Topology is concerned with the intrinsic properties of shapes of
spaces. One class of spaces which plays a central role in mathematics,
and whose topology is extensively studied, are the
n dimensional manifolds. These are spaces which locally look like Euclidean n-dimensional space.
Historically, topology has been a nexus point where algebraic
geometry, differential geometry and partial differential equations meet
and influence each other, influence topology, and are influencedby
topology. More recently, topology and differential geometry have
provided the language in which to formulate much of modern theoretical
high energy physics. This interaction has brought topology, and
mathematics more generally, a whole host of new questions and ideas.
Because of its central place in a broad spectrum of mathematics there
has always been a great deal of interaction between work in topology
and work in these neighboring disciplines. That is true of the topology
group at Columbia, which has enjoyed a close connection with the
algebraic geomety group, the geometric PDE group, and the mathematical
physics group at Columbia. In addition to close connection to the other
research groups, our topology group also enjoys close collaboration
with the symplectic geometers at Stony Brook and Courant, running a
thrice-per-semester joint symplectic geometry seminar.
Ironically, in topology, the case of manifolds of dimensions 3 and
4, the physical dimensions in which we live, has eluded undestanding
for the longest time. The case of manifolds of dimension
n=1 is straightforward, and the case where
n=2 was understood thoroughly in the 19
th
century. Moreover, intense activity in the 1960's (including the
pioneering work of Browder, Milnor, Novikov, and Smale) expresses the
topology of manifolds of dimension
n>4 in terms of an elaborate but purely algebraic description.
The study of manifolds of dimension
n=3 and
4 is quite different from the higher-dimensional cases; and, though both cases
n=3 and
4 are quite different in their overall character, both are generally referred to as
low-dimensional topology.
Low-dimensional topology is currently a very active part of
mathematics, benefiting greatly from its interactions with the fields
of partial differential equations, differential geometry, algebraic
geometry, modern physics, representation theory, number theory, and
algebra.
For the case of manifolds of dimension
n=3, a conjectural
classification picture emerged in the 1970's, thanks to the work of
William Thurston, in terms of symmetric geometries. Specifically,
Thurston conjectured that every three-manifold can be decomposed
canonically into pieces, each of which can be endowed with one of eight
possible geometries. Elements of this vast picture are presently
unfolding thanks to the Ricci Flow equations introduced by Richard
Hamilton, which have been used by Grigory Perelman to solve the
century-old Poincaré conjecture, and have also shed light on Thurston's
more general geometrization conjecture. The central role of Thurston's
conjecture in three-manifold topology has helped place hyperbolic
geometry, the richest of the eight geometries, into the research
forefront.
The case of manifolds of dimension
n=4 remains the most
elusive. In view of the foundational results of Freedman, understanding
manifolds up to their topological equivalence is a theory which is
similar in character to the higher-dimensional manifold theory.
However, the theory of differentiable four-manifolds is quite
different. The subject was fundamentally transformed by the pioneering
work of Simon Donaldson, who was studying moduli spaces of solutions to
certain partial differential equations which came from mathematical
physics. Studying algebro-topological properties of these moduli
spaces, Donaldson came up with very interesting smooth invariants for
four-manifolds which demonstrated the unique and elusive character of
smooth four-manifold topology. In the case where the underlying
manifold is Kähler, these moduli spaces also admit an interpretation in
terms of stable bundles, and hence shed light on the differential
topology of smooth algebraic surfaces. Since Donaldson's work, the
physicists Seiberg and Witten introduced another smooth invariant of
four-manifolds. Since then, the study of four-manifolds and their
invariants has undergone several further exciting developments, tying
them deeply with ideas from symplectic geometry and pseudo-holomorphic
curves, and hence forming further bridges with algebraic and symplectic
geometry, but also connecting them more closely with knot theory and
three-manifold topology.
Topology at Columbia University has enjoyed a long tradition.
Illustrious professors from the past include Samuel Eilenberg, who is
responsible for the foundations of algebraic topology, and Lipman Bers,
whose ideas in complex variables played an influential role in
Thurston's program for three-dimensional manifolds. The senior faculty
in the topology group at Columbia are Joan Birman (emerita), Troels
Jorgenson, Mikhail Khovanov, Dusa McDuff, John Morgan, Walter Neumann,
and Peter Ozsváth; junior faculty include Robert Lipshitz and Dylan
Thurston; and there is also a number of post-doctoral researchers and
visitors. The closest connections with the research interests other
mathematicians not strictly in the topology group, include David Bayer,
Robert Friedman, Brian Greene, Richard Hamilton, Melissa Liu, and
Michael Thaddeus. The topology group has a number of informal student
seminars, a regular
Topology seminar (which often meets twice on Fridays), and also a
gauge theory seminar which meets on Fridays.
