The Mathematical Institute consists of two interacting groups, one focusing on modelling, the other on fundamental aspects of mathematics. Within the national Sectorplan 2019, the activities of the research group Fundamental Mathematics fall under the scope of the focus area "Utrecht Geometry Center".
The research and teaching activities of the group cover a wide range of topics, including algebraic and differential geometry, algebraic topology, geometric analysis, history of mathematics, mathematical logic, and number theory, as well as connections to theoretical physics. Interesting developments often occur at the borderline between these subdisciplines, and we use geometry as the ordering principle for our research.
In geometry we study the notion of space, which we now understand to be a multi-faceted concept. The history of the subject shows that different incarnations of the idea of space appear naturally, often motivated by real-world examples. For instance, the word "space” may refer to the phase space of a classical mechanical system, the spacetime of general relativity, the operator-theoretic space of quantum mechanics, or the graphs that model a network like the Internet.
Geometry has preceded modern applications too. For example, GPS-technology uses differential geometry and discrete geometric optimization, cryptographic protocols increasingly use ideas from arithmetic geometry, and search algorithms use the theory of linear spaces.
These examples show that geometry has always been intricately linked with theoretical physics, but also highlight its more recent role in the theoretical study of big data.