The PI will work on some problems in symplectic geometry (moduli of singular toric spaces and degenerations), Akhiezer-Gindikin domains (bounded realization problem), realizability of the Beauville class in Hodge theory, and the rigidity of asymptotically complex hyperbolic manifolds and of some new examples of extremal cycles in flag varieties.
Put less technically, the PI will carry out several research projects in closely related areas of mathematics in complex geometry and analysis. These are areas which are direct descendants of the work of Descartes on the geometry or shape of solution sets of equations. The motivations for these studies are primarily geometric, and most employ analysis, the descendent of the calculus. One project deals with geometry related to that of special mechanical systems in classical physics (Hamiltonian dynamics and integrable systems), and one deals obliquely with complex geometry related to quantum physics, specifically string theory, a highly speculative but geometrically attractive theory of elementary forces on the smallest scales. Two of the subprojects deal with what is called the growth of spaces as one travels far out in them towards the horizon. Properties that are visible at this scale are known as asymptotic properties, and the PI will study whether certain local geometric conditions can lead to regular patterned behavior in distant regions of these spaces. These projects refer to a kind of rigidity: if a modest amount of control over the growth or complexity of a space as it stretches out to its horizon, then it will ``freeze'' into a very specific and determined form. Finally, one of the subprojects deals with more algebraic problems. This means that ultimately they are questions about polynomials rather than more complicated functions. They deal with special spaces which are cut out of simpler spaces by equations, and which have maximal intersection with fixed reference spaces. Can one conclude the exact geometry from this maximal intersection property? The point is to use geometric differential equations, not dissimilar to the ones originating in mechanics. Most of these projects will be carried out with young researchers, colleagues or former students of the PI. Hopefully, they will serve to develop the research interests of these young mathematicians. In another direction, the PI hopes as well to further his interests in molecular biology and genetics, working with students on projects to understand the mathematical nature of the information carried by genetic molecules, though this grant application does not seek funding for such activities beyond a couple of undergraduate students who might have a summer research experience on such aspects of math and biology.
