Geometry and Topology
Created by Gauss to take account of the curvature of the Earth in surveys of large areas in Germany, differential geometry with its notion of curvature was extended to spaces of arbitrary dimension by Riemann, and found significant application in dimension 4 in Einstein's general relativity. It has also developed many connections with complex analysis, algebraic geometry, PDE, and other areas of mathematics. Members of our department study curvature and the Ricci tensor, geometry of symmetric spaces and compact 2-step nilmanifolds, rough metric tensors, Gromov-Hausdorff convergence, and connections with PDE.
Topology and Foliation Theory
A primary theme of topology is to associate algebraic structures to a topological space with the goal of determining if such structures allow one to decide whether two topological spaces are equivalent. One goal is to classify the topological structures of manifolds. Topologists in our department do research in characteristic classes, low dimensional topology, especially 3-manifolds, foliations of manifolds, connections with dynamical systems, and Lie groups.
This is the study of the images of smooth spaces under smooth maps whose derivatives are not everywhere injective, therefore giving rise to singular images. Precise studies of the nature of these singularities connect to topics such as the behavior of caustics of waves and catastrophes. Members of this group do research on the structure of singularities and stratified spaces. There is also have a joint project with members of the Computer Science Department, on connections with computer vision.
The geometrical side of algebraic geometry emphasizes complex varieties, the geometry and topology of their singular sets, and the influence of curvature, particularly the Ricci tensor. Members of our department do research on singularities of algebraic surfaces, curves on K3 surfaces, deformation theory, geometry of stratified sets, global structure of singularities, cohomology of moduli spaces, degeneracy loci, and quantum invariants.
Geometric methods in modern mathematical physics have evolved from the Kaluza-Klein unification of general relativity and electromagnetism, bringing in fiber bundles and connections to define Yang-Mills fields describing weak and strong interactions, and on to other geometrical structures. Members of this group do research in conformal field theory, and on compactifications of 10-dimensional string space to bundles whose fibers are Calabi-Yau manifolds.