Course Description: Today’s world is awash with data. Data analysis has long been an important component in all scientific study, as it plays a role in a variety of applications, including signal processing, imaging, evolutionary dynamics, and forecasting of all kinds, just to name of few. Moreover, rational decision making depends critically on the correct analysis of available data. This is becoming ever more difficult as we are flooded with a myriad of information. While extracting important information to make good decisions is complicated enough when the collected data are assumed to be correct, the problem is much more difficult when the accuracy of data is compromised, either intentionally or not. Recent political events have prompted mathematicians and scientists to further understand the effects of “rogue’’ data, for the purposes of detecting suspicious behavior, understanding its propagation, and limiting its effects. Moreover, there have always been efforts to interfere with good data acquisition efforts, for example in jamming radar signals, and these efforts are becoming increasingly sophisticated. Finally, there are natural limitations to our data acquisition devices, for example in medical devices, that may lead to false diagnoses with sometimes serious repercussions.
Topics in Summer 2018 will include signal detection from corrupted data in radar and medical imaging, and evolutionary dynamics of misinformation. 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 mathematicians from outside institutions, various Dartmouth departments, and scientists from the Cold Regions Research and Engineering Laboratory (CRREL). Students will have additional opportunities to meet with relevant Dartmouth faculty to discuss their research and get feedback on their results.
The course is designed so that Dartmouth summer students have the opportunity to participate in the summer mathematics research experience program (REU) , which is supported by the National Science Foundation (NSF) , to enhance undergraduate research activities in STEM fields. This summer 9 students from outside Dartmouth are scheduled to participate in the REU. Dartmouth students enrolled in MATH 76 are encouraged to jointly collaborate with each other and the REU participants. Postdoctoral fellows and graduate students will be on hand to assist and collaborate in research projects. Students in MATH 76 will present their findings at the end of their term.
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; Vogel, C. (2002) Computational Methods for Inverse Problems, SIAM; and Hansen, et. al. (2006) Deblurring Images: Matrices, Spectra, and Filtering. The SIAM books are available as e-books through the Dartmouth library. Supplementary material and MATLAB code can be found here. The MATLAB code in this supplementary material may be used in some assignments. A good reference for MATLAB coding is D.J. Higham and N.J. Higham (2005) MATLAB Guide, Second Edition. SIAM. There are also many other resources online as well as the MATLAB tutorial available through MATLAB.
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. We strongly recommend that you start programming assignments early.
Grading formula: (i) Attendance & Participation (10%) + (ii) Homework Problem Sets (40%) + (iii) Final Project Proposal (10%) + Final Project Report (30%) + Final Project Presentation (10%).
Tyrosine kinase inhibitors such as imatinib (IM), have significantly improved treatment of chronic myelogenous leukemia (CML). Yet, most patients are not cured for undetermined reasons. In this talk we will describe our recent work on modeling the autologous immune response to CML. Along the way, we will discuss our previous results on cancer vaccines, drug resistance, and cancer stem cells.
Given a piecewise smooth function, it is possible to construct a global expansion in some complete orthogonal basis, such as the Fourier basis. However, the local discontinuities of the function will destroy the convergence of global approximations, even in regions for which the underlying function is smooth. The global expansions are contaminated by the presence of a local discontinuity, and the result is that the partial sums are oscillatory and do not converge uniformly. This behavior is called the Gibbs phenomenon. However, David Gottlieb and Chi-Wang Shu showed that these slowly and non-uniformly convergent global approximations retain within them high order information which can be recovered with suitable post- processing. In this talk I review the history of the Gibbs phenomenon and the story of its resolution, and present some evidence that these postprocessing approaches work for other approaches, such as weighted essentially non-oscillatory methods and radial basis functions approximations.
Mathematical models describe how we think a system should behave under certain conditions. Observational data, on the other hand, tells us how the system actually behaved. Inverse problems provide a natural way of bringing models and data together. Given the observations, the solution to an inverse problem will tell us what conditions are likely to have existed. This talk will focus on the Bayesian formulation of inverse problems, which allow us to rigorously quantify uncertainty stemming from ill-posedness and noisy observations. I will start by describing the relationship between deterministic and Bayesian approaches and will finish will several examples of Bayesian inverse problems that arise in my work at the US Army Engineer Research and Development Center (ERDC).
My research group has been developing a comprehensive model for the evolution of ocean oil spills. I will give an overview of the oil fate model, highlighting modeling and computational challenges and how these challenges are met by tightly coupling physics and computing. One of the successes of the model is the prediction of nearshore sticky waters. Nearshore sticky waters refers to the propensity of certain pollutants and biological tracers to slow down and park as they approach a shore. Determining under what conditions stickiness arises can inform first responders trying to protect a beach from oil landfall. In the second part of the talk I will describe the basic dynamics of advection and dispersion of buoyant oil, as it is affected by ocean flows in the neighborhood of a beach, and how the oil fate model explains nearshore sticky waters.
| Week | Lecture |
|---|---|
| Week 1 | (Gelb) Numerical methods for inverse problems: solving linear systems, least squares, SVD, regularization. |
| Week 2 | (Gelb) Numerical methods for inverse problems: compression, noise, bad actors in the data. |
| Week 3 | Fu |
| Week 4 | Fu |
| Week 5 | (Gelb) Numerical Methods for inverse problems: Fourier data, Bayesian approach. |
| Week 6 | (Gelb) Numerical Methods for inverse problems: Bayesian approach, multiple measurements, bad data. |
| Week 7 | Fu |
| Week 8 | Fu |
| Week 9 | Final project presentations |
The Department of Mathematics at Dartmouth College enthusiastically welcomes the following 8 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 23, with dormitory move in on June 22. Participants must commit to being on campus through August 10. 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 |
|---|---|
| Aaron Alphonsus | South Dakota |
| Adam Baldoni | Dickinson College |
| Mariah Boudreau | Saint Michael’s College |
| Ruxuan (Sissi) Chen | Case Western |
| Ruyin Hing | UT Arlington |
| Javier Salazar | UT Arlington |
| Chi Zhang | Wisconsin @ Madison |
| Carley Walker | Southern Mississippi |
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.
Here are some potential project ideas related to material taught by Professor Gelb. You are also encouraged to come up with your own project idea. Please consult with either instructor early in the term so that you have the time to complete a meaningful research project. At the end of the term, each student's project will be listed below.