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Undergraduate Program:
Research Experiences for Undergraduates (REU)
The summer 2001 REU program involved even more undergraduates
than in summer 2000, and we had weekly presentations of students' work,
after Friday pizza lunch. Scot Adams' group, Filip Matejka, Andrew Liesch,
and Michael Lieberman, modeled traffic to understand whether metering
is effective in improving the flow of traffic. They began by modeling
traffic on a simple Y-shaped road system, with cars traveling down the
Y, so that two lanes of traffic would merge into one, using Mathematica.
Cars were produced at both sides of the top of the Y accordin g
to an exponentially distributed waiting time. If the rate of production
was sufficiently large, then a traffic jam would occur at the merge point,
as in real life. Thus the cars would leave the merge point with an initial
velocity of zero, and if the maximum acceleration is set small enough,
then cars leaving the merge point would be widely spaced from one another
producing a low flow off the bottom of the Y. They found that by implementing
metering this flow could be increased. They then analyzed more complicated
road models, developing a Java program with which a user could draw a
road system using the mouse, and be prompted for parameters such as maximal
acceleration, speed limits, rate of production of cars, rate of metering.
Their basic result is that, while in some situations metering is clearly
advantageous, in complicated road systems choosing efficient metering
is hard, and perhaps not feasible.
Vic Reiner's group, Hans Christianson and Hyung Kim, continued
a project that his Summer 2000 REU group (Paul Bendich and Tristram Bogart)
began. In 2000, his students performed computer experiments to determine
the structure of the critical group (a finite abelian group which is a
subtle graph-isomorphism invariant) for a certain class of graphs (threshold
graphs). They did a great job on this, and conjectured a description for
"almost all" threshold graphs. This summer, Hans Christianson
produced an elegant proof of their conjecture, and Hyung Kim did more
experiments which led them to extend the statemen t
of their conjecture to all threshold graphs. Christianson and Vic have
written a paper based on this work, submitted to Linear Algebra and Its
Applications. It is linked-to from Vic's web page at www.math.umn.edu/~reiner/.
In Rachel Kuske's group, Jessica Myers (University of Minnesota)
combined linear stability analysis with numerical simulations to study
the effects of coupling on a canonical model of neurons with bursting
(alternating active/silent) dynamics. She developed predictive rules for
synchronization and considered different models for coupling. Justin Douglas
(University of Minnesota) developed code for studying stochastic effects
on models of elasto-plastic behavior. His code involved determining the
stability of equilibria and then testing the effects of external random
effects on transitions between equilibria. Michael Hsieh (University of
California, Berkeley) analyzed pricing of the American put option. He
incorporated asymptotic results for the behavior near expiry into an iterative
method adapted for the Black-Scholes partial differential equation.
Paul Garrett's group, Laura Chasman (Caltech), Ben Chastek
(St. Mary's), Kevin Costello (Caltech), Lee Dicker (McGill), Michael D'Sa
(U of M), Ali Elgindi (U of Wisconsin), McKenzie Lamb (Beloit College),
Natalie Linnell (U of M), Christina Mulligan (Harvard), and Mohammad Zaki
(MN State U), studied a variety of problems related to number theory and
its applications, such as random number generation, distribution of prime
numbers, local-to-global principles (Hasse-Minkowski theorem), fixed-point
theorems (aiming toward Weil conjectures), partitions, and general structure
of algebraic number fields. We started by developing a common ground of
Fourier analysis, complex analysis, abstract algebra, and some specific
number-theoretic ideas such as the gamma function, zeta-functions and
L-functions, Gauss sums, quadratic reciprocity, algebraic integers, elliptic
functions, theta series, and modular functions. We continued talking about
modern developments in number theory and its applications while individuals
worked on their own projects, to get a little idea about the Wiles-Taylor
proof of Fermat's last theorem via the Taniyama-Shimura conjecture, as
well as trying to think of parts of Langlands' conjectures as being extensions
of quadratic reciprocity and other (known) reciprocity laws of classfield
theory.
For details about the Summer 2002 program please see the
department web page at www.math.umn.edu/arb/reu.
Paul Garrett,
Professor of Mathematics and REU Coordinator
www@math.umn.edu
URL http://www.math.umn.edu/index.shtml
The University of Minnesota
is an equal opportunity educator and employer.
2000, The Regents of the University of Minnesota
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