Geometric Methods in the Analytic Theory of Differential Equations

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Theory of differential equations is a fundamental mathematical tool of physics and engineering. Few differential equations can be solved explicitly, and approximate numerical solutions do not always give essential features of the behavior of exact solutions. A class of simple differential equations that frequently occur in applications has been studied by mathematicians for centuries; their solutions are called special functions of mathematical physics, and they are widely used in science. The general goal of this project is to extend this class of well-understood equations. Building on earlier work relevant for applications to physics, control theory, and materials science, this project aims to apply a variety of recently-developed methods to study longstanding questions of intrinsic mathematical interest. The work is anticipated to further improve understanding of the qualitative features of analytic functions defined by some basic differential equations arising in mathematical physics and geometry.Most of the special functions of mathematical physics are defined by linear differential equations with at most three singularities. Solutions of the Heun equation (with four regular singularities) and the Painlevé VI equation (non-linear, with four fixed singularities and no movable singularities) lie on the boundary of the class of special functions. Because of their intrinsic mathematical interest and numerous applications in science, they have been extensively studied since the beginning of the 20th century. This research project aims to advance understanding of these important functions through the use of new geometric methods. The main topics of this project are the qualitative study of real solutions of the Painlevé VI equation, the study of Riemannian metrics of constant positive curvature with conic singularities, with the emphasis on the metrics with four singularities closely related to the Heun equation, and finally the study of the eigenvalues of some parity-time symmetric anharmonic oscillators.