Recently, spectacular progress in the field of conformally invariant processes has been achieved, in particular with the introduction of the SLE processes. Thanks to these advances, it appears now possible to understand at an unprecedented level of resolution a wide class of models, and to construct their universal limits. The Ising model, which is probably the most studied model for phase transitions, has seen recent breakthroughs, and its further investigation seems very promising: it is one of the models that offer the richest behaviors and is among the best understood from a mathematical point of view. A unified picture, describing the limit of this model as random curves and fields, capturing subtle geometry and boundary conditions effects, seems to be rigorously obtainable. This picture will also deepen the comprehension of other models and offer many applications.

Phase transitions are abrupt changes in the nature of systems: vapor that suddently condensates into water, metals that gain supraconductivity, social networks whose activity explode once critical mass has been reached. A major question is to understand how such global-scale phase transitions occur in large systems. We plan to develop tools and techniques to study models where random macroscopic geometries arise. Such models have found applications for instance in chemistry, image processing, ecology, economics or machine learning. Thanks to new ideas and methods introduced in the recent years, it now appears possible to give a more complete description of the phase transition of such models, which will hopefully become useful tools for both theoretical and applied researchers.

Project Report

The proposal suggested to tackle about 30 problems in the field of statistical mechanics, conformal invariance and field theory. The goal was to study new structures that arise in lattice models, in particular the Ising model, and to improve the mathematical development of the theory. In particular, it was asked how some symmetries and ideas coming from quantum physics can be rigorously implemented to study lattice models at thephase transition point. Such questions are useful to understand complex problems and to find an efficient language to describe them. Most of these 30 problems have been solved completely, in particular the most fundamental ones (for instance the conjectures considering the scaling limit of the spin correlations have been solved completely). This project has also given new insights and surprising results. In particular, interesting connections between various mathematical objects arising at the small scale and the large scale have been found, which were not expected. This project has resulted in several publications, in particular some published in the Journal of the American Mathematical Society, Acta Mathematica and the Annals of Mathematics. The PI has been awarded the 2014 Blavatnik Award for Young Scientists (physical sciences category) for his research, which was mostly supported by this NSF award. Concerning the broader impacts, the PI has given a few dozen talks at seminars and international conferences and organized a summer research project for undergraduates.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Tomek Bartoszynski
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Columbia University
New York
United States
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