EMSW21-RTG: Interdisciplinary Training in the Applications of Probability

该项目将支持对各种领域中出现的概率问题的研究。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。在金融领域，理解市场泡沫、波动期权和与次级抵押贷款相关的信用风险需要来自控制理论、后向随机微分方程和扩大过滤的复杂数学思想。对金融市场、保险风险、通信和计算机网络的研究导致了表现出长期依赖和重尾的随机模型。对这些过程及其呈现的新现象的分析，涉及到与经典的独立随机变量理论截然不同的技术，利用了包括遍历理论在内的各种来源的思想。此外，概率与代数、几何和分析（包括使用分析技术来研究马尔可夫链的收敛率）中产生的各种主题之间存在着丰富的相互作用。概率论诞生于对赌博游戏的分析，现在在生物学、物理学、经济学、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。这一建议的主要目的是培养更多的概率研究人员，以解决由其应用产生的许多重要问题。康奈尔大学是开展这些活动的理想场所，因为它在概率论研究和应用数学领域的跨学科工作方面有着悠久的传统。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，有许多重要的应用。在研究生阶段，增加奖学金支持将加强博士研究生的培训。博士后职位将帮助新毕业生拓展视野，而热门话题会议和年度暑期学校将成为扩大研究人员（年轻和年老）教育的国家资源。在所有这些活动中，重点将放在发展理论来处理应用中出现的问题。

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.