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，这些问题的目标是在某种约束下最小化应用于一类对象的函数，识别极值对象，并研究极值的稳定性。相关的例子范围从连续世界中的一些（凸几何中的等周现象，泛函解析不等式）到离散世界中的一些（加性组合学中的结构现象），以及两者兼而有之（统计学中的最大熵问题，概率论中的极限定理）。所有这些问题类的自然语言是概率论，以及信息论（尽管不明显）。该项目将根据信息论不等式从这些领域中开发出极值问题的新公式，然后使用分析、概率论、凸几何、组合学和信息论等各种工具，在这些领域取得进展。所采用的视角的统一性质将使用一组公共工具来连接离散和连续的问题，并实现重要的交叉受精。此外，与统计决策理论相结合的一些信息理论不等式将应用于涉及工程、经济学和生物学中出现的多个参与者的新统计挑战（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在凸几何、算术组合和概率论这三个看似不相干的领域的挑战性数学问题上取得进展。除了这些领域在数学中的内在意义外，它们还有很多实际意义——凸几何在医学断层扫描中有应用，算术组合在计算机科学中有应用，概率作为统计推断的基础无处不在。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，为学生的动机创造了极好的机会。新开发的关于极端问题的信息理论方法的课程和资源网站将利用这些机会。研究者将通过调查文章传播关键发现，组织跨学科研讨会，并通过非学术性的公开讲座来传达研究的兴奋，以吸引有前途的学生进入数学科学。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提出定价机制（例如，通过搜索引擎为广告定价），并改善传感器网络收集和使用各种应用（例如，用于灾难恢复协调或智能幼儿园）数据的方式。

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).