Doctor of Science in Pure Mathematics - Geometry and Topology in Cambridge United States | Massachusetts Institute of Technology

The field of modern geometry is transforming nearly every area of mathematics and is currently experiencing remarkable advancements. Long-standing challenges are being addressed through innovative approaches from this domain (most notably Perelman's solution to the Poincaré conjecture), while novel research directions are being explored. A key focus has been applying advanced PDE methods to examine naturally occurring geometric equations. The groundbreaking Atiyah-Singer index theorem, established in the 1960s, bridges linear PDE theory with topological and geometric concepts. While progress in nonlinear PDE tools for geometry has been more gradual, these developments have yielded some of mathematics' most spectacular achievements, including Donaldson's revolutionary work on four-dimensional manifolds using Yang-Mills equations from particle physics.

Investigations into lower-dimensional manifolds (particularly dimensions four and below) hold special importance for theoretical physics and numerous practical applications. Floer homology provides a rigorous mathematical framework for constructing components of quantum field theories. Another expanding frontier involves mathematical approaches to general relativity, where the Lorentz formulation of Einstein's equations represents the current pinnacle of hyperbolic PDE techniques. One aspect of geometric analysis focuses on reconstructing functions from their integrals across various domains, with Computed Tomography (CT scans) being a prominent real-world implementation.

This research group's focus spans geometric analysis, symplectic topology's role in mirror symmetry, low-dimensional topology and gauge theory, Riemannian geometry with minimal surfaces, and connections to mathematical physics.