An Excursion Through Geometry, Topology and Analysis in 4 Dimensions and Beyond

The study of differential geometric structures on manifolds has evolved from elementary geometry and calculus to the more complex structures prominent in current research. Work in this area of research has led to significant advances in theoretical physics.

In this lecture, Simon K. Donaldson will explain some basic concepts in modern differential geometry and their historical development. He will discuss connections with analysis of complex variables and illustrate how fundamental existence questions lead to nonlinear partial differential equations. The central section of the talk will discuss four-dimensional K3 surfaces. Donaldson will discuss the special solutions of Einstein’s equation on such surfaces, connected to the quaternions. He will outline some of the analytical techniques used to establish the existence of these. He will also sketch some related questions in higher dimensions which are the scene for much current research activity.

Donaldson studied mathematics at Pembroke College, Cambridge, earning his bachelor’s degree in 1979. In 1983, he received his doctoral degree from Oxford University. He went on to hold positions at Oxford University and Stanford University and is currently a professor at Imperial College London and a permanent member of the Simons Center for Geometry and Physics at Stony Brook University in New York. Donaldson’s research centers on differential geometry and its connections with topology, algebraic geometry and theoretical physics. He received a Fields Medal in 1986, is a fellow of the Royal Society and a foreign member of the US, French and Swedish academies of science.