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Matilde Marcolli's Math Page


Matilde Marcolli
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Courtesy Appointment
Ph.D., University of Chicago, USA, 1997

Detailed description of research

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.