MANIFOLDS WITH DENSITY AND PARTITIONING PROBLEMS
Frank Morgan, Williams College
Friday, August 3, 1:00 pm - 4:00 pm
Perelman's stunning 2006 proof of the million-dollar Poincaré Conjecture needed to consider not just manifolds but "manifolds with density" (like the density in physics you integrate to compute mass). Yet much of the basic geometry of such spaces remains unexplored. Partitioning problems provide a good place to start. Speakers will include: Frank Morgan, Williams College, Michael Hutchings, University of California at Berkeley; Neil Hoffman, University of Texas at Austin; members of the Williams College undergraduate research Geometry Group, and Joseph Corneli, PlanetMath.org.
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MATHEMATICAL QUESTIONS IN BIOINFORMATICS
Jennifer Galovich, St. John's University
Laurie Heyer, Davidson College.
Friday, August 3, 1:00 pm - 4:00 pm
The speakers and their topics in this session will include Laurie Heyer, Davidson College, "Molecular Computing"; Laura Salter Kubatko, The Ohio State University, "Phylogenetics"; Glen Tesler, University of California at San Diego, "Genome Rearrangements"; Stephen Billups, University of Colorado at Denver, Microarray Analysis"; Stephen Hartke, University of Illinois, " DNA Codewords"; and Stephen Hartke, University of Illinois, "DNA Code Words and DeBruijn Sequences." The session is sponsored by the MAA SIGMAA on Mathematical and Computational Biology.
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GEMS IN APPLIED MATHEMATICS
Kay Somers, Moravian College
Saturday, August 4, 8:30 am - 10:30 am
The speakers and topics in this session will be Annalisa Crannell, Franklin & Marshall College, "Size Matters"; Michael A. Jones, Montclair State University, "A Voting Theory Approach to Golf Scoring"; Nathan Shank, Moravian College, "Unsolved Gems in Random Graphs"; and Jennifer Wilson, Eugene Lang College, The New School for Liberal Arts, "Algebraic Models in Kinship Systems".
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MATHEMATICAL AND THEORETICAL BIOLOGY INSTITUTE / INSTITUTE FOR STRENGTHENING THE UNDERSTANDING OF MATHEMATICS AND SCIENCE (MTBI/SUMS) UNDERGRADAUE RESEARCH PROGRAM
Carlos Castillo-Chavez, Arizona State University
Saturday, August 4, 8:30 am – 11:30 am
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RESEARCH WITH UNDERGRADUATES
Mario Martelli, Claremont-McKenna College
Saturday, August 4, 1:00 pm - 3:30 pm
The speakers will present research in Mathematics completed in collaboration with undergraduates and, possibly, submitted for publication to a professional journal. Each speaker will describe in detail how the research was done and will highlight the undergraduates' participation. The speakers and the titles of their talks are as follows: Estelle Basor, California Polytechnic San Luis Obispo, "Eigenvalues of Random Matrices"; Frank Morgan, Williams College, "The Double Bubble Theorem"; Mike O'Neill, Claremont McKenna College, "An Inverse Theorem in Additive Number Theory"; and Aldolpho Rumbos, Pomona College, "Solvability of Semi-Linear Two-Point Boundary Value Problems".
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PRIME NUMBERS --- NEW DEVELOPMENTS ON ANCIENT PROBLEMS
Dan Goldston, San Jose State University and Carl Pomerance, Dartmouth College
Saturday, August 4, 1:00 pm - 4:00 pm
1:00 - 1:40 PRIMAL SCREENS
Carl Pomerance, Dartmouth College
Prime numbers are as old as Euclid yet we are still discovering new things about them. The world of computational number theory was stunned a few years ago when Agrawal, Kayal, and Saxena announced their "polynomial time" test for primality. We were stunned not only because this long-sought algorithm turned to be so simple, but because Kayal and Saxena had done their research as undergrads.
1:45 - 2:25 THE RIEMANN HYPOTHESIS, RANDOM MATRICES, AND PRIMES
Brian Conrey, American Institute of Mathematics
Perhaps the most famous problem in mathematics is the Riemann Hypothesis. (Aside from fame, there is also a million dollar prize for a proof of this.) The conjecture is that the complex zeros of the zeta-function all lie on a vertical line, but it is also equivalent to the primes having a random but regular distribution. Starting with the initial discovery by Montgomery in the early 70's, the zeros of the zeta function are apparently distributed according to the same laws as the eigenvalues of large random matrices, a subject of research by physicists. This is now an energetic field of research with input both from number theory and physics.
2:30 - 3:10 PROGRESSIONS OF PRIMES
Kannan Soundararajan, Stanford University
The prime numbers appear to be randomly sprinkled among the natural numbers subject only to their generally thinning out the higher you go. But certain patterns of primes seem to occur and then recur, and then again.
It is easy to notice these things, but it is an entirely different issue to prove it. In 2004 Green and Tao proved the seemingly intractable conjecture that there are arbitrarily long arithmetic progressions of primes. Previously the best we knew in a result from about 70 years ago was that there are infinitely many progressions of 3 primes. It is interesting that the Green--Tao proof rests in part on a tool that Goldston and Yildirim had just developed to study the also thorny twin-prime problem.
3:15 - 3:55 PRIMES, RESEARCH, ACADEMIC FREEDOM, AND HOW THE NSA GOT WHAT IT WANTED (BUT NOT WHAT IT ASKED FOR)
Susan Landau, Sun Microsystems
In the mid 1970's, academic researchers discovered how prime numbers could be used to design strong cryptographic systems. Cryptography had been the sole purview of governments and the National Security Agency fought to stop the research in the public sector; it lost. But from losing the battle, NSA has reaped mighty benefits. The fruits of public research are now fundamental to government systems.
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MATH CIRCLES
Tom Davis, Silicon Graphics; Tatiana Shubin, San JoseStateUniversity; and Joshua Zucker, Castilleja School.
Sunday, August 5, 9:00 a.m. – 11:00 a.m.
Our mathematical circles are modeled after those in Eastern Europeand are as successful here as they were there. Circles bring mathematicians into direct contact with middle or high school students who work together on problems that require deep thinking rather than rote solutions. Circles give students who enjoy studying mathematics a social context for doing so. This demonstration will be an actual math circle run by Tom Davis and Josh Zucker.
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GRAPH THEORY IDEAS FOR UNDERGRADUATE RESEARCH
Aparna Higgins, University of Dayton
Sunday, August 5, 1:00 pm - 4:00 pm
This session will highlight some topics in graph theory that are intriguing to undergraduate researchers. The speakers have successfully guided undergraduate students in research by directing undergraduate research in intensive summer experiences or in undergraduate thesis activities. The session will provide insight into what makes a topic in graph theory suitable for investigations by undergraduates, and will provide additional avenues of research.
