# Fundamental Mathematics

The three broad pillars of this section are algebra, geometry and analysis. We also explore their relation to logic, number theory and physics, with an eye for the history of the subject.

Interesting developments often occur at the borderline between these subdisciplines, and we use geometry as the ordering principle for our research.

Geometry is the theory of space, and the history of the subject shows that “space” is a multi-faceted concept; it refers equally well to the phase space of a classical mechanical system, the differential-geometrical space of general relativity or the operator-theoretic space of quantum mechanics from physics, as to the internet as a huge graph of connections, or to a big relational data base. In mathematics, we unearth, conceive and study different possible concepts of space, even where they at first seem unexpected.

Geometry has often preceded modern applications: for example, GPS-technology uses differential geometry and discrete geometric optimization; cryptographic protocols increasingly use ideas from arithmetic geometry, and search algorithms use linear spaces. Geometry has always been intricately linked with theoretical physics, but also the recent theoretical study of the structure of data involves more and more geometry.

## Our research areas

- Algebraic and arithmetic geometry, where algebra and geometry meet; including number theory, model theory, (higher) category theory, algebraic topology, and mathematical physics.
- Differential geometry and geometric analysis, where analysis and geometry meet; including symplectic and Poisson geometry, Lie theory, automorphic forms, integrable systems, noncommutative geometry, and mathematical physics.
- History of mathematics, including cultural and societal aspects.

## Our people

Our section has four KNAW-members and one KHMW-member. Since 2003, we have won 16 veni/Marie Curie grants, three vidi and two vici grants, and one ERC starting grant, with a total of more than 16 million euro in external grant income. Since 2006, we have appointed about 40 PhD students. At a national level, we coordinate the NWO-research cluster GQT “Geometry and Quantum Theory”. Within the Graduate School of Natural Sciences, we run an (NWO-sponsored) Graduate Programme, the Utrecht Geometry Centre.