Geometric Analysis investigates the geometric and topological properties of smooth manifolds using tools from modern analysis, PDE and measure theory. The broader area of differential geometry also includes the geometries defined via the corresponding groups of invariances, e.g. General relativity, affine, projective, and symplectic geometries.
Our research group specializes on the following topics. The complex geometry of Kähler manifolds and holomorphic vector bundles, function theory and complex structure of noncompact complete Kähler manifolds, and notions of positivity in Kähler geometry and algebraic geometry. Heat flows methods, Ricci flow, Harmonic map heat flow, and Moving curves, surfaces and hypersurfaces. Harmonic functions and harmonic maps, related variational problems. Fully nonlinear elliptic and parabolic equations. Lie groups/algebras and geometries invariant under group actions. Isoperimetric inequalities, curvature independent geometric inequalities, Geometry of convex bodies. Entropy monotonicity and Hessian (differential Harnack) estimates for parabolic flows. Regularity aspects of solutions to geometric variational problems, geodesics, minimal surfaces, and free-boundary problems. Applications to Physics, particularly quantum mechanics, general relativity, condense matters, and string theory. Image processing, quantum computing/representation theory and information geometry. Reaction-diffusion systems in mathematical biology.
Image Credit: Costa-Hoffman-Meeks Surface, 3DXM Virtual Math Museum, https://virtualmathmuseum.org/Surface/costa-h-m/costa-h-m.html, Public Domain.