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Algebraic Geometry and Mathematical Physics

This line covers various topics in the field of Algebraic Geometry and its interactions with Differential Geometry, Symplectic Geometry, Topology, Mathematical Physics and Number Theory. The institute has world leading experts working on themes such as the study of moduli spaces parametrizing geometric structures of various kinds, and the study of singularities. These are all topics of much international activity in mathematical research.

 

Moduli Spaces and Geometric Structures

Moduli spaces are geometric versions of parameter spaces. That is, they are geometric spaces which parametrise something --- each point represents one of the objects being parametrised, such as the solution of a particular equation, or a geometric structure on some other object. In the language of physics, a moduli space is a model of the degrees of freedom of some system of equations.

Moduli spaces of holomorphic vectors bundles over Riemann surfaces have been studied for more than 50 years, specially since the fundamental work of Mumford on Geometric Invariant Theory, and Narasimhan and Seshadri. In the last three decades, very much through the work of Atiyah--Bott, Donaldson, Hitchin and others, there has been a lot of work in the study of the relations between moduli spaces of vector bundles, principal bundles and similar objects like Higgs bundles ---also on higher dimensional Kaehler manifolds---, with other moduli spaces, like the moduli spaces of solutions of non-linear partial differential equations emerging from gauge theory in Particle Physics, and moduli spaces of representations of the fundamental group of a Kaehler manifolds in reductive Lie groups. The study of these relations is the content of the Hitchin--Kobayashi correspondences and non-abelian Hodge theory, much studied by the members of the ICMAT team, who have made important contributions also in the study of the topology and geometry of these moduli spaces.

Singularity Theory

Singularity theory is at the crossroad of several mathematical disciplines, such as algebraic and analytic geometry, algebraic and geometric topology and differential geometry. Lately even connections with number theory have been found. It studied phnomena which undergo qualitative changes or bifurcations under small perturbation of the parameters of the problem in question. It finds aplycations in physcal sciences, robotics, criptography...

Singularity Theory research at ICMAT deals mainly with the algebro-geometric and topologycal aspects. It involves the study of algebraic and analytic singularities from different perspectives. Specific topics are the vanishing cohomology of singularities with its D-module and Hodge structure, the topology of the Milnor fibrarion, the embedded topology of the link, equisingularity, resolution in characterisric zero and p (including algorithmic aspects), hypersesolutions and descent categories, versal deformations and their base spaces with additional structure, invariants of surface singularities and of complex analytic spaces (including relations with gauge theoretic invariants), rational cuspidal curves, arrangements, arcs spaces and motivic integration, and relations with contact and symplectic geometry.