Brown University Center for Statistical Sciences Seminar
Abstract: The main focus of this talk is on the conceptual and methodological contributions of the social network community going back over eighty years. The field is, and has been, broadly multidisciplinary with significant contributions from the social, natural and mathematical sciences. This has lead to a plethora of terminology, and network conceptualizations commensurate with the varied objectives of network analysis. As a primary focus of the social sciences has been the representation of social relations with the objective of understanding social structure, social scientists have been central to this development. Exponential family random graph models attempt to represent the complex dependencies in networks in a parsimonious, tractable and interpretable way. A major barrier to the application of such models has been lack of understanding of model behavior and a sound statistical theory to evaluate model fit. This problem has at least three aspects: the specification of realistic models; the algorithmic difficulties of the inferential methods; and the assessment of the degree to which the network structure produced by the models matches that of the data. In this talk I will show how insight can be gained by considering model degeneracy and inferential degeneracy for commonly used estimators. I will illustrate these methods using the "statnet" software suite (http://statnetproject.org).
Probability Seminar
Abstract: The Maslov idempotent calculus provides a framework in which various asymptotic problems can be considered, including large deviations for stochastic processes. The asymptotic limit is typically described through a deterministic optimization problem. However, the limit still retains a kind of "max-plus" probabilistic interpretation, in the Maslov setting. Max-plus stochastic control problems with either terminal cost or max-plus additive cost criteria are considered. For terminal cost criterion, dynamic programming leads to an Isaacs PDE for an associated differential game. For max-plus additive running cost, dynamic programming leads to a variational inequality. An illustrative example from mathematical finance is given.
***SPECIAL*** Applied Math Colloquium
Center for Fluid Mechanics Seminar
Abstract: Concentrated suspensions of particles have many applications in engineering and nature, ranging from the composite material forming the fuel of the booster engines of the space shuttle to the slurry of blood cells flowing through our arteries. Despite this, it is only in recent years that the mechanisms leading to non-uniformities in concentration distributions have become fully understood. In this talk we will explore the behavior of concentrated suspensions by focusing on the curious behavior of a suspension flowing behind an advancing meniscus. Through experiment and theory we show how the nature of particle interactions in a sheared suspension lead to asymmetric local structure, non-Newtonian rheology, particle migration, convective instabilities, and macroscopic concentration non-uniformities. Examples of such behavior will be demonstrated.
Center for Computational Molecular Biology Seminar
Abstract: Apart from a relatively small number of variants the genomes of two individuals are identical. These small differences explain much the human diversity. The most common form of variation between individuals are single nucleotide polymorphisms (SNPs), representing a single letter change in an otherwise conserved sequence. In the past couple of years, tremendous progress has been made in identifying the genetic causes of a number of disease and phenotypic traits using arrays, such as those made by Illumina, that simultaneously assay a large number of SNPs. Copy number variations (CNVs) occur when a segment of the genome is either copied or deleted so that individuals have different number of copies of that variant. Although copy number variations are much less common than SNPs they have the potential to have a much greater impact on an individual, since a functional element of the genome may be missing or occur more often than desired. We use the Illumina arrays to detect copy number variations and designed an array of SNP and univariant probes in the genome. We have assayed a large number of individuals for these variations. In this talk we consider the problem of translating these assay results into a determination of the number of copies an individual has of a copy number variation.
Department of Mathematics Colloquium
Probability Seminar
Abstract: We consider a continuous-time model of financial market with proportional transaction costs. Our result is a dual description of the set of initial endowments of self-financing portfolios super-replicating American-type contingent claim. As it was observed by Chalasani and Jha, already in the simplest discrete-time models consistent price systems form a class which is too narrow to evaluate American claims correctly. The phenomenon appears because one cannot prohibit the option buyer to toss a coin and take a decision, to exercise or not, in dependence of the outcome. A financial intuition suggests that the expected ''value" of an American claim is an expectation of the weighted average of ''values" of assets obtained by the option holder for a variety of exercise dates. This expected ''value" should dominate the ''value" of the initial endowment. The main question is what is the class of price systems which should be involved to calculate ''values" to be compared. We show that a suitable class is the class of price systems for which the expected weighted average of future prices knowing the past is again a price system.
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Applied Mathematics Colloquium
Abstract: We illustrate the use of measure-valued processes in the study of stochastic networks via the specific example of a many-server queue. Many-server queueing models arise in a variety of applications, including computer data networks and call centers, and are typically much harder to analyze than single-server queues. We establish strong-law and central-limit approximations that are shown to be accurate in the limit as the number of servers goes to infinity. The characterization of the approximations entails the study of (distributional) partial differential equations and stochastic partial differential equations with boundary conditions. We also discuss the implications of our analysis for the design of multi-server systems.
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