Topology is the study of shapes and spaces. What happens if one allows geometric objects to be stretched or squeezed but not broken? In fact there’s quite a bit of structure in what remains, which is the principal subject of study in topology.
The modern field of topology draws from a diverse collection of core areas of mathematics. Much of basic topology is most profitably described in the language of algebra – groups, rings, modules, and exact sequences. But topology has close connections with many other fields, including analysis (analytical constructions such as differential forms play a crucial role in topology), differential geometry and partial differential equations (through the modern subject of gauge theory), algebraic geometry (the topology of algebraic varieties), combinatorics (knot theory), and theoretical physics (general relativity and the shape of the universe, string theory). In addition, topology can strikingly be applied to study the structure of large data sets.
Many of these threads of topology are represented by the faculty at Duke. For instance:
- Richard Hain studies the topology of complex algebraic varieties and moduli spaces, applying techniques from mapping class groups, Hodge theory, and Galois theory;
- John Harer uses computational topology to study a wide range of problems of an applied flavor;
- Lenhard Ng and Adam Levine study low-dimensional topology, the topology of three- and four-dimensional spaces, via knot theory and symplectic geometry;
- David Kraines has worked on higher algebraic operations in homology and cohomology;
- William Pardon studies the algebraic topology of varieties and singular spaces using tools such as algebraic K-theory;
- Leslie Saper works in aspects of topology related to analysis and number theory, in particular studying automorphic forms and singularities in algebraic varieties.
Other faculty that work in related areas include Paul Aspinwall (string theory) and Hubert Bray, Robert Bryant, and Mark Stern (geometric analysis).