Under this award, the principal investigator plans to continue his work in geometric partial differential equations. In particular he will investigate the role of level sets of solutions of differential equations. Special cases of level sets are given by the nodal sets, the singular sets and the branch sets. An important part of the study is the investigation of the asymptotic behavior of solutions near these sets or the asymptotic behavior of these sets themselves. The differential equations may be elliptic, hyperbolic or of mixed type. Some problems have close connections with other fields in mathematics, including several complex variables and algebraic geometry. Another class of problems that the PI will continue to work on involves the effect of level sets of known functions in the equations on the existence and properties of solutions. A particular problem is the isometric embedding of 2-dim Riemannian manifolds in 3-space when the zero set of Gauss curvature is well behaved. This problem is in the form of degenerate Monge-Ampere equations.
The problems involving nodal sets or the singular sets originate from materials science and control theory. Singular sets, as the name suggests, are those sets where singularities occur. Precise definitions vary depending on the problems where they arise. It often is impossible to eliminate the singular sets, the so-called "bad sets". One of the central tasks is to identify the conditions under which the singular sets can be controlled and the conditions under which the singular sets are small. The proposed problems concerning singular sets in the project are in their simplest forms. They are related to the Erickson's model for liquid crystals and the Ginzburg-Landau equation in the superconductivity. The PI believes that the discussion of these mathematical problems will help scientists work with singular sets in various applications.