Advisor: Allison Pacelli
Project Description:
The Fundamental Theorem of Arithmetic states that the ring of integers is a unique factorization domain, that is, every integer can be uniquely factored into a product of primes. In other rings, there are analogues of prime numbers, but factorization into primes is not necessarily unique!
In order to determine whether factorization into primes is unique in the ring of integers of a number field or function field, it is useful to study the associated class group - the group of equivalence classes of ideals. The class group is trivial if and only if the ring is a unique factorization domain. Although the study of class groups dates back to Gauss and played a key role in the history of Fermat's Last Theorem, many basic questions remain open. This summer, we will investigate some of these questions in both the number field and function field realms.
Some knowledge of abstract algebra is required.
Advisor: Satyan Devadoss
Project Description:
Project Description: The study of geometric objects which are discrete or combinatorial in nature is not only as old as antiquity but currently one of the hottest areas of mathematical research. There is a rich world of unsolved problems [1,2] with direct applications to areas such as robot motion planning, cartography, efficient packings, origami, protein foldings, and computer animations. Our group will begin by focusing on some unsolved problems in the area of origami and polyhedral geometry [3] but will venture into other areas as the summer progresses.
References:
1. See http://cs.smith.edu/~orourke/TOPP/ to get a taste.
2. "Research Problems in Discrete Geometry" by Brass, Moser, and Pach.
3. Beautiful book "Geometric Folding Algorithms" by Demaine and O'Rourke.
Advisor: Cesar Silva
Project Description:
Ergodic theory studies dynamical systems from a probabilistic point of view. A discrete-time dynamical system can be given by the iteration of a self-map defined on some measure space. An interesting class of examples is given by continuous maps defined on cantor spaces. Another interesting class of examples is given by polynomial maps defined on some compact and open subsets of the p-adic numbers. We will consider three projects. One is the study of measurable dynamics of polynomial maps in the p-adics. For an introduction to measurable p-adic dynamics see Measurable dynamics of simple p-adic polynomials. Amer. Math. Monthly, Vol. 112 (2005), no. 3, 212-232."
The second topic will study mixing and ergodic properties of cellular automata. The third project will consider ergodic properties of nonsingular transformations. Results of the 2006 summer project are in http://arxiv.org/abs/math.DS/0612480 (to appear in Proceedings of the Americam Mathematical Society). Results of the 2005 summer project are in http://arxiv.org/abs/math/0701899 (to appear in Indagationes Mathematicae) and http://arxiv.org/abs/0710.5562 (to appear in Transactions of the Americam Mathematical Society).
Advisor: Frank Morgan
Project Description:
Perelman's stunning 2006 proof of the million-dollar Poincare Conjecture needed to consider not just manifolds but "manifolds with density" (like the density in physics you integrate to computer mass). Yet much of the basic geometry of such spaces remains unexplored. This will be our first research topic. See the first three references below.
In 1999 Hales [4,5] proved that regular hexagons provide the least-perimeter way to partition the place into equal areas. In 2002 he showed [6] that on the unit sphere, regular pentagons provide the least-perimeter partition into twelve equal areas. We'll continue Geometry Group work on such partitions of the sphere and of compact hyperbolic manifolds; see [7].
We will participate in the MAA Mathfest in Madison, Wisconsin, July 31-August 2, 2008 or possibly spend a month in Granada, Spain.
References
[1] Frank Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), 853-858, http://www.ams.org/notices/200508/fea-morgan.pdf
[2] Ivan Corwin, Neil Hoffman, Stephanie Hurder, Vojislav Sesum, Ya Xu (Geometry Group), Differential geometry of manifolds with density, Rose-Hulman Und. Math. J. 7 (1) (2006), http://www.rose-hulman.edu/mathjournal/v7n1.php
[3] Colin Carroll, Adam Jacob, Conor Quinn, Robin Walters (2006 Geometry Group), The isoperimetric problem on planes with density, Bull. Austral. Math. Soc., to appear.
[4] Thomas C. Hales, The honeycomb conjecture, Discr. Comput. Geom. 25 (2001), 1-22, http://front.math.ucdavis.edu/math.MG/9906042
[5] Frank Morgan, Geometric Measure Theory, Academic Press, third edition, 2000, Chapter 15.
[6] Thomas C. Hales, The honeycomb problem on the sphere,
[7] Conor Quinn (2006 Geometry Group), Area-minimizing partitions of the sphere, Rose-Hulman Und. Math. J. 8 (2) (2007).
Advisor: Colin Adams
Project Description:
In summer, 2008, we will be investigating superinvariants of knots. These are related to traditional invariants of knots, such as bridge number, crossing number and braid number. The superbridge number is related to the stick index of a knot, which is the least number of sticks glued end-to-end to make the knot. Supercrossing number was investigated by the knot theory group from SMALL '00 (see "An Introduction to the Supercrossing Index of Knots and the Crossing Map", C. Adams, C. Lefever, J. Othmer, S. Pahk, A. Stier and J. Tripp, Journal of Knoy t Theory and its Ramifications, Vol. 11, No. 3(2002) 445-459.) There is a fair amount of previous work on superbridge number(e.g. see the papers of Gyo Taek Jin). A variety of results on superinvariants were obtained by the SMALL knot theory group in 2007, including their utilization for the determination of stick index for many new knots. We hope to extend these results and determine stick indices for many other knots.