The object of this proposal is to continue our research on the topics below using techniques such as complex analysis, spectral theory, non-linear analysis and ergodic theory: i) Spectral theory of the Laplacian on functions and forms - connections with ergodic theory. ii) Global injectivity theorems. iii) Topology of umbilics. iv) The generalized Hilbert theorem. Topic i) is a collection of problems coming from (and relating to) spectral and ergodic theory, dynamics, vanishing theorems and the topology of manifolds of non-positive curvature. Topic ii) is an outgrowth of our efforts to understand embeddedness of minimal surfaces. It centers on the problem of finding conditions for a local diffeomorphism between non-compact manifolds to be injective. This type of question arises in areas of mathematics as diverse as algebraic geometry and mathematical economics. Our methods here come from geometry, topology and global analysis. Topic iii) is a classical problem in differential geometry namely, understanding the topology of umbilical singularities. The plan is to continue studying this problem as a question about the blow-up of certain hyperbolic partial differential equations. Topic iv) is also a classical problem, dealing with isometric immersions of hyperbolic spaces. As in iii), we plan to approach this question as a blow-up problem.
In our approach to research, starting with the earlier work on minimal surface theory, we have always sought to achieve a balance between technique and creative mathematics. This proposal is full of problems and ideas coming from diverse areas, such as spectral theory, dynamical systems, hyperbolic equations, algebraic and Riemannian geometry. We have already contributed to all these questions. Given the variety of topics, we would like to believe that this proposal advances the "internal" conversation of mathematics. On the other hand, the work on Global Inversion is potentially of interest to applied disciplines, since it addresses the question of solvability of systems of non-linear equations.
