Complex analytic geometry is a very broad area of mathematics straddling differential geometry, algebraic geometry and analysis. Much of the interactions between mathematics and theoretical physics, especially string theory, channels through complex analytic geometry. Since 1950, it has remained one of the most active areas of research in mathematics. Some of the high points of research in this topic are: Yau's proof of Calabi's conjecture, Donaldson-Uhlenbeck-Yau's theorem that polystable vector bundles are precisely the solutions of the Hermitian-Einstein equation, Demailly's work of Kobayashi hyperbolicity.
