Lagrangian submanifolds

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\underline{Geometry of Lagrangian submanifolds}
 
The purpose of this talk is to see some nontrivial examples of Lagrangian
submanifolds, and to acquaint oneself with the classical topological
invariants. There's obviously too much literature to summarize here. This
talk has no strict prerequisites, except for a knowledge of characteristic
classes. However, it still requires quite a bit of independent work in its
latter part, which is where the meat is. A basic reference is
[McDuff-Salamon, Introduction to symplectic topology].
 
\underline{Plan}
 
{\it Definition.} Having defined Lagrangian submanifolds $L \subset M$, we
want to see the first basic invariant, the relative symplectic class
$[\omega] \in H^2(M,L;\R)$. We want to see how this varies under isotopies
of $L$, leading to the crucial distinction between exact (Hamiltonian) and
non-exact Lagrangian isotopies. Gromov's theorem about nonexistence of
exact Lagrangian submanifolds in $\R^{2n}$ (only the statement).
 
{\it First examples.} Lagrangian sections of cotangent bundles. The
Lagrangian tubular neighbourhood theorem. Lagrangian tori in $\C^n$ and
$\C P^n$. Fixed point sets of antisymplectic involutions (real loci in
K{\"a}hler geometry), for instance the Lagrangian sphere in $\C P^1 \times
\C P^1$, or $\R P^n \subset \C P^n$.
 
{\it The Maslov class.} The selfintersection number of an (oriented)
Lagrangian submanifold versus its Euler characteristic. The relative first
Chern class $2 c_1 \in H_2(M,L;\Z)$ (or $c_1$ for an oriented Lagrangian
submanifold). The Maslov class of a disc $(D,\partial D) \rightarrow
(M,L)$. Should be explained using the examples above, such as $T^n,\; \R
P^n \subset \C P^n$.
 
{\it Lagrangian surgery.} This is a way of constructing new Lagrangian
submanifolds [Polterovich, The surgery of Lagrange submanifolds] or
[Seidel, Lagrangian two-spheres can be symplectically knotted]. We should
experiment a little with this, to see what kinds of Lagrangian surfaces
one can get in $\R^4$ or in $\C P^2$.
 
{\it Example: the Chekanov torus.} Define the notion of monotone
Lagrangian submanifold in $\R^{2n}$, for instance [Buhovski, The Maslov
class of Lagrangian tori and quantum products in Floer cohomology] or
references therein. Besides the standard tori, there is another
(Lagrangian isotopic but not exact isotopic) example, the Chekanov torus
[Chekanov, Lagrangian tori in a symplectic vector space and global
symplectomorphisms]. Both tori are actually obtained by surgery on the
same immersed Lagrangian two-sphere, described in [Seidel, op. cit.]. They
are distinguished by a simple invariant counting mod $2$ the number of
holomorphic discs going through a generic point. At this point we want to
see some holomorphic disc theory: the compactness of the moduli space of
lowest-energy discs, and the degree mod $2$ of the evaluation map.
Computing this for the standard torus is straightforward (there are two
families of discs, one in each coordinate plane). Computation for the
Chekanov torus (where there are three families) is not obvious, but I can
offer some advice.
 
\underline{Notes} 
 
There is an interesting preprint by Schlenk et al. on Lagrangian
submanifolds of $\C P^2$ but I can't seem to find it anywhere. Other
vaguely related literature would be [Schwarz et al., Mean curvature flow
of monotone Lagrangian submanifolds], [Albers-Frauenfelder, A
non-displaceable Lagrangian torus in $T^*\!S^2$]. There is a lot of
literature on Lagrangian tori in $\R^{2n}$ (Viterbo, Polterovich, etc.)
surrounding the Audin conjecture, whose proof has been work in progress
for a while.
 
\underline{Dependencies}
 
This should be the first talk, given while we're still all wide awake.
 
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