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** General description:**

The positions are associated to the reseach project of Marius Crainic, who will act as supervisor/host. The project is rooted in Poisson Geometry, but rapidly exceeds the boundaries of Poisson Geometry to neighboring areas. We are looking for:

- __PhD candidates __ with a good background in Differential Geometry/Topology. We have 2-3 such positions to be filled in 2015 or 2016. These are standard, 4 years PhD positions at Utrecht University.

- __ PostDoc candidates __ whose expertise comes with one of the following key-words:
contact geometry, low-dimensional topology, K3 surfaces and Kahler geometry,
hyperkahler geometry, Lefschetz fibrations (classical as well as variations, up to broken ones),
open-book decompositions, symplectic geometry/topology, toric geometry, moment maps,
geometric aspects of gerbes, integral affine geometry, nil and solv-manifolds,
the standard conjectures from affine geometry (Auslander conjecture, Markus conjecture),
foliations and leafwise geometric structures, G-structures and the geometry of PDEs.
We have 2-3 such PhD positions, they are for 2-3 years and they come with very low teaching duties (if at all).

- for the PhD positions: please send your CV, including a list of courses that you followed and the marks, a letter of motivation and make sure that 2 recommendation letters are sent as well. All these should be sent to Marius Crainic (m.crainic@uu.nl).

- for the PostDoc positions: please send your CV, including a list of publications and preprints, and your research plans, a letter of motivation and make sure that 2 recommendation letters are sent as well. All these should be sent to Marius Crainic (m.crainic@uu.nl).

The deadline, for both the PhD positions as well as the PostDoc ones, is

The project is divided into three sub-projects. Below I give an indication of their flavor.

The bottom line of this direction is a duality:

The corank 1 Poisson structure can also be thought of as codimension one symplectic foliations (i.e. endowed with symplectic forms varying smoothly from leaf to leaf). There are clear signs of the existence of a highly non-trivial relationship of this form (maybe as non-trivial as mirror-type duality). This project calls for making this duality more precise.

Some evidence of such a duality (or just relationship):

- the two share the same "almost" structures (an almost complex corank 1 sub-bundle of the tangent bundle).

- such a duality does exist in dimension 3- and this comes with the Confoliations of Thurston and Eliashberg. From this point of view, our point is that the confoliations in dimension 3 should become, in higher dimensions, corank 1 Poisson structures.

- various constructions/results from Contact Geometry seem to have, once one turns the page "by 180 degrees" (whatever that means), analogues in Poisson Geometry, and the other way around.

- these analogy is however non-trivial. For instance, the principal S^1-bundles over symplectic manifolds from Contact Geometry (with the Boothby-Wang structure) are very similar to the symplectic fibrations over a circle (with the obvious symplectic foliation). Such S^1-bundles with symplectic base should be in duality with bundles over S^1 with symplectic fibers.

- also, in this analogy, the techniques used to prove some of the theorems, are very similar in spirit, although the details are not (they should be dual). For instance, proving the a 4-manifold times the circle admits a contact structure, or that it admits a corank 1 Poisson structure, are both based on decompositions of the 4-manifold (dual, we believe), related to various Lefschetz-type fibrations (the fibrations are on manifolds with boundary in both cases but, as part of the duality, the behavior at the boundary is really complementary in the two cases (tangent in one, transversal in the other)).

- etc etc.

Note that finding the analogue of a result from one side, on the other side, is non-trivial. One exciting thing is some of the results that are trivial on one side become non-trivial on the other side (but do hold!). Hence, even if this duality is used only at the level of "slogan", there is something to learn (expect/predict and prove).

This project calls for making this duality more precise, understanding it or just taking advantage of it. Some of the things to do:

- complete the (mysterious for now) dictionary between Contact and Poisson. This can be done by inspecting several existing results, finding their analogues in the other side, testing them.

- go through some of the basic results of Contact geometry, search for their Poisson analogues and prove them; then understand the duality at the level of the proofs. This will give rise to several questions, e.g. regarding the behavior of Lefschetz fibrations (classical or broken) under various deformations.

- study the possibility of deforming the two types of structures into each other- similar to (and as a generalization of) the 3-dimensional case and confoliations.

- the study of open book decompositions adapted to a symplectic foliations, and understanding such open books with symplectic pages with boundary of "cosymplectic type" foliation (note: as in contact Geometry, such a decomposition does give rise, rather explicitly, to a corank 1-Poisson structurs). Understand how such decompositions relate (via appropriate deformations) to the similar decompisitions relevant to Contact Geometry.

- apply such ideas to settle questions regarding the existence of symplectic foliations on compact manifolds. Symplectic (and related) fibrations seem to offer a very powerful technique. Oppen books as well.

- Can one prove that, also for corank 1 Poisson structure, the existence of "almost structures" is enough?

- At least understand the existence of symplectic foliations on spheres (only dim 3 and 5 are understood at this point).

- use the insight to prove the following conjecture: on S^5, there are no codim 1 symplectic foliations whose leafwise symplectic forms are induced from a global closed 2-form.

- etc etc.

The aim here is to understand compactness phenomena in Poisson Geometry. Such compactness phenomena are best described in terms of the associated symplectic groupoids, in the same way that the compactness of a Lie algebra is about the compactness of (some or all) Lie groups associates to it. If you do not know (or do not like) groupoids, then never mind, there are other characterizations, at least in particular case.

This question hides a lot of beautiful geometry. Already for the simplest case of Poisson structures which come from fibrations over S^1; one is searching for symplectic forms along the fibers that vary from fiber to fiber non-trivially, in a very compact fashion (really the opposite of symplectic fibrations). Unraveling the condition, this is related to S^1-valued moment maps and to a problem that was open in symplectic geometry for a while (due to MacDuff and solved by Kotschick) on free S^1-actions with contractible orbits;with that, the search boils down to the search of symplectic manifolds that admit very nicely behaved families of symplectic structures. K3-surfaces and their Kahler structures come in as the best candidates (and they do work beautifully). But this is a very particular case. One can try to do things a bit more general (but still staying rather particular); that calls for understanding what made the K3 work, and brings in hyperkahler geometry into discussion (not understood at this point).

Another piece of geometry that comes in the discussion (which was trivial above- but hidden behind the S^1) is that of integral affine structures. Indeed, already for the zero (!!!) Poisson structure, the groupoids that should be compact would be just bundles of tori, with a symplectic form on the total space, compatible with the multiplication; and that turns out to be related (as for integrable systems) to an integral affine structure on the manifold (and this is really a 1-1 correspondence between such structures and groupoids that make the zero Poisson structure compact). Interesting enough: the relevance of IAS (integral affine structures) continues all the way to general Poisson structures of compact type (through transversal IAS, and then IAS on orbifolds), through an IAS-version of MOlino's structure theorem for Riemannian foliations (which, by the way, is based also on a "resolution" that is very much related to Grothendieck's simultaneous resolution, and the Weyl resolution G/T\times_{W} \mathfrak{t} \to \mathfrak{g} for compact Lie groups). Of course, the entire discussion brings us to some fundamental conjectures related to (integral) affine Geometry such as the Auslander conjecture and the Markus conjecture. Most intriguing is the case of nil-manifolds which is the one that is best understood.

Of course, the integral affine geometry (related to the space of symplectic leaves) and the hyperkahler-type geometry (related to the leaves) ultimately have to interact in a non-trivial way. This bring in questions regarding the symplectic and the Kahler cone and their integral affine Geometry (already the case of base nil-manifolds is very intriguing; to give you an idea of where we are, I should also say that the only case in which we can really carry out everything and produce example of the strongest possible compact type, is when the base if a circle or a torus).

Yet another piece of structures that shows up is very similar to the appearance of gerbes in geometric instances; just that the circle is replaced by a bundle of tori, and there is a symplectic structure around (due to the IAS). I.e., what we would call "symplectic gerbes". Such theory was not carried out yet. Further complications that may arise in the most general case is the fact that some of these structure (IAS, gerbes, etc) live over an orbifold rather than a manifold. This is another problem (but a doable one).

These are just a few indications of the structures/interactions/problems/phenomena that arise in this project. We do not understand all of them and, moreover, there is clearly even more that still have to be found.

etc etc.

Very very briefly, the leading question here (but not the only one) is whether there is a "smooth version of the Cartan-Kahler theorem" which ensures the existence of local solutions of PDEs (or integrals of an EDSs) under certain involutivity conditions. Our idea/guess/hope is that:

- Recall that, in the classical Cartan-Kahler, involutivity can be expressed cohomologically, in terms of vanishing of certain Spencer cohomology groups (purely algebraic objects).

- For the smooth case, the key point is to make sense of "smooth involutivity" by making sense of "smooth Cartan-Kahler" (and give up on "Cartan's test, etc). This is very much supported by examples- e.g. deformation or rigidity problems when the controlling cohomology for the formal or analytic problem is usually a rather algebraic cohomology (e.g. think about Lie algebra cohomology with coefficients in polynomials), while for the smooth problem the cohomology becomes more analytical (e.g. polynomials are replaced by smooth functions and the vanishing of the cohomology is a "strong vanishing", in the sense that it comes with "tame control" on the homotopy operators).

- The proof of such a result will no longer be an inductive, rather algebraic, process (e.g. building formal power series and proving convergence, or building integrals of higher and and dimension using Cauchy-Kovalevskaya), but becomes much more analytical- using the Nash-Moser fast convergence method.

For more details, please look at the project.