CAREER: An integrated probabilistic approach to discrete and continuous extremal problems via information theory

数学中有大量的极值问题，其目标是在一定的约束条件下最小化应用于一类对象的泛函，识别极值对象，并研究极值的稳定性。 相关的例子从连续世界中的一些（凸几何中的等周现象，泛函分析不等式），到离散世界中的一些（加法组合学中的结构现象），以及两者中的一些（统计学中的最大熵问题，概率极限定理）。 所有这些问题类的自然语言是概率，尽管不是显而易见的，信息论。 该项目将从信息理论不等式的角度开发这些领域的极值问题的新公式，然后使用分析，概率，凸几何，组合学和信息论的各种工具来取得进展。 所采用的观点的统一性将使用一套共同的工具来弥合离散和连续的问题，并实现重要的交叉施肥。 此外，一些信息理论不等式的发展，结合统计决策理论，将被应用到新的统计挑战，涉及多个球员出现在工程，经济学和生物学（特别是，数据定价和分布式推理问题的理论基础）。 该项目将利用信息论思维，从凸几何、算术组合学和概率这三个看似不同的领域，对具有挑战性的数学问题进行研究。除了这些领域在数学中的内在意义外，它们具有很大的实际意义--凸几何在医学断层摄影术、计算机科学中的算术组合学中得到应用，而概率作为统计推断的基础是无处不在的。 拟议研究的可解释性和统一性，以及它使用的工具的多样性，创造了极好的机会 学生的动机。 新开发的课程和资源网站上的信息理论方法的极值问题将利用这些机会。 调查员将通过调查文章传播关键发现，组织跨学科研讨会，并通过非学术公开讲座传达研究的兴奋，以吸引有前途的学生到数学科学。 研究的应用部分也将产生广泛的影响，有助于数据收集者和供应商如何提出定价机制（例如，用于搜索引擎的广告定价），以及通过改进传感器网络收集和使用用于各种应用的数据的方式（例如， 用于灾难恢复协调或智能幼儿园）。

Mathematics abounds with extremal problems problems where the goal is to minimize some functional applied to a class of objects under some constraint, identify the extremal objects, and investigate the stability of extrema. Relevant examples range from some in the continuous world (isoperimetric phenomena in convex geometry, functional analytic inequalities), to some in thediscrete world (structural phenomena in additive combinatorics), and some in both (maximum entropy problems in statistics, limit theorems in probability). A natural language for all of these problem classes is probability, and, although not obvious, information theory. The project will develop new formulations of extremal problems from each of these fields in terms of information-theoretic inequalities, and then use a variety of tools from analysis, probability, convex geometry, combinatorics, and information theory, to make progress on them. The unifying nature of the perspective adopted will bridge discrete and continuous problems using a common set of tools, and enable significant cross-fertilization. Furthermore, some of the information-theoretic inequalities developed, combined with statistical decision theory, will be applied to novel statistical challenges involving multiple players that arise in engineering, economics, and biology (specifically, theoretical foundations for the problems of data pricing and distributed inference). The project will use information-theoretic thinking to make advances on challenging mathematical problems from the three seemingly disparate fields of convex geometry, arithmetic combinatorics, and probability.Apart from the intrinsic significance of these areas within mathematics, they have much practical significance - convex geometry finds applications in medical tomography, arithmetic combinatorics in computer science, and probability is ubiquitous as the foundation of statistical inference. The interpretability and unifying nature of the proposed research, and the diversity of tools it uses, create wonderful opportunities for student motivation. Newly developed courses and a resource website on information theoretic approaches to extremal problems will exploit these opportunities. The investigator will disseminate key findings through survey articles, organize an interdisciplinary workshop, and communicate the excitement of research through non-academic public lectures to attract promising students to the mathematical sciences. The applied component of the research would also have broad impact, by contributing to how data collectors and vendors come up with pricing mechanisms (e.g., for pricing of advertisements by search engines), and by improving the way networks of sensors collect and use data for various applications (e.g., for disaster recovery coordination or smart kindergartens).