Principal Investigator: Daniel M. Burns
Some of the themes emphasized in these research projects are modern aspects of the Bohr-Sommerfeld theory from the early days of quantum mechanics in the 1920s, value distribution theory and Ahlfors currents, and Grauert tubes. In physics the Bohr-Sommerfeld theory was created as a method for quantizing an integrable classical mechanical system. The principal investigator interprets the Bohr-Sommerfeld conditions for quantization as a geometric property, name a triviality condition on the holonomy of a certain flat bundle. Ongoing work seeks to understand the connections between singularities of Hamiltonians and singularities of underlying complex spaces, and the geometry of the Bohr-Sommerfeld construction seems to be a central focus of the story.
Value distribution theory is a framework for attempting to count solutions to an equation in complex variables. The Fundamental Theorem of Algebra tells us that a polynomial equation of degree n in one variable has exactly n solutions over the complex numbers, if you count repeated roots with care. More complicated equations in a single complex variable can have infinitely many solutions, but these are spread around the complex plane and the number of solutions contained in a ball of radius R about the origin can only grow at a limited rate as R becomes large -- a situation greatly clarified by ideas introduced by the Finnish mathematician Rolf Nevanlinna in the 1920s, at about the same time that the Bohr-Sommerfeld theory was developed in physics. One of the projects supported by this award is a project that applies modern geometric tools to study the distribution of solutions to equations in several variables.
. Complex analysis revolves around special properties of imaginary numbers in calculus. The geometry harkens back to coordinate geometry as in the Cartesian plane. In recent years that has taken the PI into areas involving Hamiltonian mechanics and symplectic geometry, the natural setting for such mechanics. Much of this is motivated, however, not by the mechanical applications, but the underlying geometric structures. The specifics of the project, as it evolved and it became clearer what could be achieved in the lifetime of the grant, were 1. Singular integrable systems 2. Analytic mappings and cycles 3. Complexifications of real manifolds 4. Rigidity of non-compact Kaehler manifolds 5. (Pluri-)potential theory associated to real convex sets. Some significant results were achieved in these areas during the course of the grant. A theorem describing the property of being a locus of zeroes of polynomials in complex number space in terms of its growth measured by a potential theoretic function (homogeneous Monge-Ampere solution) was obtained, as well as uniqueness results on the ways a simple topological space can be realized as a zero locus of polynomials [about 3.]. Results were obtained on the regularity (i.e., how differentiable it is) of the complex geometry analogue of the Newtonian potential carried by a real source in real number space [about 5.]. Other results were obtained on an interpretation of imaginary quantum mechanics for geometric purposes [about 1.]. Work was begun on the asymptotic behavior of cycles (geometric loci) related to the periodicities of integrals on algebraic loci [about 2.] and on the topology of leaves of complex foliations (this includes the qualitative behavior for long time of complex ordinary differential equations with rational coefficients) [about 2.]. The problem in (4.) has not yielded to our efforts during the lifetime of this grant, but remains interesting. The PI, in work only indirectly supported by this grant, works also with a group of mathematicians, engineers and biologists on problems related to DNA dynamics and control ("4D-genome"), which might have implications for the medicine of the future. There is very interesting geometry in such problems, especially when trying to devise appropriate measures for statistical comparisons of disparate data in time series. In addition, as part of outreach associated with this grant, the PI has worked to raise the level of scientific and mathematical human capital in Africa, through organizing conferences and support networks for African scientists and mathematicians. He sought to work with University-level colleagues in the Detroit public schools for similar goals, but the situation proved politically catastrophic, and he has put that aside for now.