Course Description: Developing models to solve 21st century problems in physical, natural, and social sciences requires both theoretical and empirical understanding. Sophisticated numerical algorithms for extracting important information from data and for running long term model simulations are also critical for effectively utilizing these models. In this course, students will learn fundamental concepts and cutting-edge methods in modeling and numerical computation through engagement with data-driven problems drawn from applications of great societal impact. In particular, by motivating the mathematical solvers using real world applications, and by involving research scientists from other disciplines, this course will give students a comprehensive experience in problem solving.
Topics will include image reconstruction for ultrasound, MRI, and radar, and game theory with applications to real-world cooperation problems. Instruction will be combined with individual hands-on research experiences and projects, and will include on-site visits to various campus research facilities, as well as guest lectures from the Department of Psychology, the Department of Biology, the Thayer School of Engineering, and the Geisel School of Medicine. Students will have additional opportunities to meet with relevant Dartmouth faculty to discuss their research and get feedback on their results.
In particular, this course has an REU component with support from NSF and Dartmouth College. There are 7 REU students enrolled from outside Dartmouth. All work will be in a collaborative environment with fellow participants, graduate students, and postdocs. Students will undertake research projects and present their findings at the end of the program.
Prerequisites: Math 22 and Math 23, or per instructor approval. Previous course work in linear algebra, probability, and ordinary differential equations is highly recommended, and course work in at least one of these topics is required. Some experience in a programming language such as MATLAB is also highly beneficial.
Textbooks: For game theory, we recommend: Nowak, M.A. (2006). Evolutionary Dynamics. Harvard University Press. For numerical methods, we recommend: O’Leary, D (2009) Scientific Computing with Case Studies. SIAM.
Grading: Grades in the class will be based on homework sets which will ensure mastery of modeling and computational skills and the successful completion of a research project, which will include an end of term presentation. Students may work together on the research project, but each student will be solely responsible for part of its completion and all must participate in the research project presentation.
Grading formula: (i) Attendance & Participation (10%) + (ii) Homework Problem Sets (40%) + (iii) Final Project Proposal (10%) + Final Project Report (30%) + Final Project Presentation (10%).
Arguably, machine learning is one of the most important developments in data analysis. With its ability to recognize nonlinear and high-order interactions among features, deep convolutional neural networks have led to breakthroughs in image processing, video, audio and computer vision, whereas recurrent nets shed light on sequential data such as text and speech. Machine learning based approaches are advantageous because of their ability to handle very large data sets and nonlinear relationships in physically derived descriptors. I will specifically review graph-based semi-supervised learning algorithms for image classification and segmentation. The involved mathematical issues and potential applications to other fields will be discussed.
| Week | Lecture |
|---|---|
| Week 1 | Mathematical Modeling: Introduction & Overview; Introduction to MATLAB |
| Week 2 | Numerical Techniques for Linear Systems |
| Week 3-4 | Real World Application -- Medical Imaging: MRI, Tomography and Ultrasound |
| Week 5 | Evolutionary Games & Replicator Equations |
| Week 6 | Social Dilemmas of Cooperation |
| Week 7 | Real world Applications: Climate Change Games, Vaccine Compliance, Antibiotic Resistance |
| Week 8 | Games in Structured populations |
| Week 9 | Finite Populations |
The Department of Mathematics at Dartmouth College enthusiastically welcomes the following 7 undergraduate students from outside Dartmouth to participate in this Research Experience for Undergraduates (REU) program for mathematical modeling in science and engineering. This program is sponsored in part by the National Science Foundation and Dartmouth College. REU students will receive room and board on the Dartmouth Campus along with a small stipend and travel allowance. The program starts on June 22, with dormitory move in on June 21. Participants must commit to being on campus for at least six weeks, through August 3, and may stay through August 17. Students are expected to participate each weekday (except for July 4). As part of the program, students will participate in this Math 76.1 course team-taught by Professors Feng Fu and Anne Gelb.
| Name | Affiliation |
|---|---|
| Imani Carson | Spelman College |
| Olivia Conway | University of Oklahoma |
| Alexander Ginsberg | Michigan State University |
| Xiaoguang Huo | Cornell University |
| Haoran Liu | Arizona State University |
| Jingjong Tian | New York University |
| Yijia Zhang | Case Western Reserve University |
Approximately 5 weeks are given to complete the project. The instructors will suggest project ideas from the very beginning of the term, but you are allowed to propose your own, which has to be approved by the instructors in the fourth week at the latest. Each project presentation is limited to 15 minutes and preferably in the style of TED talks.
Course projects are listed by student in alphabetical order. We will update as needed.
Below are some papers and some references that may help you with ideas for projects. Professor Gelb also has some ideas on expanding the research in these papers. (You will note that the papers often don't correspond directly to the research topic. That is because we are interested in expanding the ideas from these papers to new applications!) The referenced papers are all available for download through the Dartmouth library system.