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.

Healthcare fragmentation is a main cause for delay in cancer diagnosis and treatment, contributing to high and steadily increasing mortality rates in Latin America(LA), particularly among disadvantaged populations. Building on Equity-LA I (GA223123) and II (GA305197), this research focuses on integrated care interventions, which have proven effective at improving early diagnosis of cancer, mostly in HIC, and are also promoted by national cancer plans in LA, but limitedly implemented or evaluated. The objective is to evaluate the contextual effectiveness of scaling-up a multicomponent integrated care intervention to improve early diagnosis of frequent cancers in healthcare networks of Chile, Colombia and Ecuador. Method: This participatory, interdisciplinary and mix-methods implementation research is two-pronged: a) a quasi-experimental design (controlled before and after) with an intervention and a control healthcare network; b) a case study design. Focussing on the most vulnerable socioeconomic population, it develops in four phases: 1) analysis of delays, related factors and contextual barriers to early diagnosis (base-line); 2) adaptation and scaling-up of the intervention (PC training, fast-track referral pathway and patient information, adapting available ICT tools) in real life; 3) intra-country evaluation of intervention; 4) cross-country analysis. ICT tools will be also adopted in research activities as needed in a Covid-19 on-going or post- pandemic context. Relevance: EquityCancer-LA contributes to H2020 call objectives advancing cancer control policies by generating: 1) robust evidence on contextual effectiveness and costs-effectiveness of an affordable, tailored intervention to reduce diagnostic delays; and a validated strategy for its large-scale implementation in LA and LMICs; 2) novel data on delays and key barriers and facilitators to early diagnosis and inequalities in access; 3) e-tools to improve clinical practice and research on early diagnosis.