MAXIMA is an intensive six-week REU program in interdisciplinary mathematics, funded by a three-year NSF REU grant and the IMA.

Students will work in teams of four on an open research problem in applied mathematics. Each team will be advised by a Macalester College Mathematics, Statistics, and Computer Science (MSCS) faculty member. In summer 2013, problems will be posed by University of Minnesota (UMN) researchers from computer science and electrical engineering. Students will also be mentored by a postdoctoral fellow at the IMA.

The program is designed to allow students to experience the excitement of doing research that is relevant to another field. Students will learn how to formulate domain-specific questions in mathematical ways and assimilate the ability to communicate across disciplines. The program will develop mathematical skills and expository argumentation. By the end of the summer, each team will produce a written report, an oral presentation, and a research poster.

Eligibility

• Students must be current sophomores and juniors who will be full-time undergraduate students as of September 2013, majoring in mathematics or a related field at a U.S. college or university.
• Students must have taken multivariable calculus, linear algebra, and at least one upper-level undergraduate course (and preferably more). Exposure to computer science or statistics is also preferred. See the problem descriptions for further background requirements.
• Students must be fully committed to the REU. Students may not engage in any other course work or employment for the duration of the program.
• Nine spots are funded by the NSF; these are open to U.S. citizens and permanent residents. Three spots are funded by the IMA; these spots are open to U.S. citizens, permanent residents, and international students studying at U.S. institutions.

Remuneration
Students receive a stipend of about $3,000 and a travel allowance to/from the REU site. UMN campus housing and meals are provided at no additional cost. Support will also be provided to attend a national mathematics meeting to present the student research.

Questions about the program should be sent to reu@ima.umn.edu.

Non-destructive testing (NDT) describes a host of techniques designed to interrogate properties of the material without causing material damage. NDT techniques based on guided acoustic waves have become popular due to their sensitivity to a variety of damage types and their ability to identify defects located far from available actuation and sensing points.

In traditional implementations, guided waves are generated and received by transmitter-receiver pairs distributed over and affixed to the test material. The signatures of wave scattering caused by material defects are captured along the transmitter-receiver path, and the positions of defects can be inferred. Recently, a new and powerful class of laser-enabled inspection methodologies have emerged. Scanning Laser Doppler Vibrometers (SLDV’s) provide precise measurements across the entire material domain (rather than a collection of paths). The data produced by SLDV’s enables high-resolution reconstruction of the propagating wavefield, which may be used for subsequent processing, visualization, or inference.

In this project we will develop techniques for inferring the locations of structural defects and anomalies from this type of spatio-temporal propagating wavefield data. We will use emerging techniques from the theory of compressive sensing and sparse representations to identify locations of potential defects or anomalies. We will evaluate our techniques on simulated and real SLDV data.

Required Background: linear algebra, statistics, Matlab experience

Useful Background: Fourier Representations, Norms and Vector Spaces, Signal Processing, Machine Learning, Optimization

Bar codes are ubiquitous -- they are used to identify products in a store, parts in a warehouse, and books in a library, etc. They contain information that are critical in supply chains, manufacturing, and monitoring. Modern handheld bar code scanners are camera-based. These include specially built scanners, multipurpose devices, and smart phones. To encourage their adoption, scanners need to be easy to use. Much of the burden of information processing of bar codes must rely on mathematical algorithms that are efficient and robust.

In this project, we will develop a technique that first determine if a bar code is present in an image. If a bar code is detected, it must then draw a bounding box around the bar code. These steps represent the critical initial stages of image processing in a scanner, and are usually followed by a decoding algorithm. We will explore existing methods from image processing including machine learning and transform methods, as well as developing new approaches that specifically take advantage of the structures of bar code images.

Required background: differential equations, Fourier series, Matlab experience. Useful background: machine learning, wavelet theory, signal processing, image processing.

Our research problem will model the control of an autonomous team of robots who are defending against one or more intruders. Current research focusses on a wide variety of these "good guy versus bad guy" games. There are many variants of these games based on the objectives of the agents, the environment (e.g., a polygon, graph), information available to the players (e.g., can they see each other at all times?), motion constraints (e.g., a car chasing an evader cannot turn arbitrarily). Examples include pursuit-evasion (aka cops and robbers) where the good guys must catch the bad guy; the firefighter problem, where the good guys must contain a dispersive fire; and spies and revolutionaries, where the good guys must prevent a meeting of the bad guys.

We will investigate one of these games in a polygonal environment, a typical setting for robotics applications. Inspired by recent results on graphs and large scale networks, we will prove analogous results in this geometric setting.

Required background: discrete mathematics, with advanced study of graph theory or combinatorics.

Useful background: algorithms, computational geometry.