MATH 76.1: Data Driven Modeling with Real World Applications

Course Information:

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%).

Important Dates

Guest Lectures

  • Title: Machine learning analysis of images
  • Abstract:

    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.


Tentative lecture plan which may be subject to further changes.

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

REU Program

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

Homework Sets

    Homework sets will be handed out in class.

Course Projects and Presentation Schedule

Final Projects

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.

Project ideas:

Papers related to projects suggested by Professor Gelb

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.

Course Policies

Honor Principle

Students are encouraged to work together to understand course material. This includes helping each other by providing insight into homework problems. However, each student is responsible for his/her own assignment, and any homework problem solution that appears to result from a team effort will result in zero points awarded for all parties involved. It is also important to avoid plagiarism in your final project report, and to cite all work appropriately. When in doubt, please ask Professor Fu or Gelb what the proper protocol is. You should also refer to Academic Honor Principle.

Accessibility Policy

Students needing special accommodations are encouraged to make an office appointment with Professors Gelb and Fu prior to the end of the second week of the term. At this time, students should provide copies of disability registration forms, which list the particular accommodations recommended Student Accessibility Services within the Academic Skills Center. The Director of Student Accessibility is Ward Newmeyer. Office 205 Collis Center; Phone (603) 646-9900.

Student Religious Observances

Some students may wish to take part in religious observances that fall during this academic term. Should you have a religious observance that conflicts with your participation in the course, please come speak with your instructor before the end of the second week of the term to discuss appropriate accommodations.

Late Policy

Deadlines will be strictly enforced. In extreme cases late final project reports will be accepted with a penalty of 5% incurring each day. The maximum extension period underal all circumstances is 4 days. Students must request extensions for the final project with instructors one week prior to the due date. Students requesting special accommodations should inform the instructors well in advance so that the instructors will have sufficient time to work with Student Accessibility Services to ensure appropriate accommodation.