REU Site: Mathematics with Applications to Medical Sciences, Biophysics, and Inverse Problems at IUPUI

印第安纳大学-普渡大学印第安纳波利斯分校（IUPUI）的本科生研究经验项目（REU）将为来自美国各地的8名本科生提供在医学科学、生物物理学和反问题方面进行应用数学研究的机会。学生们将在夏季花八周的时间与IUPUI数学科学系的教师导师一起研究四个主要项目之一：(i)模拟主动脉闭塞后腿部血流和压力的再分布，（ii）调查滥用药物对人脑的影响，（iii）分析生物分子的扩散动力学，（iv）揭示解决相位恢复、引力场和声波逆问题的方法。数学研究为学生提供了重要的技能，他们可以用来分析和解决所有学科和环境中的问题。该项目还将训练学生回答血管外科、急诊医学、生物物理学和天文学方面的具体问题。随着世界继续以令人难以置信的速度取得技术进步，对科学家和工程师的需求也在增加。让学生接触应用数学研究将鼓励这些学生中的许多人在stem相关领域从事职业，并最终为美国提供拥有深厚现代科学知识并具备在公共和私营部门对科学和技术产生影响的能力的个人。该奖项由数学和物理科学局（MPS）的数学科学部（DMS）和生物科学局（BIO）的生物基础设施部（DBI）支持。REU项目解决应用数学中的新问题和开放问题，学生在暑期项目中取得的进展将推动这些研究领域的发展。参加第一个REU项目的学生将与数学家和血管生理学家密切合作，根据在小鼠后肢进行的实验，开发一个模型，模拟血管数量或直径增加对主要动脉闭塞后血流的影响，以确定血管生成或动脉生成治疗是否提供最大的益处。第二个项目涉及开发人类大脑对药物反应的数学模型。该模型将以电生理实验为基础，对多巴胺的输入、内在特性和释放进行建模。在第三个项目中，学生将探索扩散动力学的框架，其动机是观察到催化反应释放的热量通过引起酶的突然质心平移来增强酶的扩散系数，以及各种类型和大小的颗粒和生物分子在细胞内以活跃的新陈代谢扩散得更快。学生们将探索一个基于规则的Fokker-Planck方程（描述由力和随机布朗运动驱动的扩散）的新框架，以确定惯性效应对蛋白质的重要性。从事第四个项目的学生将学习各种基于代数和解析算法的反问题和解。学生将解决在实验中使用衍射来确定内部结构时出现的相位恢复问题，以及引力场和声波的反问题。

This program of Research Experiences for Undergraduates (REU) at Indiana University-Purdue University Indianapolis (IUPUI) will provide eight undergraduate students from across the United States with the opportunity to conduct applied mathematics research in the medical sciences, biophysics, and inverse problems. The students will spend eight weeks during the summer working with faculty mentors from the IUPUI Department of Mathematical Sciences on one of four main projects: (i) modeling the redistribution of blood flow and pressure in the leg following a major arterial occlusion, (ii) investigating the influence of abused drugs on the human brain, (iii) analyzing the diffusion dynamics of biomolecules, and (iv) uncovering methods to solve inverse problems for phase retrieval, gravitational fields, and acoustic waves. Research in mathematics provides students with important skills that they can use to analyze and solve problems in all disciplines and environments. The projects in this program will also train students to answer specific questions in vascular surgery, emergency medicine, biophysics, and astronomy. As the world continues to make technological advances at an incredibly rapid pace, there is an increased need for scientists and engineers. Exposing the students to applied mathematics research will encourage many of these students to pursue careers in STEM-related fields, and will ultimately provide the United States with individuals who possess a deep knowledge of modern science and who are well-equipped to have an impact on science and technology in the public and private sectors. The award is supported by the Division of Mathematical Sciences (DMS) in the Directorate for Mathematical and Physical Sciences (MPS) and the Division of Biological Infrastructure (DBI) in the Directorate for Biological Sciences (BIO).The REU projects address new and open problems in applied mathematics, and the progress that the students make on the summer projects will advance these research areas. Students participating in the first REU project will work closely with a mathematician and vascular physiologist to develop a model, based on experiments conducted in the mouse hind limb, to simulate the effects of increased vascular number or diameter on blood flow following a major arterial occlusion to determine whether angiogenic or arteriogenic therapies provide the maximum benefit. The second project involves the development of a mathematical model for the response of the human brain to drugs. The model will be based on electrophysiological experiments, and the inputs, intrinsic properties, and release of dopamine will be modeled. In the third project, students will explore a framework for diffusion dynamics motivated by the observations that the heat released from a catalytic reaction enhances an enzyme's diffusion coefficient by causing a sudden center of mass translation of the enzyme, and that particles and biomolecules of various types and sizes diffuse faster inside cells with an active metabolism. The students will explore a novel framework based on the regular Fokker-Planck equation (which describes diffusion driven by forces and random Brownian motion) to determine the importance of inertial effects for a protein. Students working on the fourth project will be studying a variety of inverse problems and solutions based on algebraic and analytic algorithms. The students will tackle problems of phase retrieval that arise in the experimental use of diffraction to determine internal structure, as well as inverse problems for gravitational fields and acoustical waves.