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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 5-week period.
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:
00 stipend, campus housing and meals, travel allowance of up to 0 each. Participants also have a 0 allowance for travel to a national meeting during the 2010-2011 academic year to present results of their REU work.
Requirements:
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.
Faculty Advisor: Andrew Beveridge, Department of Mathematics and Computer Science, Macalester College
Problem Poser: Volkan Isler, Department of Computer Science and Engineering, University of Minnesota
Project description
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 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 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 captured when one of the cops moves onto his current location. The lion-and-man 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:
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?
References:
M. Aigner; M. Fromme; A game of cops and robbers, Discrete Appl. Math. 8 (1984), 1–12.
V. 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: 10.1016/j.ipl.2005.04.012. Keywords: Computational geometry; Pursuit evasion game; Lion-man problem
S. Alexander, R. Bishop, and R. Ghrist. Capture pursuit games on unbounded domains, (posted 1/2008) to appear, Enseign. Math.
Faculty Advisor: Daniel Flath, Department of Mathematics and Computer Science, Macalester College
Problem Poser: Satish Kumar, Department of Chemical Engineering and Materials Science, 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, 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 characteristic vertical 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.
References
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 strictly valid.
Faculty Advisor: Olaf Hall-Holt, Department of Mathematics, Statistics, and Computer Science, St. Olaf College
Problem Poser: Gilad Lerman, School of Mathematics, University of Minnesota
Consultant (June 14-25): Guangliang Chen, Department of Mathematics, Duke University
Project description
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.
References
J. Ho, M. Yang, J. Lim, K. Lee, D. Kriegman. Clustering appearances of objects under varying illumination conditions.
R. Vidal, Y. Ma, and S. Sastry Generalized principal component analysis (GPCA), IEEE Trans. Patterns Analysis and Machine Intelligence, 27 (2005).
Y. Ma, H. Derksen, W. Hong, J. Wright. Segmentation of multivariate mixed data via lossy data coding and compression, IEEE Trans. Patterns Analysis and Machine Intelligence, 29 (2007).
G. Chen and G. Lerman. Spectral curvature clustering, Int. J. Computer Vision, 81 (2009)
Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday Monday | Tuesday | Wednesday | Thursday | Friday | | |||
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Monday June 14, 2010 | |||
8:30am-9:00am | Registration and coffee | Lind Hall 400 | |
9:00am-9:15am | Welcome to the IMA | Fadil Santosa (University of Minnesota, Twin Cities) | Lind 409 |
9:15am-9:55am | Intro to Project 1: Pursuit-evasion games with multiple pursuers (all groups) | Volkan Isler (University of Minnesota, Twin Cities) | Lind 409 |
9:55am-10:35am | Intro to Project 2: Long-wave models for elastohydrodynamic instabilities (all groups) | Daniel Flath (Macalester College) | Lind 409 |
10:35am-11:00am | Break | Lind Hall 400 | |
11:00am-11:40am | Intro to Project 3: Hybrid linear modeling (all groups) | Gilad Lerman (University of Minnesota, Twin Cities) | Lind 409 |
11:40am-12:00pm | Announcements | Lind 409 | |
12:00pm-1:00pm | Lunch at the IMA, getting acquainted | ||
1:00pm-4:00pm | Groups meet separately with faculty advisors and mentors | Lind Hall 400 | |
Tuesday June 15, 2010 | |||
Work all day on projects. Lind Hall 400 | |||
Wednesday June 16, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-4:00pm | Seminar - Modeling swarms | Chad Topaz (Macalester College) | Lind 409 |
Thursday June 17, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-6:00pm | Pizza and Movie (Amount due will be announced at a later time) Social outing leader - Fadil Santosa | Lind 409 | |
Friday June 18, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-4:30pm | Each group makes progress reports | Lind 409 | |
Saturday June 19, 2010 | |||
No scheduled activity. | |||
Sunday June 20, 2010 | |||
No scheduled activity. | |||
Monday June 21, 2010 | |||
Work all day on projects. Lind Hall 400 | |||
Tuesday June 22, 2010 | |||
Work all day on projects. Lind Hall 400 | |||
Wednesday June 23, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-4:00pm | Seminar - Snail robots, adhesion, and complex fluids | Randy Ewoldt (University of Minnesota, Twin Cities) | Lind 409 |
Thursday June 24, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-6:00pm | Social outing leader - Daniel Flath | Como Zoo/ Conservatory | |
Friday June 25, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-4:30pm | Each group makes progress reports | Lind 409 | |
Saturday June 26, 2010 | |||
No scheduled activity. | |||
Sunday June 27, 2010 | |||
No scheduled activity. | |||
Monday June 28, 2010 | |||
Work all day on projects. Lind Hall 400 | |||
Tuesday June 29, 2010 | |||
Work all day on projects. Lind Hall 400 | |||
Wednesday June 30, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-4:00pm | Seminar - Title:TBA | Richard McGehee (University of Minnesota, Twin Cities) | Lind 409 |
Thursday July 01, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-6:00pm | Social outing - Mill City Museum (please bring $10 - $15 with you for entrance and transportation) Social outing leader - Andrew Beveridge | Mill City Museum 704 South Second Street Minneapolis, MN 55401 612-341-7555 | |
Friday July 02, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-4:30pm | Each group makes progress reports | Lind 409 | |
Saturday July 03, 2010 | |||
No scheduled activity. | |||
Sunday July 04, 2010 | |||
No scheduled activity. | |||
Monday July 05, 2010 | |||
Work all day on projects. Lind Hall 400 | |||
Tuesday July 06, 2010 | |||
Work all day on projects. Lind Hall 400 | |||
Wednesday July 07, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-4:00pm | Seminar - Squishy and frozen interfaces: Instabilities and applications | Satish Kumar (University of Minnesota, Twin Cities) | Lind 409 |
Thursday July 08, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-6:00pm | Social outing - St. Anthony Falls Laboratory Tour Social outing leader - Kara Lee Maki | St. Anthony Falls Laboratory | |
Friday July 09, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-4:30pm | Each group makes progress reports | Lind 409 | |
Saturday July 10, 2010 | |||
No scheduled activity. | |||
Sunday July 11, 2010 | |||
No scheduled activity. | |||
Monday July 12, 2010 | |||
Work all day on projects. Lind Hall 400 | |||
Tuesday July 13, 2010 | |||
Work all day on projects. Lind Hall 400 | |||
Wednesday July 14, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-4:00pm | Seminar - From pebbles dropped in a pond to tidal waves | Daniel Spirn (University of Minnesota, Twin Cities) | Lind 409 |
Thursday July 15, 2010 | |||
Work on projects. Lind Hall 400 | |||
3:00pm-6:00pm | Social outing - Canoe at Calhoun (please bring $10 - $15 with you for rental and transportation) In case of rain Walker Art Museum. Social outing leader - Olaf Hall-Holt | Calhoun or Walker Art Museum | |
Friday July 16, 2010 | |||
10:00am-12:00pm | Final presentations | Lind 409 | |
12:00pm-1:00pm | Poster session and pizza | Lind Hall 400 |
NAME | DEPARTMENT | AFFILIATION |
---|---|---|
Andrew Beveridge | Department of Mathematics, Statistics and Computer Science | Macalester College |
Guangliang Chen | Department of Mathematics | Duke University |
Laura Chihara | Department of Mathematics | Carleton College |
Randy Ewoldt | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |
Daniel Flath | Department of Mathematics, Statistics and Computer Science | Macalester College |
Olaf Hall-Holt | Department of Mathematics, Statistics and Computer Science | St. Olaf College |
Yan Huang | Macalester College | |
Volkan Isler | Department of Computer Science | University of Minnesota, Twin Cities |
Catherine Kealey | Department of Mathematics | Beloit College |
Markus Keel | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |
Satish Kumar | Department of Chemical Engineering and Materials Science | University of Minnesota, Twin Cities |
Gilad Lerman | School of Mathematics | University of Minnesota, Twin Cities |
Wanyi Li | School of Management | Macalester College |
Kara Maki | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |
Aaron Maurer | Department of Mathematics | Carleton College |
John McCauley | Department of Mathematics | Haverford College |
Richard McGehee | School of Mathematics | University of Minnesota, Twin Cities |
Fadil Santosa | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |
Vishal Saraswat | School of Mathematics | University of Minnesota, Twin Cities |
Daniel Spirn | School of Mathematics | University of Minnesota, Twin Cities |
Charles Talbot | Department of Mathematics | University of Connecticut |
Chad Topaz | Department of Mathematics, Statistics, and Computer Science | Macalester College |
Silviya Valeva | Department of Mathematics | Mount Holyoke College |
Peter VanKoughnett | Department of Mathematics | Oberlin College |
Yi Wang | Department of Mathematics | University of Minnesota, Twin Cities |
Andrew White | Department of Computer Science | St. Olaf College |
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