Gauge Theory:
Gauge Theory first emerged as a physical theory modeling the electroweak and
strong interactions of elementary particles. More recently, a new
gauge theory was discovered in the context of the still mysterious
symmetries of string theory: Seiberg-Witten gauge theory is the
result of one of these string theoretic symmetries known as
S-duality. In collaboration with Dr. Bai-Ling Wang of the
University of Adelaide, we have constructed an invariant
of 3-manifold from the new Seiberg-Witten gauge theory
(equivariant Seiberg-Witten Floer homology) and studied its
fundamental properties. Our long term project is understanding
its behavior under surgery of 3-manifolds and how it is related
to other invariants.
Noncommutative geometry
Connes' theory of noncommutative geometry lies at the
crossroad of many different mathematical disciplines. Drawing from
the language and techniques of functional analysis (operator
algebras), it provides a framework for quantization of the
classical notions of calculus and Riemannian geometry. My
research in noncommutative geometry is articulated in
different directions.
Fractional Quantum Hall Effect:
The Quantum Hall Effect is one of the mysteries of the
quantum world. In recent years a successful model for the Integer Quantum
Hall Effect was developed using Connes' non-commutative
geometry. However, this seemed unsuitable to accommodate the
more subtle Fractional Quantum Hall Effect,
because it is based on a single electron model, whereas the Fractional
Effect depends on the interaction of many electrons. In a
collaboration with Prof. Varghese Mathai of the University of Adelaide,
we observed that simulating the interaction of electrons by altering
the geometry to the negatively curved hyperbolic geometry it is
possible to accommodate fractions in the non-commutative geometry
model. The mathematics underlying the origin of these fractions is
related to orbifold singularities.
Noncommutative Modular Curves:
In collaboration with Prof. Yuri I. Manin (Max Planck
Institute) we showed how the algebro-geometric compactification
of modular curves by cusp points can be replaced by a
compactification by a noncommutative space which encodes
basic arithmetic information like the modular symbols and
the modular complex. This noncommutative boundary
is related to the dynamical system given by the Gauss
map on the continued fraction expansion. This suggests
other forms of noncommutative compactifications
of classical algebro geometric moduli spaces.
Spectral Triples and Arithmetic Geometry:
A spectral triple is the notion that in Connes' noncommutative
geometry replaces and generalizes that of a Riemannian (spin) manifold.
It is expected that several nontrivial constructions of spectral
triples should arise from number theory and arithmetic geometry.
In a collaboration with Prof. Caterina Consani
(University of Toronto), we showed that spectral triples provide
a natural framework where two preexisting descriptions of
arithmetic geometry at the archimedean places of an
Arakelov surface can be unified. The theories considered
are the cohomological theory of Deninger and Manin's model
of Arakelov geometry at the infinite places as 3-dimensional
hyperbolic geometry. Our construction is based on the
arithmetic theory of Hodge-Lefschetz modules and on a
dynamical system associated to the action of a Schottky
group on its limit set. |