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 hands-on 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 publication-quality written
report, and to give an oral presentation by the end of the
Project 1: Pursuit-Evasion Games with Multiple Pursuers
Problem poser: Volkan Isler,
Department of Computer Science and Engineering, University of
Faculty advisor: Andrew Beveridge,
Department of Mathematics and Computer Science, Macalester
Saraswat, School of Mathematics, University of
In a pursuit-evasion game, one or more pursuers try to capture
evader who in turn tries to avoid capture. There are many
pursuit evasion games based on the environment (e.g. a polygon,
graph), information available to the players (e.g. can they see
other at all times?), motion constraints (e.g. a car chasing an
can not turn arbitrarily) and the definition of capture (in
games, the pursuer captures the evader if the distance between
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 pursuit-evasion games.
The cops-and-robbers game takes place on a graph. At each turn,
the players move along the edges. The evader (robber) is
when one of the cops moves onto his current location.
game is a geometric version of the cops and robbers game.
In the original version, the game takes place inside a circular
The players have the same maximum speed. The objective of the
is to capture the man by moving onto the man's current
Given a planar graph G, what is the number of cops necessary
and sufficient to
capture the robber on G? (the answer should
be 1,2 or 3)
- Given a polygon with obstacles, what is the
number of lions necessary to capture the man?
M. Aigner; M. Fromme; A game of cops and robbers, Discrete
Appl. Math. 8 (1984), 1–12.
Isler, S. Kannan, and S. Khanna.
Randomized Pursuit-Evasion in a Polygonal Environment. IEEE
Transactions on Robotics, 5(21):864--875, 2005
Swastik Kopparty, Chinya V. Ravishankar. A framework for
pursuit evasion games in , Information Processing Letters,
Volume 96, Issue 3,
15 November 2005, Pages 114-122, ISSN 0020-0190, DOI:
Keywords: Computational geometry; Pursuit evasion game;
Alexander, R. Bishop, and R. Ghrist.
Capture pursuit games on unbounded domains, (posted 1/2008) to
appear, Enseign. Math.
Project 2: Long-wave Models for Elastohydrodynamic
Problem poser: Satish
Kumar, Department of
Chemical Engineering and Materials Science, University of
Faculty advisor: Daniel Flath,
Department of Mathematics and Computer Science,
Lee Maki, Institute for Mathematics and its
Applications, University of Minnesota
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, time-dependent 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 flow-induced 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 liquid-gel
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
fluid-fluid interfaces, it is known that one effective way of
understanding nonlinear aspects
of instability is the development and analysis of long-wave
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
distance to the instability wavelength. The goal of this
project is to derive and analyze
long-wave 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 long-wave model captures the linear
aspects of the instability.
The linear aspects of the elastohydrodynamic instability
described above are discussed in:
V. Kumaran, G. H. Fredrickson, and P. Pincus, Flow-induced
instability at the interface between a fluid and a gel at low
Reynolds number, J. Phys. Paris II 4, 893-911 (1994).
V. Gkanis and S. Kumar, Instability of creeping Couette flow
past a neo-Hookean solid, Phys. Fluids 15, 2864-2871 (2003).
V. Gkanis and S. Kumar, Stability of pressure-driven creeping
flows in channels lined with a nonlinear elastic solid, J.
Fluid Mech. 524, 357-375 (2005).
References to related experiments and weakly nonlinear analysis
can be found in the above papers.
A general discussion of long-wave models is given in:
A. Oron, S. G. Bankoff, and S. H. Davis, Long-scale evolution
of thin liquid films, Rev. Mod. Phys. 69, 931-980 (1997).
A long-wave model for a system involving an interface between a
liquid and a deformable solid is presented in:
O. K. Matar, V. Gkanis, and S. Kumar, Nonlinear evolution of
thin liquid films dewetting near soft elastomeric layers, J.
Colloid Interface Sci. 286, 319-332 (2005).
The approach taken in this paper can be adapted to the problem
described above if the liquid-air 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 long-wave 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 long-wave descriptions sometimes work
surprisingly well outside of the regimes in which they are
Project 3: Hybrid Linear Modeling
Gilad Lerman, School of Mathematics,
University of Minnesota
Olaf Hall-Holt, Department of Mathematics, Statistics,
and Computer Science, St. Olaf College
Wang, School of Mathematics, University of
Consultant (June 14-25): Guangliang
Chen, Department of Mathematics,
Efficient processing and analysis of massive and
high-dimensional 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.
Ho, M. Yang, J. Lim, K. Lee, D. Kriegman.
Clustering appearances of objects under varying illumination
R. Vidal, Y. Ma, and S. Sastry
Generalized principal component analysis (GPCA), IEEE Trans.
Patterns Analysis and Machine Intelligence, 27 (2005).
Ma, H. Derksen, W. Hong, J. Wright.
Segmentation of multivariate mixed data via lossy data coding
IEEE Trans. Patterns Analysis and Machine Intelligence, 29
G. Chen and G. Lerman.
Spectral curvature clustering, Int. J. Computer Vision, 81
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
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 2010-2011 academic year to
present results of their REU work.
Must be full-time 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.