Topology

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 studies 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).

Faculty

Michael Abel

Visiting Assistant Professor of Mathematics

Keywords in this area
Knot Theory, Quantum Topology

Other research areas
Algebra & Combinatorics Topology

Paul L Bendich

Assistant Research Professor in the Department of Mathematics

Keywords in this area
topological data analysis, applied topology

Robert Bryant

Philip Griffiths Professor of Mathematics

Keywords in this area
curvature, Lie groups, homology

Justin Curry

Visiting Assistant Professor of Mathematics

Keywords in this area
applied algebraic topology, category theory, Sheaf theory

Richard Hain

Professor of Mathematics

Keywords in this area
topology of varieties, mapping class groups

John Harer

Professor of Mathematics

Keywords in this area
Topological Data Analysis

David P. Kraines

Associate Professor Emeritus of Mathematics

Keywords in this area
cohomology operations

Other research areas
Biological Modeling Topology

Sayan Mukherjee

Professor of Statistical Science

Keywords in this area
stochastic topology, topological data analysis

Lenhard Lee Ng

Eads Family Professor

Keywords in this area
low-dimensional topology, knot theory

William L. Pardon

Professor of Mathematics

Keywords in this area
Singular spaces

Arlie O. Petters

Benjamin Powell Professor of Mathematics in Trinity College of Arts and Sciences

Keywords in this area
geometric lensing, black holes, singularities

Leslie Saper

Professor of Mathematics

Keywords in this area
locally symmetric spaces, intersection cohomology, topology of compactifications, K-theory

Mark A. Stern

Professor of Mathematics

Keywords in this area
Index theory