1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

IAS /Park City Mathematics S eries Volume 7 , 1999 Introduction to Symplectic Topology Dusa McDuff Introduction Instead of t ryi ...

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LECTURE 1 Basics Symplectic geometry is the geometry of a skew-symmetric form. Let M be a manifold of dimension 2n. A symplectic ...

10 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGY shows that XH is t a ngent to the level sets of H. Thus the flow of <Pf pre ...

LECTURE l. BASICS The basic example is R^2 n with the form wo considered as a bilinear form. Given a subspace W define its sympl ...

12 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGY The cotangent bundle This is another basic example of a symplectic manifold. T ...

LECTURE 2 Moser's Argument In this lecture I will show you a powerful argument due to Moser [M] which exhibits the "flabbiness" ...

14 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGY the symplectic case because there is never any problem in constructing homotop ...

LECTURE 2. MOSER'S ARGUMENT 15 is exact near Q. If we find a 1-form u that vanishes at all points of Q and is such that du = w 1 ...

16 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGY Exercise 2.8. Given any two diffeomorphic closed smooth domains U, V in R n th ...

LECTURE 3 The Linear Theory We will consider the vector space R^2 n with its standard symplectic form w 0. This may be written i ...

18 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGY Lemma 3.3. Sp(2n, R) n 0(2n) = Sp(2n, R) n GL(n, C) = 0(2n) n GL(n, C) = U(n). ...

LECTURE 3. THE LINEAR THEORY 19 w-compatible almost complex structures An almost complex structure on a vector space V is a line ...

20 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGY pull back the standard form on R^2 n x p to the given form u(p). Then the tran ...

LECTURE 3. THE LINEAR THEORY 21 subspace L is the image A(Lo) of Lo under the unitary transformation A that takes 8 ~J. to Uj an ...

22 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGY Therefore, we may set L(3 + s) = A(L(2 - s)), 0::; s::; 1. The Maslov index μ ...

LECTURE 4 The N onsqueezing Theorem and Capacities In Lecture 2, I explained various results that showed how flexible symplecto- ...

24 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGY However , below we will use the strong form (iii). The interesting question is ...

LECTURE 4. THE NONSQUEEZING THEOREM AND CAPACITIES 25 It is now easy to check that er maps the unit ball into the cylinder Z(> ...

26 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGY why there is an interesting theory of symplectic topology. There is much recen ...