Principal Investigator: Mark McLean
The subject area of this project is the symplectic geometry of Stein manifolds. A Stein manifold is a properly embedded complex submanifold of complex affine space. This has a symplectic form induced from the standard one in affine space. If we take a large sphere and intersect it with this Stein manifold then we get another manifold called a Stein fillable contact manifold. Important progress in studying Stein manifolds symplectically was achieved by Eliashberg and Gromov. The primary aim of this project is to find exotic Stein structures and Stein fillable contact structures. The PI intends to prove that there are uncountably many symplectically different Stein structures diffeomorphic to an even dimensional manifold admitting a proper and bounded from below Morse function with only finitely many critical points of index at most half its dimension. The PI also intends to prove that there are infinitely many Stein fillable contact structures on each odd dimensional sphere of dimension 5 and higher, and more generally on Stein fillable contact manifolds obtained from affine varieties. The PI will use an invariant of Stein manifolds called symplectic homology to distinguish these. The PI also aims to show that there is no algorithm to tell you whether one Stein manifold diffeomorphic to affine space is symplectomorphic to another one diffeomorphic to affine space of complex dimension greater than 6. The PI also aims prove a similar undecidability result for contact structures on all odd dimensional spheres of dimension greater than 13. The PI will use an invariant called the growth rate of symplectic homology to achieve this. The PI will use growth rates to show that certain cotangent bundles have many Reeb orbits (even degenerate ones). This generalizes the Gromoll-Meyer theorem. The PI will show that the cotangent bundle of a rationally hyperbolic manifold is not symplectomorphic to a smooth affine variety using growth rates.
If we have some classical system such as a pendulum then at any point in time it has a particular position and momentum. If this system has many moving parts such as a double pendulum or a collection free particles then it has many positions and momenta. The set of all such positions and momenta can be encoded in an object called a symplectic manifold. For example the symplectic manifold associated to a pendulum turns out to be a cylinder. Symplectic manifolds are important in many areas of physics such as quantum mechanics and String theory. The PI will study a large class of symplectic manifolds obtained from objects called Stein manifolds. The PI will construct a large list of Stein manifolds called exotic Stein manifolds which look very similar to the symplectic manifold coming from a set of free particles but are actually different if we look at the motion of their respective classical systems. The PI intends to show that there is no computer algorithm telling you if two given exotic Stein manifolds come from the same classical system. This result is useful because it tells us that certain classical systems are very hard to study in general.
I have written multiple papers relating two areas of mathematics. One area is called symplectic geometry which is a branch of geometry related to classical physics. The other area is called algebraic geometry, and this is a branch of geometry closely related to algebra. Both are very active areas of mathematics that are also used by theoretical physicists. Because of this grant I have gained a much better understanding of the relationship between these two subjects, and this will lead to many more breathroughs in the future as well. There are now new projects that have started recently by myself and other people as a result of this NSF grant. My first NSF publication actually showed that two families of geometric objects are very different from each other. This result is a symplectic analogue of a result in algebraic geometry. Actually I think that the techniques used in this paper are more important than the results as these ideas have now been used by several people around the US. At the same time I released a minor publication which was designed to compute certain special properties of objects in symplectic geometry. In another result I showed that there are certain problems in symplectic geometry that cannot be solved with a computer. This demonstrates the fact that some problems in symplectic geometry are too hard to solve forcing us to focus on simpler examples. Another publication generalized a result from the 60's. This result was dynamical in nature and showed that there are a huge number of physical systems that after some time come back to where they started and repeat themselves. In algebraic geometry, one thing people like to do is start with some fairly complicated algebraic object and then try to simplify it as much as possible. During this simplification process some properties remain constant and one of them is called log Kodaira dimension. My final result shows that in some circumstances, log Kodaira dimension has an interpretation in symplectic geometry. Having said that there is still a long way to go in this direction.