Random Structures and Integrable Systems: Analysis and Applications

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The overarching goal of the projects supported by this award is to develop mathematical tools that can describe the structure and properties of networks and surfaces that model random evolution in random environments. Examples of this are contact processes which model infections that spread by contact with an infected neighbor. Other example are voter models which describe the spread of opinions where an individual's opinions are affected by his or her neighbor's opinions. Other motivations arise from extending methods of statistical mechanics classically done on regular lattices to the setting of random lattices which we may think of as random graphs or networks on a surface. In this work the PI will provide an improved understanding of such models in terms of random metrics. One may imagine these metrics are determined by patterns of disease propagation or social contact. Interestingly the general study of random metrics arose from physical investigations of two-dimensional (2D) quantum gravity.The central problem of 2D quantum gravity, in mathematical terms, is to rigorously construct a measure on the space of metrics on a Riemann surface. This is a long-standing problem of both geometric and physical relevance. But aside from, and perhaps even beyond this, it serves as a rich source of novel problems and ideas that are at the interface between random structures and integrable systems. The focus of this proposal is on emerging crosscurrents of research between probability theory, with an emphasis on random geometry and combinatorics, and integrable systems theory with an emphasis on classical analysis, complex function theory, dynamical systems and conservative partial differential equations.