Symplectic Cobordism Relations on Contact Manifolds

Differential geometry is the study of "smooth shapes", e.g. curved surfaces that have no rough edges or sharp bends. A surface is a 2-dimensional object, and one can similarly imagine smooth shapes that are 1-dimensional, such as a line, or curve, or circle. What is much harder to imagine, but can nonetheless be described in precise mathematical terms, is a smooth shape in an arbitrary number of dimensions: these objects are called "manifolds".

A specific example of a 2-dimensional manifold is a disk, i.e. the region inside a circle, and its "boundary" is a 1-dimensional manifold, namely the circle. Similarly, for any positive integer n, an n-dimensional manifold may have a boundary which is an (n-1)-dimensional manifold. All the 3-dimensional manifolds that we can easily picture are of this type: e.g. if we imagine any surface in 3-dimensional space, such as a sphere or a "torus" (the shape of the surface of a doughnut), then the region inside that surface is a 3-dimensional manifold whose boundary is the surface.

We can now ask one of the most basic questions concerning manifolds: given an n-dimensional manifold, is it the boundary of something? This is actually not just a geometric question, but really a question of "topology", which is a certain way of studying the "overall shape" of geometric objects. As in the example given above, most 2-dimensional manifolds that we can easily imagine are boundaries of the 3-dimensional regions they enclose. But for a more interesting example, we can try to imagine a "Klein bottle": this is a surface formed by taking an ordinary bottle and bending its opening around and through the glass into the inside, then connecting the opening to the floor of the bottle by curving the floor upward. The result is a surface that is not a boundary of anything, as its inside is not distinct from its outside; like a Moebius strip, but closed in on itself.

The subject of this proposal concerns a more elaborate version of the above question about boundaries: we deal with a particular type of manifold in an even number of dimensions, called "symplectic" manifolds, and their odd-dimensional boundaries are called "contact" manifolds. The idea of a symplectic manifold comes originally from physics: a century ago, symplectic manifolds were understood to be the natural geometric setting in which to study Hamilton's 19th century reformulation of Newton's classical mechanics. Today symplectic manifolds are considered interesting in their own right, and they retain a connection to physics, but of a very different and non-classical sort: by studying certain special surfaces in symplectic manifolds with contact boundary, one can define a so-called "Symplectic Field Theory" (or "SFT" for short), which bears a strong but mysterious resemblance to some of the theories that modern physics uses to describe elementary particles and their interactions. Unlike those theories, SFT does not help us to predict what will happen in a particle accelerator, but it can help us answer a basic question in the area of "Symplectic and Contact Topology": given a contact manifold, is it the boundary of any symplectic manifold?

More generally, one way to study contact manifolds themselves is to consider the following relation: we say that two such manifolds are "symplectically cobordant" if they form two separate pieces of the boundary of a symplectic manifold. The question of whether two given contact manifolds are cobordant helps us understand what kinds of contact manifolds can exist in the first place, and Symplectic Field Theory is one of the most powerful methods we have for studying this. The goal of this project is thus to use this and related tools to learn as much as we can about the symplectic cobordism relation on contact manifolds. Since most previous results on this subject have focused on 4-dimensional manifolds with 3-dimensional boundaries, we aim especially to gain new insights in higher dimensions.

The project proposed here is fundamental research. As with all research in pure mathematics, it may eventually have a considerable impact on society at large, but it would be impossible to predict when or how. A classic example of this phenomenon is the 19th century development of Riemannian geometry, which existed for decades as a field of pure mathematics, largely without application, before Einstein related it to the real world through his profound theory of gravitation.

The immediate impact of this research will be on mathematicians in symplectic/contact topology and other fields, as detailed in the "Academic Beneficiaries" section of the proposal. Further impact beyond academia may come about due to the work of those academic beneficiaries, and I will describe two speculative scenarios for this below. One further point regarding the academic impact pertains in particular to young researchers and PhD students: among the activities planned in conjunction with this project is the formation of a London Symplectic Working Group or "Symplectic Learning Seminar", the point of which will be for local researchers in this field and occasional invited guests to gather informally and explain current research topics (not necessarily their own research) to each other at a level suitable for PhD students. The planned workshop, to be held during the second year of the project, will also include an "educational" component of this sort, the purpose being to generate more interest in and knowledge of the topics of this proposal among early stage researchers. At the same time, we will be counteracting the current "brain drain" among young British mathematicians by attracting to the UK, or retaining in the UK, a highly skilled scientist, namely the Research Associate. And furthermore, we plan to take full advantage of London's proximity to the Continent through regular seminar visits to Paris and Brussels and collaboration with mathematicians in France, maximising the international exchange of ideas. All of this may be seen to have a cultural impact, as it contributes to upholding the great intellectural tradition of pure mathematics in the UK, and in London especially.

We now mention two speculative possibilities as to the further impact on society that may eventually result from this research:

(1) The study of both symplectic manifolds and contact manifolds originates to some degree in Hamiltonian dynamics, and results in this field (including some of our objectives described under "Problem 5" in the Case for Support) often have dynamical consequences that might have real world applications. An exciting recent example is the work of Frauenfelder, van Koert and collaborators who have shown that the restricted 3-body problem can be understood as defining a contact type hypersurface: existence results for periodic orbits on this hypersurface can then be proved using holomorphic curve techniques, and may have a practical use in finding low energy trajectories for a space craft travelling from the Earth to the moon.

(2) Symplectic Field Theory and other types of "topological" quantum field theories share many formal properties with some of the mathematics behind the "standard model" of elementary particle physics, but the mathematical foundations of this model are not fully understood, and the project to solidify them has made very little progress in the last half century. One may hope that the further development of these purely mathematical analogues such as SFT will eventually lend useful insights to theoretical physics, which may be necessary before the coveted "unified" field theory can be found. (Note: this is surely a minority view within the physics community, but I am not a physicist.)