Topology, Geometry and Mathematical Physics
Geometry and topology are core areas of mathematics that have recently experienced rapid development, leading to the solution of the century-old Poincaré conjecture and providing key ideas that underlie general relativity, string theory and high-energy physics.
What these branches of mathematics have in common is their concern with the fundamental notion of shape. NC State’s geometers and topologists draw on tools from many other areas of mathematics — algebra, analysis, combinatorics, differential equations and representation theory — to answer questions about the nature of shape. Research directions of particular interest include low-dimensional topology and knot theory, symplectic geometry and topology, homotopy theory and geometric flows.
Geometry and topology have also emerged as rich sources of methods and ideas for solving problems in other fields of mathematics as well as in contemporary science and engineering, Members of our group are using their expertise to answer questions ranging from how to teach a computer to recognize images to determining the shape of the universe.
Mathematical physics, Lie algebras, vertex algebras, integrable systems.
Algebraic and topological combinatorics.
Representation theory, quantum groups, infinite-dimensional Lie algebras and groups, vertex algebras, algebraic combinatorics, quantum computation.
Mathematical physics; classical and quantum gravity, quantum cosmology, gauge theories, general relativity.
Geometric study of differential equations and variational problems; equivalence and symmetry problems; computational invariant theory.
Mathematical physics; general relativity, gauge theories, unified field theories; generalized symplectic geometry.