- Workshop. We plan to invite five to eight people who are making significant contributions to contact geometry to AIM and Stanford University from August 21 to December 21 of 2000. Together with approximately ten people at Stanford University, or in the Bay Area, we shall run several weekly seminars and mini-courses. There shall be ample time for more informal discussions and collaborations. We shall also have eight to ten people visiting for shorter periods of time.
- Conference. In December we shall hold a week long conference to publicize our work and stimulate further work in the field.

Contact geometry is an odd dimensional analog of symplectic geometry. Specifically, if
*M* is an oriented *2n+1*-manifold then an oriented **
contact structure** *\xi* is a
*2n*-dimensional distribution (sub-bundle of the tangent bundle) for which there is a
1-form *\alpha* for which *\xi=\hbox{ker}(\alpha)* and *\alpha\wedge(d\alpha)^n\not=0.*

Contact structures first appeared in an explicit form in the work of Sophus Lie, and later were intensively studied by E. Cartan, Darboux, and others. The first result of topological flavor, existence of contact structures on all orientable 3-manifolds, was first obtained by Martinet [35] in 1971. The birth of symplectic topology in eighties (see [3, 29, 7, 21], et al.), greatly influenced by Arnold's conjectures (see [2]) brought an exciting development in contact topology as well. However, this odd-dimensional cousin of symplectic topology did not get as much attention as its symplectic counterpart. We hope that this program, besides its immediate mathematical consequences, will help to attract more attention to this exciting and important area.

In 3 dimensions, there are two
fundamentally different types of
contact structures: tight and overtwisted. This dichotomy, first
observed in [11], grew out of Bennequin's study
[3] of transversal knots (knots transverse to the
contact planes). Bennequin proved that the contact structures on
*R ^{3}* defined by

and\alpha_1=dz + r^2\, d\theta,

are not the same by showing they support different transversal knots. The contact structure\alpha_2=\cos r\, dz + r\sin r\, d\theta,

We hope that the answer to this fundamental question will be given as one of the results of the program.Does any irreducible 3-manifold admit a tight contact structure with one orientation or another?

The contact structure *\xi* on *M* is said
to be filled by a symplectic manifold *(X,\omega)* if *\partial
X=M* (as oriented manifolds) and *\omega\vert_\xi* is a symplectic
form on *\xi.* Gromov [29] and Eliashberg
[10] have used the theory of holomorphic curves in
*X* to show that a fillable contact structure is tight. With this
result one can constructed many tight contact structures on
3-manifolds
(see [9, 12,
28]).
Since many (maybe most) currently known tight contact
structures come from this construction one is prompted to ask:

Are all tight contact manifolds symplectically fillable?

Either answer to this question would be serious progress in understanding the relation between 3 dimensional contact and the 4 dimensional symplectic topology. Providing an answer to this question is a focal point of this project. The only other general construction of tight contact structures is due to Eliashberg and Thurston [16] who have shown how to perturb a Reebless foliation into a tight contact structure. Some of the tight structures constructed this way should shed light on the above question.

Ultimately, one would like to:

Through the work of many mathematicians we understand tight structures on many simple manifolds. Specifically they are classified on the 3-ball, the 3-sphere [11], lens spaces [18, 25, 32], torical annuli, solid tori, torus bundles overClassify tight contact structures on 3-manifolds.

Though there are many new techniques for understanding contact structures, the most
striking is contact homology (see [13])
or its still evolving generalization, symplectic field
theory.
Not only are these powerful new invariants of contact structures but they
also demonstrate an intimate connection between contact geometry, Hamiltonian dynamics and
Gromov-Witten invariants.
Contact homology is a systematic way to bring Gromov's very successful theory of
pseudoholomorphic curves [29] in symplectic manifolds into the arena of contact topology.
Given a contact manifold *(M,\xi)* one may form its symplectization
*(W,\omega)=(M\times R, d(e^t\alpha))* where *\alpha* is a contact form for *\xi.*
In *W* one now considers pseudoholomorphic curves with respect to a special almost
complex structure. Since *W* is noncompact, ``Gromov compactness'' of the space of
pseudoholomorphic curves does not hold but
the way in which it fails can be understood. In particular, compactness fails essentially
because a sequence of surfaces may limit to a surface with extra punctures, and near
each of these punctures the surface is asymptotic to *\gamma\times R* where *\gamma* is
a closed orbit in the Reeb vector field of *\alpha* (recall the Reeb vector field *X*
is uniquely defined by *\alpha(X)=1* and *d\alpha(X,\cdot)=0*). Thus understanding the
periodic orbits of the Reeb field allows one to understand the compactness issues for
pseudoholomorphic curves in *(W,\omega)* and vice versa. For example Hofer [30] used these
ideas to show, among other things, that any Reeb vector field on *S ^{3}* must have a
closed orbit. The contact homology of a contact manifold

These new methods should lead to better understanding of tight contact geometry in 3-dimensions.Develop methods to compute the contact homology of contact structures on 3-manifolds.

In trying to understand contact structures it is useful to study transversal knots (ones transversal to the contact planes) and Legendrian knots (ones tangent to the contact planes). As mentioned above, the genesis of the tight vs.\ overtwisted dichotomy was through the study of transversal knots [3]. Moreover, transversal and Legendrian knots have been used to distinguish tight contact structures on the 3-torus [33] and homology spheres [1]. The study of these special knots also provide an interesting and rich theory in their own right. Though the importance of these knots is evident, there is little known about them. In [12, 14] it was shown that simple invariants can be used to classify transversal and Legendrian unknots, while in [20] the same was shown for torus knots and the figure eight knot. In contrast, Chekanov [4] has shown that there are knots that cannot be classified in such a simple way. During this project we shall:

To prove his above mentioned result Chekanov used a simple combinatorial version of contact homology. Eliashberg and Hofer have shown how to define a relative contact homology for pairsExtend this understanding of transversal and Legendrian knots.

There is an interesting conjecture of Arnold, the ``chord conjecture,'' that would provide some insight into the contact homology of a pair. The conjecture is:Find effective methods to calculate the relative contact homology and better understand what it has to say about Legendrian knots.

There is a Reeb orbit connecting two points on any Legendrian knot.

Not much
is known concerning contact structures
on 5-manifolds, and we hope that as a result of this program, the situation shall be
corrected.
With the new invariants of contact homology
and symplectic field theory we hope to lay a foundation for
the theory of contact structures on
5-manifolds. One of promising results along these lines was recently
obtained by I. Ustilovsky [36]. He showed that there are infinitely many
non-isomorphic contact structures on *R ^{5}*, which are standard at infinity,
and homotopically
standard.
An important class of contact

We also hope that progress will be achieved in the problem
of existence of contact structures on 5-dimensional manifolds.
In 1991, Geiges [23]
showed for simply connected 5-manifolds that the
vanishing of the 3-rd Stieffel--Whitney class *W _{3}(M)*, which is a
necessary homotopical condition
for the existence of a contact structure on a 5-manifold

Last modified: Wed Jun 28 10:34:56 PDT 2000