EMSW21-RTG: Interdisciplinary Training in the Applications of Probability

该项目将支持对来自广泛领域的概率问题的研究。遗传学的主题包括调控序列进化、基因复制和癌症基因组计划产生的数据分析。在金融学中，要理解市场泡沫、回旋期权以及与次级抵押贷款相关的信用风险，需要来自控制论、倒向随机微分方程式和过滤扩大的复杂数学思想。对金融市场、保险风险以及通信和计算机网络的研究导致了随机模型的出现，这些模型表现出长期相关性和重尾。对这些过程及其呈现的新现象的分析，涉及到与独立随机变量的经典理论有很大不同的技术，利用了包括遍历理论在内的各种来源的想法。此外，概率与代数、几何和分析中产生的各种主题之间存在着丰富的相互作用，包括使用分析技术来研究马尔可夫链的收敛速度。概率理论诞生于赌博博弈的分析，现在在生物、物理、经济、金融、保险、通信网络以及数学中的许多主题中发挥着重要作用。这项提议的主要目的是培训更多的概率研究人员，以解决其应用中出现的许多重要问题。康奈尔大学是开展这些活动的理想场所，因为在概率研究和应用数学的跨学科工作方面有着深厚的传统。对暑期研究项目的支持将向本科生展示，概率是一个有趣的研究领域，具有许多重要的应用。在研究生层面，增加奖学金支持将加强我们博士生的培训。博士后职位将帮助应届毕业生开阔眼界，而热点话题会议和年度暑期学校将成为拓宽研究人员(无论老少)教育的全国性资源。在所有这些活动中，重点将是开发理论来处理应用程序中出现的问题。

This project will support research on probability problems that arise from a wide variety of fields. Topics from genetics include regulatory sequence evolution, gene duplication, and the analysis of data generated by cancer genome projects. In finance, understanding bubbles in markets, swing options, and credit risk associated with subprime mortgages require sophisticated mathematical ideas from control theory, backward stochastic differential equations, and enlargement of filtration. The study of financial markets, insurance risk, and communication and computer networks lead to stochastic models that exhibit long range dependence and heavy tails. The analysis of these processes and the new phenomena they present, involves techniques much different from the classical theory of independent random variables, making use of ideas from a wide variety of sources including ergodic theory. In addition, there is a fertile interaction between probability and a variety of topics that arise from algebra, geometry, and analysis, including the use of analytic techniques to study convergence rates of Markov chains.Probability theory, born in the analysis of gambling games, now plays an important role in biology, physics, economics, finance, insurance, communication networks, and in many topics within mathematics. The main aim of this proposal is to train more researchers in probability to tackle the many important problems that arise from its applications. Cornell is an ideal place for these activities because of the strong traditions of research in probability and interdisciplinary work in applied mathematics. Support for summer research projects will show undergraduates that probability is an interesting research area with many important applications. At the graduate level, increased fellowship support will enhance the training of our Ph.D. students. Postdoctoral positions will help new graduates expand their horizons, while hot topics conferences and annual summer schools will be a national resource for broadening the education of researchers (young and old). In all of these activities, the focus will be on developing theory to treat problems that arise from applications.