (cf Geiges)
This course should hopefully provide good preparation for the program
on
Symplectic and Contact Geometry and Topology at MSRI from August
2009 to May 2010.
Final project
Each student is expected to research a topic of interest and either
write a 510 page expository article about it, or give a short
presentation to the class. The articles will be posted here (if I am
given permission). The books by McDuffSalamon and Geiges suggest a
number of good starting points with many references. I can give more
references. (Since McDuffSalamon is a few years old, it does not
cite the latest works.)
Lecture summaries and references
 (1/20) Basic notions: symplectic manifolds, Lagrangian
submanifolds, Hamiltonian vector fields, Poisson brackets. The
cotangent bundle example. References: McDuffSalamon, chapters 1 and
3. There is also a nice overview of symplectic geometry by Ana Cannas
da Silva which you can download here.
 (1/22) Continuing the introduction (cf McDuffSalamon
chapter 1), we discussed:
 The origins of symplectic geometry in
classical mechanics. Reference: V. Arnold, Mathematical methods of
classical mechanics.
 Introduction to the Weinstein conjecture.
Here is a draft of a survey article I
am writing about this.
 (1/27)
 More about the Weinstein conjecture.
 Introduction to Gromov's nonsqueezing theorem and related topics. See McDuffSalamon, sections 1.2 and 2.4.
 (1/29)
 Basics of almost complex structures and
holomorphic spheres. See McDuffSalamon, Section 2.5 and Chapter 4.
 Outline of Gromov's proof of nonsqueezing. For some more
details, see Ka
Choi's lecture notes on the course I taught a year ago.
 (2/3) More about holomorphic curves and Gromov's proof of
nonsqueezing. Digression on the first Chern class. This material is discussed
from a different perspective in sections 4.4 and 2.6 respectively of
McDuffSalamon. For more about obstruction theory (which I used to define
the first Chern class) I have some old lecture notes
here. (These notes explain the Euler class of an oriented ksphere bundle, which in the case k=1 determines the first Chern class of a complex line bundle.)
 (2/5)
 More about holomorphic curves and Gromov's proof of nonsqueezing. For many more details, see McDuff and Salamon,
Jholomorphic curves and symplectic topology.
 Brief review of the Hodge decomposition. (We will use this just a little bit next time, but it's good to know anyway.)
 (2/10) Darboux's theorem, Moser trick, etc. See eg
McDuffSalamon sections 3.2 and 3.3.
 (2/12)

More applications of the Moser trick to prove various "no
local symplectic geometry" results.

Introduction to contact
manifolds. See McDuffSalamon section 3.4, and Geiges chapters 1 and
2.
 (2/17) More about contact manifolds, hypersurfaces of contact
type, and the Weinstein conjecture.
 (2/19)
 Contact structures on circle bundles.
 Introduction to threedimensional contact geometry. There is a very nice introduction to this by John Etnyre here. This is the basic reference for the next two lectures. More details can be found in the book by Geiges.
 (2/24)
 Tight and overtwisted contact structures in three dimensions.
 Legendrian knots, introduction to the Bennequin equality and adjunction inequality for tight contact structures.
 (2/26)

Introduction to symplectic fillings.
 Outline of the proof of the adjunction inequality.
 (3/3)

Statement of the Arnold conjecture on fixed points of Hamiltonian symplectomorphisms.
 Flux. (See McDuffSalamon chapter 11)
 (3/5) Review of Morse homology. See for example the book by Matthias Schwarz, or my mistakeridden lecture notes.
 (3/10) More about Morse homology.
 (3/12) Introduction to Floer homology of Hamiltonian
symplectomorphisms. See for example Dietmar Salamon, Lectures
on Floer homology.
 (3/17) Index and spectral flow. For the basic idea see chapter 2
of Salamon's lecture notes, and for the technical details see Robbin and
Salamon, Spectral flow and the Maslov index.
 (3/19) The ConleyZehnder index and the grading on Floer
homology. See chapter 2 of Salamon's lecture notes.
 (3/31) More about the ConleyZehnder index and Floer homology:
the mapping torus picture and Gromov compactness.
 (4/2) Details of the computation of the Floer homology of
Hamiltonian symplectomorphisms in the monotone case. Reference:
D. Salamon and E. Zehnder, Morse theory for periodic solutions of
Hamiltonian systems and the Maslov index.
 (4/7) Floer homology for Lagrangian intersections.
 (4/9)
 How Floer homology for symplectomorphisms is a
special case of Floer homology for Lagrangian intersections.

Symplectic capacities. Definition of the HoferZehnder capacity,
and how it leads to a proof of the Weinstein conjecture for contact
type hypersurfaces in R^(2n). See McDuffSalamon, chapter 12 and
Symplectic invariants and Hamiltonian dynamics by Hofer and
Zehnder.
 (4/14) Examples of the HoferZehnder capacity. Some analytic
background.
 (4/16) Outline of the proof of the hard part of the fact that
the HoferZehnder capacity is a capacity, following the
HoferZehnder book. This is explained in McDuffSalamon using
discretized (finite dimensional) loop spaces instead. This can be
reproved and generalized using versions of Floer homology that keep
track of the values of the symplectic action functional; see eg
Ginzburg and Gurel, Relative HoferZehnder capacity and periodic
orbits in twisted cotangent bundles.
 (4/21) Introduction to symplectic reduction. See McDuffSalamon
chapter 5. This material is also covered in lots of other books; some
useful online references which I am partly following are by Cannas here and
Cieliebak here
 (4/23) Basic lemmas about moment maps.
 (4/30) MarsdenWeinstein reduction.
 (5/5) Some examples. Reference for the harder example: Atiyah and
Bott, The Yang Mills equations over Riemann surfaces.
 (5/7) AtiyahGuilleminSternberg convexity.
 (5/8) extra meeting, 10am1pm in room 736 Evans, for final project talks.