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Mathematics, Carleton College 

Mathematics and Computer Science, Macalester College 

Mathematics, Statistics, and Computer Science, St. Olaf College 
The IMA summer REU gives students an experience in working on an interdisciplinary project involving mathematics. Students will work in teams of three with a faculty advisor and a postdoctoral mentor. This is a handson experience. There will be a few formal lectures. However, students will spend most of their time doing actual research. Students are expected to produce a poster and a publicationquality written report, and to give an oral presentation by the end of the 5week period.
Project 1: PursuitEvasion Games with Multiple Pursuers
Problem poser: Volkan Isler, Department of Computer Science and Engineering, University of Minnesota
Faculty advisor: Andrew Beveridge, Department of Mathematics and Computer Science, Macalester College
Mentor: Vishal Saraswat, School of Mathematics, University of Minnesota
Project description
In a pursuitevasion 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 can not turn arbitrarily) and the definition of capture (in some games, the pursuer captures the evader if the distance between them is less than a threshold. In other games, the pursuers must see or surround the evader in order to capture it.)
We will study two fundamental pursuitevasion games. The copsandrobbers game takes place on a graph. At each turn, the players move along the edges. The evader (robber) is captured when one of the cops moves onto his current location. The lionandman game is a geometric version of the cops and robbers 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 by moving onto the man's current location.
Research Questions:
Project 2: Longwave Models for Elastohydrodynamic Instabilities
Problem poser: Satish Kumar, Department of Chemical Engineering and Materials Science, University of Minnesota
Faculty advisor: Daniel Flath, Department of Mathematics and Computer Science, Macalester College
Mentor: Kara Lee Maki, Institute for Mathematics and its Applications, University of Minnesota
Project description
In contrast to rigid boundaries, flexible solid boundaries can deform under the action of shear and normal stresses, resulting in the creation of surface waves. If the stresses are exerted by an adjacent flowing fluid, these waves may lead to a complicated, timedependent flow. Important consequences of this modified flow include the alteration of mass and heat transfer rates and alteration of the stresses exerted on the solid surface. Such elastohydrodynamic instabilities, if better understood, could find application in a variety of areas including microfluidic mixers, membrane separations, and the rheology of complex fluids that undergo flowinduced gelation.
The schematic below shows a liquid flowing past a gel, a type of deformable solid. The liquid flow may be driven by a combination of boundary motion and externally applied pressure gradients. In the situation pictured, the flexible boundary is the interface between the liquid and gel. At a critical liquid flow rate, the initially flat liquidgel interface becomes unstable, leading to a state in which waves travel along the interface. As a consequence, the liquid flow, which initially had parallel streamlines, becomes more complicated. This instability occurs even when inertia is completely absent; it is purely a consequence of having a deformable boundary.
Whereas there has been much theoretical work concerning the linear aspects of this instability, relatively little is known about its nonlinear aspects. For systems with fluidfluid interfaces, it is known that one effective way of understanding nonlinear aspects of instability is the development and analysis of longwave equations. These equations are essentially the leading order problem in an asymptotic expansion of the full governing equations, where the expansion parameter (assumed small) is the ratio of a characteristic vertical distance to the instability wavelength. The goal of this project is to derive and analyze longwave equations for the system shown in the above schematic. After the equations have been derived, it will be of interest to perform a linear stability analysis, a weakly nonlinear analysis, and direct numerical simulations. It will also be of interest to compare the linear stability analysis results with the results of a similar analysis of the full governing equations to determine how well the longwave model captures the linear aspects of the instability.
References
The linear aspects of the elastohydrodynamic instability described above are discussed in:
References to related experiments and weakly nonlinear analysis can be found in the above papers.
A general discussion of longwave models is given in:
The approach taken in this paper can be adapted to the problem described above if the liquidair interface is replaced by a rigid solid boundary. As a first step, it would be worthwhile to (i) completely neglect inertia, (ii) assume that a linear constitutive model for the gel is appropriate, and to (iii) suppose that a longwave description is appropriate. The reason for (i) is that the instability is known to occur in the absence of inertia. The reason for (ii) and (iii) is that linear models and longwave descriptions sometimes work surprisingly well outside of the regimes in which they are strictly valid.
Project 3: Hybrid Linear Modeling
Problem poser: Gilad Lerman, School of Mathematics, University of Minnesota
Faculty advisor: Olaf HallHolt, Department of Mathematics, Statistics, and Computer Science, St. Olaf College
Mentor: Yi
Wang, School of Mathematics, University of
Minnesota
Consultant (June 1425): Guangliang
Chen, Department of Mathematics,
Duke University
Project description
Efficient processing and analysis of massive and highdimensional data requires its reduction by a simpler model. The simplest and most common geometric data modeling uses a single affine subspace, but it does not represent well many types of data. The next one uses a combination of several affine subspaces and is often referred to as Hybrid Linear Modeling (HLM). Various HLM algorithms have been suggested and applied in diverse problems such as segmenting motions in video sequences and clustering faces under varying illuminating conditions. In this project we will review all algorithms for hybrid linear modeling we know of and carefully compare them on artificial and real data, while trying to explain their successes and failures and possibly suggest improvements or effective combination of several algorithms.
References
Reading and supporting material will be provided to successful applicants prior to the start of the program.
Deadline March 31, 2010. Applications are invited from students who will be entering Junior and Senior years in the Fall of 2010. There is no citizenship requirement. Only 9 students will be admitted to the program.
Stipend and allowance:
$3500 stipend, campus housing and meals, travel allowance of up to $400 each. Participants also have a $400 allowance for travel to a national meeting during the 20102011 academic year to present results of their REU work.
Requirements:
Must be fulltime undergraduate mathematics major and must devote full time to the program and may not engage in other course work or employment during the 5 week period.
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