Consider a hyperboloid, a plane and a round sphere as in the Figure 1. Although the geometry of these three objects is fundamentally different, we can transition from between them. Flattening the hyperboloid gives the plane. Alternatively, inflating a sphere to a very large radius makes it locally look like a plane. Both of these are examples of a geometric transition: a continuous path of geometric structures that changes the geometric type in the limit, producing a limit of geometries. Geometric transitions arise in the seminal program for 3-dimensional manifolds of Thurston, and deform general relativity into special relativity, or quantum mechanics into Newtonian mechanics.

Figure 1: Hyperboloid (t = −1), plane (t = 0), round sphere (t = 1)

Studying unexplored fundamental research trajectories

The above-mentioned geometries are modelled over our usual number system, the real numbers. An alternative is the p-adic numbers, where p is a prime number and the digits used are from 0 to p − 1. The p-adic numbers and p-adic geometries are at the core of 2018 Fields medallist Peter Scholze’s ground-breaking work, and are fundamental tools in number theory, algebraic geometry, representation theory, arithmetic dynamics, and cryptography. So far limits of geometries over the p-adic numbers have been studied very little, a direction of research that is not yet explored.

The objective of Corina-Gabriela Ciobotaru’s five year Villum Young Investigator project entitled ‘Limits of geometries over the p-adic numbers’ is to answer fundamental questions for many of the possible p-adic geometries: Given a family of geometries can one compute all its possible limits of geometries? Can one provide precise geometric transitions towards those limits?

“The Villum grant is an exciting opportunity for me to combine my unique background in p-adic metric geometry, buildings and Chabauty topology with my knowledge of symmetric spaces and limits of geometries over the reals. It enables me to conduct a different type of research on limits of geometries than has been done before,” said Corina-Gabriela Ciobotaru, AIAS Fellow and assistant professor at the Department of Mathematics at Aarhus University.

Ciobotaru’s scientific key idea is to use purely metric geometric techniques, an approach that has not been employed so far, and to provide a much deeper understanding of the p-adic geometries and the interconnection between them.

The results of the project will find applications to yet unsolved fundamental classification problems in the representation theory of Lie groups, i.e. continuous symmetries of physical systems, which are of great importance in modern Mathematics and Theoretical Physics.

Title of the project: Limits of geometries over the p-adic numbers
Project period: 01 August 2023 to 31 July 2028

About the Villum Young Investigator programme

The aim of the Villum Young Investigator to support early career researchers with ambitions of creating their own, independent research identity, and with the potential to significantly contribute to research in technical and natural sciences at a Danish research institution.
Read more about The Villum Young Investigator.
See all YIP grantees: https://veluxfoundations.dk/en/villum-young-investigators-2023

Contact

Corina-Gabriela Ciobotaru
Email: cociobotaru@aias.au.dk

Department of Mathematics and
Aarhus Institute of Advanced Studies (AIAS)
Aarhus University
Denmark