Objective
The interplay between Geometry and Analysis has been among the most fruitful mathematical ideas in recent years, the most obvious example being Perelman's proof of Poincare' conjecture. I plan to pursue further this approach and make distinct progress in two different problems.
Scalar Curvature: A classical theorem in Riemannian Geometry states that nonnegative scalar curvature metrics which are flat outside a compact set must be Euclidean. The equivalent problem for positive scalar curvature is known as the Min-Oo conjecture and was recently disproven by Brendle, Marques, and myself.
I plan to show uniform area bounds for minimal surfaces in manifolds with positive scalar curvature where the bounds are attained if and only if we are on a round sphere. I also plan to show that those manifolds have an infinite number of minimal surfaces (Yau's conjecture). My approach consists of studying min-max methods in order to obtain existence of higher-index minimal surfaces.
Mean curvature flow: An hard open problem consists in determining which Lagrangians in a Calabi-Yau admit a minimal Lagrangian (SLag) in their isotopy class. A complete answer would be a breakthrough of considerable size. A possible approach consists of deforming a given Lagrangian in the direction which decreases area the most and hope to show convergence to a SLag. The difficulty with this method is that finite-time singularities can occur.
I plan to study the regularity theory for this flow and show that, for surfaces, singularities are isolated in space. My approach consists in classifying the possible blow-ups and find monotone quantities which will rule out non SLag blow-ups.
In October of last year I completed 9 years in the USA where the last 2 were spent as an Assistant Professor at Princeton University. Due to personal reasons I decided to move back to Europe. Hence this grant will provide me with the necessary financial support to continue my research.
Fields of science (EuroSciVoc)
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CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
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Programme(s)
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Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Call for proposal
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
FP7-PEOPLE-2010-RG
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Funding Scheme
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
Coordinator
SW7 2AZ London
United Kingdom
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