[+] Team 1: Pursuit-Evasion in Polygonal Environments

**Mentor** Brendan Ames, University of Minnesota, Twin Cities- Rosalie Carlson, Harvey Mudd College
- Claire Djang, Oberlin College
- Stephen Ragain, Pomona College
- Maxray Savage, Macalester College

Faculty Advisor: Andrew Beveridge, Mathematics, Statistics and Computer Science, Macalester College

Problem Poser: Volkan Isler, Department of Computer Science and Engineering, University of Minnesota

Pursuit-evasion games are models for controlling autonomous teams of robots. In a pursuit-evasion game, one or more pursuers try to capture an evader who in turn tries to avoid capture. There are many variants of pursuit-evasion games based on 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), and the definition of capture (e.g., confinement, threshold distance, line of sight).

The lion and man game is a geometric pursuit game. In the original version, the game takes place inside a circular arena. The players have the same maximum speed. The objective of the lions (pursuers) is to capture the man (evader) by moving onto the man's current location. The team will study this game played in a polygonal environment, a typical setting for robotics applications. In the turn-based version, one lion can always catch the man in a simply connected polygonal region. Recently Isler and Bhadauria showed that three lions can always catch a man in a polygonal region with polygonal holes. The research team will characterize the polygonal environments with holes where two lions suffice. The team will also investigate the effect of more practical models of robot movement, taking into account robot turning speed.

Required background: discrete mathematics, combinatorics. Useful background: computational geometry.

[+] Team 2: Modeling the Dynamics of Coupled Laser Cavities

**Mentor** Yulia Hristova, University of Minnesota, Twin Cities- Philip Burnham, Villanova University
- Delani Cele, Ithaca College
- Hyunmoon Kim, Princeton University
- Tim Moon, Rice University

Faculty Advisor: Dan Flath, Department of Mathematics, Statistics and Computer Science, Macalester College

Problem Poser: James Leger, Department of Electrical and Computer Engineering, University of Minnesota

The ability to scale a laser to high power is generally limited by one or more physical effects. For example, single-spatial-mode semiconductor lasers are limited by catastrophic facet damage to approximately 1 watt. Fiber lasers, on the other hand, can produce powers at the kilo-watt level before they are limited by nonlinear effects, such as stimulated Brillouin scattering. To achieve higher powers within a single spatial mode, it is possible to generate smaller amounts of power in separate modules and combine these powers in a separate optical system. This generally involves coupling many lasers together to form a coupled oscillator. Laser systems are inherently nonlinear; this system of coupled oscillators gives rise to complex dynamics. Indeed, self-pulsation, stable regions, and chaos have all been observed.

The research team will develop a mathematical model to explain the system dynamics observed in two custom-made fiber lasers operating at high power (100 watts). This model will be used to determine the optimal performance of this approach to high-power laser operation.

Required background: differential equations, linear algebra, and an interest in physics. Useful background: waves and optics.

[+] Team 3: Measuring the Health of Online Communities

**Mentor** Paolo Codenotti, - Alexander Bristol, University of Massachusetts
- Mary Huffman, Virginia Polytechnic Institute and State University
- Nathan Leech, Macalester College
- Guanyu Wang, The University of Iowa

Faculty Advisor: Shilad Sen, Department of Mathematics, Statistics and Computer Science, Macalester College

Problem Poser: Loren Terveen, Department of Computer Science and Engineering, University of Minnesota

Member contributions power online communities. Users edit Wikipedia pages, upload videos to YouTube, and tweet on Twitter. Our work and play depend on the health of these online communities, but they may not be perfectly fit. For example, the number of active Wikipedia editors is declining, and these editors must deal with an increasing amount of vandalism on articles.

This project seeks to 1) develop mathematical metrics that capture the health of an online community and 2) apply those metrics to identify specific behaviors that hurt and harm a community.

Students in this REU will analyze huge data sets representing online behavior in communities, such as Twitter and Wikipedia. They will develop hypotheses about effective behaviors for a community (i.e., that users follow a diverse set of other users) and analyze user-level data to determine whether those behaviors do, in fact, lead to positive outcomes. Students will draw together these findings to develop health metrics that quantify the overall value of a community and study the implications of these metrics.

Required background: data structures, algorithms, and/or statistical analysis. Useful background: experience with Python, parallel computing, social psychology, and graph theory.