The main goal of this project is to understand the geometry of the deeply influential topological phase transitions which were discovered in the 70's by Berezinskii, Kosterlitz and Thouless. The archetypal example of such phase transitions arises in the 2d XY model in which topological defects, called vortices, behave very differently at small and high temperature. The mathematical understanding of this rich phenomenon goes back to the work of Fröhlich and Spencer in the 80's and involves the 2d Coulomb gas. This project is aimed at analyzing this phase transition through the prism of random fractal geometry by associating natural percolating sets to the XY model whose behavior will depend crucially on the temperature. One constant source of inspiration will be the deep geometric content and powerful probabilistic methods gathered over the last 20 years for celebrated discrete symmetry models such as 2d critical Ising or percolation. New tools will be brought in, among which the recent works of the PI with Sepúlveda which analyze the 2d Coulomb gas and make connections with Bayesian statistics. Since the early days of topological phase transitions, topological defects have been found to arise also in some discrete symmetry spin systems as well as in Abelian lattice gauge theory in 4d. This project will explore the geometry of these by making several novel and fruitful connections with the dimer and Ising models. The new connections made with statistical reconstruction and Bayesian statistics will give access to the even more fascinating and least understood world of spin systems with non-Abelian (gauge-)symmetry. Finally, we shall investigate the mechanisms which relate the microscopic background noise with the large scale structures it induces in the contexts of Quantum Field Theory and KPZ fixed point. The impact of this project will go well beyond the current understanding of topological phase transitions in a wide variety of settings where they arise.

The current Copernicus snow water equivalent (SWE) service lacks coverage in mountains and the southern hemisphere. The SNOWCOP addresses this gap by proposing an innovative re-analysis approach that assimilate the data from the Copernicus program into a physically based snow model. SNOWCOP will produce high-resolution maps of SWE and ice melt rates. The maps, which are produced for the last 20+ years over the extra-tropical Andes, will have a spatial resolution of 50 m and a daily temporal resolution. SNOWCOP will leverage on the Copernicus Data Space Ecosystem (CDSE) platform to perform the processing directly where the data is stored. SNOWCOP commits to open science making all results and code available. SNOWCOP will push the use of EGNSS technology to power snow station able to measure simultaneously SWE and liquid water content (LWC). The overall aim of SNOWCOP is to improve the quantification of meltwater from snow and ice in the Andes by unlocking the full potential of EU Copernicus Data and Infrastructure through four specific objectives: -Develop and evaluate new SWE and ice melt reanalysis datasets. -Promote the use of CDSE to foster innovation and excellence in the development of Copernicus solutions. -Raise awareness of Copernicus and EGNSS technologies among water management stakeholders. -Attract investment for developing innovative water management solutions. By achieving these objectives SNOWCOP is expected to open an innovative window for stakeholders, showcasing meltwater resources in the Andes and beyond with unprecedented spatial and temporal detail. By harnessing the power of Copernicus data and infrastructure, and developing innovative processing tools, the project unlocks the accessibility of such information to all the interested users. This empowers stakeholders to make informed decisions about water resource management, open new marketing opportunities for SME and significantly impacting human well-being in mountainous regions.