Take a piece of paper and cut a slice out of it. It will be impossible to pull the opposite sides of the slice apart without stretching the paper. On the other hand, if you cut out a cross, its rather simple to bend each corner upwards. The piece of paper is a model of a space, and we investigate how the space can be bent inside another, or how small a size can the space be folded into. In the case of the paper with a slice, an obstruction is formed by the family of parallel lines on each side of the slice. The research group investigates such geometric problems for general spaces, where the role of lines is played by a positive modulus family of curves. Our goal is to develop a theory capable of working both at infinitesimally small scales, and quantitatively at larger scales. This requires new geometric tools. Spaces can also model data, and thus the methods are connected to algorithmic problems.

2023

2027