Mathematical Sciences: Differential Geometry and Partial Differential Equations

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The Principal Investigator will study Riemannian metrics of constant scalar curvature and related problems in nonlinear partial differential equations. He will study the geometry of singularities for solutions of differential equations arising from the Yamabe problem. And he will use the Calculus of Variations to analyze blow-ups of solutions to the parabolic equation arising from the harmonic map problem. For many years mathematicians studied geometry with techniques which have their origins in the Calculus. Other mathematicians and physicists used differential equations to understand motion and heat flow. The PI will combine the two techniques to solve various geometric problems arising from heat flow theory and cosmology.