Differential methods of Resolution of Singularities and applications to algebraic and differential geometry

Descripción del proyecto

Resolución de singularidades de foliaciones y formas diferenciales en dimensiones arbitrarias

Las singularidades de los objetos geométricos, incluidas las ecuaciones, las foliaciones y los morfismos, son puntos en que el objeto se comporta de forma inesperadamente compleja y surgen en las matemáticas y la física. Resolverlos —sustituyéndolos por objetos similares pero perfectamente «suaves»— ha permitido comprender mejor los puntos singulares en el caso de las ecuaciones algebraicas. Extender esto a objetos de geometría diferencial y sistemas dinámicos, incluyendo foliaciones y formas diferenciales, ha sido un reto abierto desde el siglo XIX, con solo resultados fragmentarios en dimensiones bajas. El proyecto DiffeRS, financiado por el Consejo Europeo de Investigación, pretende aplicar métodos de geometría diferencial nunca utilizados en este contexto para desarrollar una teoría de resolución sistemática y de alta dimensión con aplicaciones de gran alcance en geometría algebraica y diferencial.

Objetivo

A singularity of a geometric object, such as an equation, variety, foliation, morphism, etc, is, loosely speaking, a point with non-trivial local behavior. In geometry, analysis, algebra, physics, among other sciences, their occurrence is unavoidable. Resolution of singularities is one of the most successful techniques to study singular points of an algebraic equation. Arguably, Hironaka's proof of the existence of RS for algebraic varieties over a field of characteristic zero stands as one the greatest achievements of algebraic geometry in the previous century.

Extending the technique of resolution of singularities to objects of interest to differential geometry and dynamical systems, such as foliations, differential forms and metrics, has intrigued mathematicians since the 19th century. Nothing short of spectacular applications are anticipated, including to Riemannian geometry, Lipschitz geometry, sub-Riemannian geometry, global analysis, birational geometry, among others. However, the geometry of foliations involve transcendental phenomena and only low dimensional results are known. This seriously limits the potential for applications.

The proposed research will approach resolution of singularities of foliations and differential forms from a new direction, bringing to bear methods from differential geometry which have not been used before in this context. Toward this end, our key goals will be to develop methods of resolution of singularities that would encompass key examples in Lipschitz and Riemannian geometry; and to combine the acquired insight with newly developed methods in birational geometry, to produce a systematic approach to resolution of singularities of foliations and differential forms in arbitrary dimensions and its applications in algebraic and differential geometry. I believe that my preliminary works in this direction amply demonstrates the feasibility and potential of this approach.