2006 Claremont Colleges REU
    Algebra
    Number Theory
    Metric Spaces
    Knot Theory

2006 Claremont REU students and faculty on the CMC campus.

2006 Speaker Series

All of our research teams made progress on their problems and several have written papers that have been submitted to journals for publication. Each team also be presented their research findings at the Undergraduate Poster Session of the Joint Mathematical Meeting in January, 2007, in New Orleans.

Research Papers

• Sampling Lissajous Knots and Fourier Knots, A. Boocher, G. Daigle, J. Hoste, W. Zheng, (preprint), submitted for publication.

  Abstract: A Lissajous knot is one that can be parameterized by a single cosine function in each coordinate. Lissajous knots are highly symmetric, and for this reason, not all knots are Lissajous. We prove several theorems which allow us to place bounds on the number of Lissajous knot types with given frequencies and to efficiently sample all possible Lissajous knots with a given set of frequencies. In particular, we systematically tabulate all Lissajous knots with small frequencies and as a result substantially enlarge the tables of known Lissajous knots.

  Lissajous Knot Pillow!

  
  A Fourier (i, j, k) knot is similar to a Lissajous knot except that each coordinate is now described by a finite sum of i, j, and k cosine functions respectively. According to Lamm, every knot is a Fourier-(1,1,k) knot for some k. By randomly searching the set of Fourier-(1,1,2) knots we find that all 2-bridge knots up to 14 crossings are either Lissajous or Fourier-(1,1,2) knots. We show that all twist knots are Fourier-(1,1,2) knots and give evidence suggesting that all torus knots are Fourier-(1,1,2) knots.

  As a result of our computer search, several knots with relatively small crossing numbers are identified as potential counterexamples to interesting conjectures.



• Compactness and measures of noncompactness in metric trees, A. Aksoy, S. Borman, A. Westfahl, Proc. Inter. Sym. Banach and Function Spaces II, Japan, M. Kato and L. Maligranda eds., (2007)

Conference Presentations

A. Merberg in

New Orleans.



• A. Merberg, E. Udovina, Orbits of Hecke triangle groups, MAA Undergraduate Poster Session, New Orleans, LA, January 7, 2007. Prize Winner.
• A. Boocher, J. Daigle, W. Zhang, Sampling Lissajous Knots, MAA Undergraduate Poster Session, New Orleans, LA, January 7, 2007.
• B. Froehle, M. Jameson, Algorithms for computing fast Fourier transforms of the symmetric group, MAA Undergraduate Poster Session, New Orleans, LA, January 7, 2007. Prize Winner.

M. Jameson & B. Froehle

in New Orleans.




• S. Borman, A. Westfahl, Compactness and measures of noncompactness in metric trees, MAA Undergraduate Poster Session, New Orleans, LA, January 7, 2007.

  A. Aksoy & S. Borman

  in New Orleans.

• A. Boocher, J. Daigle, J. Hoste, W. Zhang, Sampling Lissajous and Fourier Knots, Spring 2007 meeting of the Southern California-Nevada MAA Section, Pomona College. Meritorious Poster award.

• A. Boocher, J. Daigle, J. Hoste, W. Zhang, Sampling Lissajous and Fourier Knots, First Joint International Meeting of the American Mathematical Society and the Polish Mathematical Society, Warsaw, Poland, August 3, 2007.

• A. Boocher, Knot Theory and Lissajous Knots , Undergraduate Work in Mathematics, A Conference Sponsored by the Notre Dame Mathematics Department, September 30, 2007.

J. Daigle, W. Zheng & A. Boocher

in New Orleans.

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Last Modified November 22, 2006.