Brown Applied Mathematics Pattern Theory and Vision Seminar
A number of scientific endeavors of current national and international interest involve populations with interacting
and/or interfering units. In these problems, a collection of partial measurements about patterns of interaction and
interference (e.g., social structure and familial relations) is available, in addition to the more traditional measurements
about unit-level outcomes and covariates. Formal statistical models for the analysis of this type of data have
emerged as a major topic of interest in diverse areas of study. Probability models on networks date back to 1959.
Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an
active community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late
1970s and 1980s, and the past decade has seen a burgeoning literature in statistical physics and computer science.
The growth of the World Wide Web and the emergence of online social networking websites such as Facebook and
LinkedIn, and a host of more specialized professional networking communities has intensified interest in the study of
networks, structured measurements and interference. In this talk, I will review a few ideas and open areas of
research that are central to this burgeoning literature, placing emphasis on inference and other core statistical issues.
Topics include elements of sampling and inference from non-ignorable (network sampling) designs, and
semi-parametric modeling, with hints to the applications to social, biological and information networks that motivate
these statistical problems.
[pizza will be provided]
Special PDE Seminar
Abstract: The Einstein-nonlinear electromagnetic system is a coupling of the Einstein field equations of general relativity to a model of nonlinear electromagnetic fields. In this talk, I will discuss the family of covariant electromagnetic models that satisfy the following criteria: i) they are derivable from a sufficiently regular Lagrangian, ii) they reduce to the linear Maxwell model in the weak-field limit, and iii) their corresponding energy-momentum tensors satisfy the dominant energy condition. I will mention several specific electromagnetic models that are of interest to researchers working in the foundations of physics and in string theory. I will then discuss my main result, which is a proof of the global nonlinear stability of the 1 + 3-dimensional Minkowski spacetime solution to the coupled system. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in a wave coordinate gauge. The analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows one to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. The analysis of the electromagnetic fields, which satisfy quasilinear first-order equations, is based on an extension of a geometric energy-method framework developed by Christodoulou, together with a collection of pointwise decay estimates for the Faraday tensor that I develop. Throughout the analysis, I work directly with the electromagnetic fields, thus avoiding the introduction of electromagnetic potentials.
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