The Principal Investigator will undertake a study of 3-, 4-, and higher-dimensional spaces using a theory called "higher-dimensional Heegaard Floer homology'' which the PI is developing with his collaborator Vincent Colin. These spaces will locally be similar to the standard (Euclidean) n-dimensional spaces and may be very complicated globally, but a local observer cannot tell the difference, just as an ant cannot tell whether it is sitting on a flat plane or a very large sphere. Higher-dimensional Heegaard Floer homology is defined through contact and symplectic geometry, which in turn are closely related to mathematical physics and string theory. These studies in higher dimensions should in turn contribute significantly towards our understanding of low-dimensional (i.e., 3- and 4-dimensional) shapes, from the knotting of DNA at the microscopic level to the shape of the universe at the macroscopic level.
The PI will study contact and symplectic geometry in higher dimensions through a higher-dimensional generalization of Heegaard Floer homology. Heegaard Floer homology, due to Ozsváth and Szabó, is a package of invariants of 3- and 4-dimensional spaces as well as knots and links in 3-space, which is extremely effective at distinguishing these spaces and knots and links from one another. The main goal of the PI's research program is to develop the theory of Heegaard Floer homology in higher dimensions, i.e., in dimensions greater than 4. This theory is expected to yield important applications to contact and symplectic geometry in higher dimensions, which in turn are related to algebraic geometry and mathematical physics. Khovanov homology, another important invariant of knots and links in 3-space, can at least conjecturally be viewed as a special case of higher-dimensional Heegaard Floer homology.
