Quantum phases and random surfaces

This project aims to advance our understanding of the mathematics underlying fundamental aspects of nature.

We will study two fascinating topics within statistical physics: phases and phase-transitions of quantum spin-systems, and random geometric structures inspired by quantum gravity.

Specifically, the project aims to (1) provide a mathematically precise description of a subtle type of phase-transition called BKT-transition in a quantum spin-system, (2) precisely describe the phases in a class of quantum spin-systems with large symmetry groups, and (3) establish the existence of new classes of random surfaces arising as scaling limits of random planar maps with causal features.The overall purpose of the project is to develop mathematical tools and techniques for analyzing complex systems.

The study of quantum spin-systems and random surfaces poses fascinating mathematical challenges that tie together various fields of mathematics, such as analysis, probability theory, and representation theory.

The project is significant because it aims to provide mathematically precise results in these areas, which can lead to new insights and techniques relevant in various fields of mathematics.Part of the funding applied for is intended to be used to support a PhD student, who would primarily work on part (2) above.

Another important aspect of the project will be collaboration with several researchers in the applicant’s network, located in Warwick UK, Vienna, Reykjavík and Paris.