There is a very strong group of topologists at OSU. There is normally an active topology seminar with good graduate student participation. The topology research at OSU is concentrated in the area of low dimensional manifolds. There is good reason to study such objects, indeed, the universe we live in is normally thought of by physicists as a four dimensional manifold.
Three- and four-dimensional manifolds are unique in many surprising ways. The biggest recent surprise was the proof by Donaldson that four dimensional Euclidean space has more than one differentiable structure. In all other dimensions, there is only one such structure. This is particularly striking when one considers the fact that we live in a four dimensional manifold. One of the most difficult problems in topology is to determine when things are different and when they are the same. There are natural group invariants associated with manifolds. Perhaps the single most interesting problem in low dimensional topology is to determine to what extent group invariants characterize a manifold. It is for example not known if the three-sphere is characterized by its group invariants; this is the classical Poincaré conjecture, which is known to be true in every dimension except three. It was thought for many years that three-dimensional manifolds were generally lacking geometric structure. Mathematicians were wrong in a big way. In the seventies Thurston showed that most (perhaps all) compact three-manifolds are in fact very rich in geometric structure. Even more surprisingly the natural geometry for the most common three-manifolds is not Euclidean, but hyperbolic. (There are infinitely many lines through a given point parallel to a given line.) The existence of this structure has allowed problems that seemed untouchable ten years ago to be attacked and solved. The classical Smith conjecture that circles left fixed by group actions on the three-sphere must be unknotted is now the Smith theorem. It is a very exciting time in low dimensional topology. The Poincaré conjecture is no longer thought to be unreachable, and Thurston's structure theorems may even lead to a classification scheme for all compact three-manifolds. Mathematicians at OSU are heavily involved in studying these problems and will be a part of their final solutions.
Faculty:
Benny Evans: Low-dimensional topology; description of
three-dimensional manifolds.
William Jaco: Low-dimensional topology, involving 3-manifolds and
combinatorial group theory.
Robert Myers: The topology of 3-dimensional manifolds: structure
and classification of compact 3-manifolds, group actions on these spaces,
and their knot theory; generalizations to non-compact 3-manifolds. Covering
spaces of compact 3-manifolds. Also geometric group theory: the application
of methods from geometry, topology, and automata theory to problems in group
theory.
Weiping Li: Low-dimensional topology, involving 3-manifolds,
Casson invariants and Floer homology, using methods from gauge theory,
nonlinear analysis on infinite dimensional manifolds, symplectic topology
and dynamical systems.