Research and Education in Physical Mathematics

这个项目解决了从流体力学到生物学的几个当前科学和技术上的重要问题。具体地说，第一个项目研究流体湍流背后的与时间相关的事件，这些事件太快了，无论是在实验还是理论上都无法解决。这些事件支撑了流体湍流中的能量级联，并具有巨大的技术重要性。求解它们是流体力学中一个长期存在的问题，这与欧拉方程的正则性问题密切相关。第二个主题是对鼻腔形状的数学和计算研究，以及它如何影响鼻部的气流和功能。人们早就知道，鼻腔的形状在进化过程中发生了戏剧性的变化，从具有复杂迷宫般的内腔的狗到具有短内腔的人类。该项目旨在开发一个数学框架，以了解这些变化的起源。第三个项目旨在开发新的数学方法来辅助药物发现。目前还没有有效的方法来搜索小分子的空间来发现与药物靶标相关的配体。这样的搜索必须既找到与相关蛋白质靶标结合的分子，又不与非预期靶标结合。高通量筛选的进步已经为蛋白质和小分子之间的相互作用产生了巨大的数据集。PI和他的合作者已经证明，一种基于随机矩阵理论的将配体与蛋白质靶标关联的方法比任何其他已发表的方法都具有更高的准确性，本项目旨在进一步开发这种方法，以影响药物发现流水线。更广泛的影响集中在研究生、本科生和博士后的人才培养，以及通过烹饪教授科学和数学的教材的开发。科学和烹饪的教学计划已经通过一个在线课程达到了近25万人。流体湍流背后的事件的研究将集中在两个反平行涡流细丝的碰撞上。PI和他的合作者最近发表了一项分析，表明这样的碰撞可能会导致规模越来越小的结构级联，这是由于渐近控制碰撞的Biot Savart方程的相似解的崩溃。这些解决方案将在可实验实现的两个涡环碰撞的问题中进行计算和实验测试。研究小组将对小尺度结构的发展进行数值跟踪，将它们与平行实验进行比较，并开发出流体湍流中最小尺度的数学描述。鼻腔流体力学的研究始于不同生物鼻子的CT图像，以及用于求解流场以产生流动的CFD程序。然后，研究小组将开发一种通过曲折空腔的流动的近似解析描述，以便理解这些结果。将在这些解的基础上制定比例定律，以得出设计原则的数学描述。蛋白质配基结合的研究围绕着随机矩阵理论的马尔琴科·帕斯图尔分布的BBP阈值的偏差。通过将更多关于配体化学的信息纳入到这一描述中，这项研究有可能提高对药物发现流水线中实用的预测能力。

This project addresses several problems of current scientific and technological importance, from fluid mechanics to biology. Specifically, the first project studies the time dependent events that underlie fluid turbulence, events that are too fast to have ever been resolved either in experiments or theory. These events underpin the energy cascade in fluid turbulence, and have immense technological importance. Resolving them is a long-standing problem in fluid mechanics, which is highly related to the question of regularity of the Euler equation. The second topic is a mathematical and computational study of the shape of the nasal cavity, and how it affects the airflow and functionality of the nose. It has long been known that the shape of the nasal cavity changes dramatically across evolution, ranging from dogs with complex labyrinth like internal cavities, to humans who have short internal cavities. This project aims to develop a mathematical framework to understand how the origin of these changes. The third project aims to develop new mathematical methods for aiding drug discovery. Currently there is no efficient way to search the space of small molecules to discover ligands that are relevant for drug targets. Such a search must both find molecules that bind to relevant protein targets and don't bind to unintended targets. Advances in high throughput screening has generated enormous datasets for the interaction between proteins and small molecules. The PI and his collaborators have shown that a method for associating ligands with protein targets based on random matrix theory has higher accuracy than any other published methods, and this project aims to develop this method further to impact the drug discovery pipeline. The broader impact centers around both personnel development of graduate students, undergraduate students and postdocs, and the development of educational materials for teaching science and mathematics through cooking. The teaching initiatives on science and cooking have reached nearly 250,000 people through an online class.The study of the events underlying fluid turbulence will focus on the collision of two antiparallel vortex filaments. The PI and his collaborators recently published an analysis suggesting that such a collision could lead to a cascade of structures on ever smaller scales, due to the breakdown of a similarity solution of the Biot Savart equations that asymptotically governs the collision. These solutions will be tested both computationally and experimentally in the experimentally realizable problem of the collision of two vortex rings. The research team will numerically track the development of small scale structure, compare them to parallel experiments, and develop a mathematical description of the smallest scales in fluid turbulence. The study of the fluid mechanics of the nasal cavity begins with CT images of noses from different organisms, together with a CFD code for solving the flow field to generate the flows. The research team then will develop an approximate analytical description of the flow through a tortuous cavity that allows understanding of these results. Scaling laws will be developed based on these solutions to arrive at a mathematical description of the design principles. The study of protein ligand binding is centered around deviations from the BBP threshold of the Marcenko Pastur distribution of random matrix theory. By incorporating more information about the chemistry of ligands into this description the research has the potential to improve predictive power for utility in the drug discovery pipeline.