A major thrust of the project concerns Cappell-Shaneson homotopy 4-spheres. These have long been considered the most likely counterexamples to the smooth 4-dimensional Poincare Conjecture. Recent work by Akbulut, and a simpler, more general approach by the PI, have shown that infinitely many of these are standard. The PI plans to extend his method further, trying to show that all CS-spheres are standard. He will also study other homotopy 4-spheres containing fibered 2-knots, accumulating evidence suggesting that the conjecture is true. The project includes various other avenues of research related to the PI's expertise. He will search for compact domains of holomorphy in complex surfaces, using one of his recent theorems. It now suffices to locate smoothly embedded compact 4-manifolds admitting suitable handle decompositions. He hopes to find (or rule out) examples such as pseudoconvex embeddings of preassigned homology spheres, and a compact domain of holomorphy in C^2 homotopy equivalent to the 2-sphere (violating a conjecture of Forstneric). A related theorem of the PI allows one to construct topological embeddings of 3-manifolds with a generalized pseudoconvexity property; he will investigate these in more detail. Another project is to study what manifolds of dimension 4 and higher can be realized as orbit spaces of vector fields in Euclidean space. Preliminary research shows that such manifolds must be simply connected, but many 4-manifolds with nontrivial 2-homology can arise. Various other investigations, concerning compact 4-manifolds, symplectic structures and Lefschetz pencils will also be pursued.
The Poincare Conjecture for 3-manifolds has now been proved by Perelman, a full century after it was first proposed. Its generalization to higher dimensions was proved in the 1960's except in dimension 4. That last version was proved around 1981 by Freedman. However, the original conjecture was made for topological manifolds, so that one is allowed to crinkle the manifold in complicated ways. When manifolds are used in practice, in geometry, analysis, physics, economics and so on, one normally wants to be able to apply calculus, so one must disallow crinkling and work entirely with smooth manifolds. The smooth analog of the Poincare Conjecture has been understood in dimensions >4 since the 1960's, and is equivalent to the topological version (hence solved) in dimensions <4. However, the smooth 4-dimensional Poincare Conjecture is still mysterious, and is the last fundamental open question remaining from the initial heyday of manifold topology a half-century ago. There have been many potential counterexamples constructed, homotopy 4-spheres that might not be the standard 4-sphere, but none has been shown to actually be nonstandard. It has also been quite difficult to show that any of these examples are standard, but the PI has been in the forefront of research in this direction. His new methods have dispensed with a large family of potential counterexamples that were constructed in the 1970's. He intends to further investigate this problem, adding evidence that the conjecture may be true after all, in spite of the prevailing belief to the contrary in recent decades. He will also study other problems involving 4-manifolds and other classical mathematical objects.
Manifolds are smooth shapes of various dimensions. A curve is a 1-dimensional manifold (1-manifold) and a surface, such as a soap bubble or inner tube, is a 2-manifold. Ironically, the least understood manifolds are those with four dimensions, such as the universe of space-time in which we live. Manifolds can be equipped with various geometric structures that have applications throughout the sciences: Structures called flows (equivalently, vector fields or ordinary differential equations) have been studied in various forms for centuries, but recent developments continue to have applications in diverse areas such as physics, chemistry, biology and engineering. Symplectic structures originated independently in algebraic geometry and physics (classical and quantum Hamiltonian mechanics and string theory). Their cousins, Stein structures, have been studied by complex analysts for nearly a century. The goal of this project has been to improve our understanding of 4-manifolds and these structures on them. Manifolds with at least 4 dimensions can often be smoothed in different ways - as if one could iron a garment in one way to get a shirt, or a different way to get a sock. This is hard to visualize, since the phenomenon cannot happen with fewer than 4 dimensions, but is fundamental to high-dimensional manifold theory, and thought by some physicists to be related to the most fundamental properties of matter in the universe. Ironically, the phenomenon is least understood in 4 dimensions, where it is not even known whether the 4-dimensional sphere has different ("exotic") smoothings. (This is the last open case of the notorious Generalized Poincare Conjecture, which Poincare first stated for 3-manifolds over a century ago.) In the course of the project described here, the Principal Investigator studied various potential counterexamples to this conjecture, showing that some were not exotic, but that other examples occurred in surprisingly large families warranting further study. He also studied open 4-manifolds (those that extend infinitely, such as 4-dimensional Euclidean space), greatly expanding the catalog of their known exotic smoothings. Regarding geometric structures on 4-manifolds, the PI developed a procedure for locating Stein manifolds inside of 4-manifolds equipped with complex structures, answering questions from the original proposal and exhibiting various surprising new phenomena. He made progress on the problem of describing symplectic 4-manifolds in a purely topological way (contrasting with the original geometric description). The use of multiple equivalent viewpoints for the same object has traditionally been a powerful approach to mathematics and physics; the case discussed here is still incomplete, but had already been exploited in fundamental progress by other researchers. The project also dealt with flows on manifolds of arbitrary dimension, through their "orbit spaces", which encode the way their flow lines fit together. Even simple flows often have orbit spaces that fail to be manifolds, due to complicated local behavior (around a single flow line). However, Jack Calcut and the PI showed that for the simplest (gradient-like) flows, the global structure of the orbit spaces is still determined in a simple way from the manifold on which the flow occurs. In contrast, the PI showed that orbit spaces that are manifolds can have complicated global behavior, even when the manifold on which the flow occurs is as simple as possible (namely, Euclidean space, the case of classical ordinary differential equations). Broader impact: During the grant period, the PI mentored an undergraduate student, three Ph.D. students (two of whom are women) and a Simons postdoctoral fellow, as well as hosting a graduate student from Japan one semester. All of these people worked on their own research problems, closely related to the NSF project described here. The undergraduate wrote a Dean's Scholar thesis, making a modest advance related to the Generalized Poincare Conjecture, and is about to begin graduate studies at Duke University. The student from Japan has since earned his Ph.D. and is continuing in academic research, as is the Simons Postdoc. The three Ph.D. students are continuing, and all making progress. (One has already posted three papers.) In addition to one-on-one mentoring, the PI served for a year as Assistant Graduate Adviser. His research experience enriches his graduate and advanced undergraduate teaching. Some of the undergraduate students plan to be secondary school teachers, for whom any insight into the process of research should benefit future generations. The PI gave talks on his research results at various conferences and seminars. The grant funds also allowed his students to attend conferences and exchange ideas with researchers from around the world. The results of his research have been (or are being) written up, posted online, and published in refereed journals. In addition, International Innovation, in collaboration with the PI, published an expository article on this research project. The target audience of that publication is a broad spectrum of scientifically literate lay readers, including various policy-making organizations internationally.
