Topology
Topology is the mathematical study of those properties that
are preserved through continuous deformations of objects.
A circle is topologically equivalent to an ellipse, a
sphere is equivalent to a cube, and a coffee cup to a
donut. Topology began with the study of curves, surfaces,
and other objects in the plane and three-space. It can be
used to abstract the inherent connectivity of objects
while ignoring their detailed form.
The "objects" of topology are formally defined as
topological spaces. If two spaces have the same
topological properties, they are said to be homeomorphic;
if one can be continuously deformed into the other,
they are said to be homotopy equivalent.
The basic language of
topology is known as point-set topology. Algebraic topology is the study of algebraic objects attached
to topological spaces. The algebraic invariants reflect some of
the topological structure of the spaces. The algebraic tools
include homology groups, cohomology rings, homotopy groups,
derived functors, and spectral sequences.
Differential topology is the
field dealing with differentiable functions on differentiable manifolds,
vector fields, and foliations. It arises naturally from the study of
differential equations, and is closely related to differential geometry.
These fields have many applications in physics, notably in the
theory of relativity.
Geometric topology is the study of manifolds and
their embeddings, with representative topics being knot theory
and braid groups. It has come over time to be almost synonymous
with low-dimensional topology, concerning in particular objects
of two, three, or four dimensions.
David Massey studies the local topology of singular spaces, especially
complex analytic singular spaces. Many of his results revolve around finding
effectively calculable algebraic data that describe or control the
topology of the singularities in the space.
Egon Schulte studies discrete structures in geometry and combinatorics, such as polytopes, maps on surfaces, tessellations on manifolds, complexes, and graphs. The classification of regular abstract polytopes by global or local topological type is a prominent part of his Abstract Regular Polytopes research monograph with Peter McMullen.
Alex Suciu's research interests are in topology, and how it relates to algebra, geometry, and combinatorics. He studies cohomology jumping loci, hyperplane arrangements, and polyhedral products, as well as various problems concerning the topology and geometry of knots, links, and manifolds, and the homology and lower central series of discrete groups.
Last modified October 29, 2009
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