MATHEMATICS RESEARCH AREAS
Homotopy theory is concerned with properties of spaces that are
invariant under continuous deformation. A branch of algebraic topology,
it has found applications in fields as diverse as theoretical physics and
computer science. The following people at Galway work in this area.
JIM CRUICKSHANK is interested in the interaction between equivariant
structures and fibrewise structures in topology and geometry. One can reduce
many problems of equivariant topology to nonequivariant fibrewise problems.
Equivariant topology is topology with group
actions. Thus all the functions must satisfy certain symmetry constraints.
Fibrewise topology theory is topology done "over a base space". For example,
instead of vector spaces, one considers vector bundles, or instead of ordinary
cohomology one considers cohomology with twisted coefficients. It is sometimes
easier to deal with the fibrewise problem rather than the original equivariant
problem.
GRAHAM ELLIS is interested
in algebraic models of low-dimensional integral homotopy types, and
in using them to obtain explicit calculations (often with the aid of a
computer). In contrast to much of homotopy theory, the algebra of these
models can be very non-abelian. Some of Graham's work is joint with JOHN
BURNS, and he is currently working with EMIL SKOLDBERG on the design of
an homotopical algebra computer package.
EMIL SKOLDBERG is
interested in homological algebra. A particular interest is the use of
techniques such as perturbation theory, discrete Morse theory and Groebner
bases in constructing resolutions for commutative algebras, Lie algebras,
monoids etc. Although homological algebra was invented as a purely theoretical
tool, the ever increasing power of computers means that it is now practical
to implement the theory on computers. Emil is currently working on the
design and implementation of a "high performance" homotopical algebra library
for the Haskell functional programming language. The library is primarily
intended for non-abelian structures such as monoids, (infinite dimensional)
Lie algebras and posets. The "high performance" aspects are being tested
on the University's super-computer.
JAMES HARRIS is working on his PhD thesis. He is interested in polytopes
as a means of constructing explicit small resolutions and cohomology rings
for groups whose Cayley graph is "almost lattice-like". Coxeter groups,
Braid groups and finite p-groups seem to be examples of such groups.
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