Geometric analysis

Geometric analysis is a fun part of mathematics. We use special equations to study shapes and how they behave. This helps us learn about surfaces and spaces.

People have used these ideas for many years. Tibor Radó and Jesse Douglas did early work on special surfaces. Later, John Forbes Nash Jr. showed how to place curved spaces into regular Euclidean space. Other mathematicians like Louis Nirenberg, Aleksandr Danilovich Aleksandrov, and Aleksei Pogorelov also found new things about shapes.

In the 1980s, geometric analysis became very interesting. Karen Uhlenbeck, Clifford Taubes, Shing-Tung Yau, Richard Schoen, and Richard Hamilton made big discoveries. Their work helped solve important questions about space.

Scope

Geometric analysis is a part of mathematics. It uses special equations to study shapes and spaces. These equations help us learn about curves, surfaces, and more complex spaces.

Some important topics in geometric analysis include gauge theory, harmonic maps, Kähler–Einstein metrics, mean curvature flow, minimal submanifolds, positive energy theorems, pseudoholomorphic curves, Ricci flow, the Yamabe problem, and Yang–Mills equations. These topics help mathematicians understand the links between shapes and equations.

Main articles: Gauge theory, Harmonic maps, Kähler–Einstein metrics, Mean curvature flow, Minimal submanifolds, Positive energy theorems, Pseudoholomorphic curves, Ricci flow, Yamabe problem, Yang–Mills equations