Geometric analysis

Geometric analysis studies geometric objects with tools from analysis, especially differential equations, partial differential equations, and variational methods.

Differential geometry: The study of smooth spaces, their curvature, and the geometric structures they carry. It supplies the manifolds, metrics, and curvature that geometric analysis investigates.
Elliptic partial differential equation: A partial differential equation whose principal part is elliptic, often governing equilibrium or regularity. Elliptic equations describe many stationary geometric configurations.
Yau's theorem: Shing-Tung Yau's proof that every closed Riemannian manifold of nonpositive sectional curvature has a contractible universal cover. Its use of harmonic maps linked curvature assumptions to global topology.
Shing-Tung Yau: A mathematician known for major contributions to differential geometry, geometric analysis, and mathematical physics. His work on the Calabi conjecture and positive mass theorem transformed geometric analysis.
Synthetic geometry: The study of geometric properties through axioms and direct geometric constructions rather than coordinates or analysis. It derives geometric conclusions without the differential equations central to this field.
Partial differential equation: An equation involving an unknown function of several variables and its partial derivatives. Many central geometric problems become equations for functions or geometric fields.
Mean curvature flow: An evolution of a hypersurface in which its normal velocity equals its mean curvature. This geometric evolution uses a parabolic equation to smooth and reshape surfaces.
Positive mass theorem: A theorem asserting nonnegative total mass for asymptotically flat initial data satisfying suitable energy conditions. Its proofs use geometric inequalities and analysis on manifolds.
Richard Hamilton: An American mathematician who introduced Ricci flow as a tool for studying three-dimensional geometry. His foundational work established the evolution equation later used in the Poincaré conjecture proof.
Algebraic geometry: The study of geometric spaces defined by polynomial equations using algebraic methods. It shares geometric questions but emphasizes algebraic structure over analytic estimates.

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