Advisor: Edward Burger
Project Description:
While almost all numbers are known to be irrational, a fundamental question remains: Given an irrational, how irrational is it? Here we will consider questions from diophantine approximation-an area in which we develop insights into the structure of certain irrational quantities by approximating them by rational numbers. Thus we learn about the mysterious irrational numbers by examining how close they come to the mundane world of rationals.
Advisor: Susan Loepp
Project Description:
Consider the set of polynomials in one variable over the complex numbers. We can define a distance between these polynomials that turns out to be a metric. The cauchy sequences with respect to this metric, however, do not all converge. So, we can complete this metric space to get a new metric space in which all cauchy sequences converge. What is this new space algebraically? Surprisingly, it turns out to be the set of formal power series in one variable over the complex numbers. The idea of completing a set of polynomials generalizes to rings. The relationship between a ring and its completion is important and mysterious. We will work to unravel this mystery by constructing examples of rings whose relationship to their completion is truly bizarre.
Advisor: Frank Morgan
Project Description:
It is well known (Schwarz 1884) that a round soap bubble is the most efficient, least-area way to enclose a given volume of air. The Double Bubble Theorem (2000) says that the familiar double soap bubble is the least-area way to enclose and separate two given volumes of air (see references below). The SMALL Geometry Group, since its 1990 proof of the planar double bubble theorem, has provided many important advances, including some results in Rn, the sphere Sn, and hyperbolic space Hn. The 2005 Geometry Group will continue this noble tradition, focusing on double bubbles in Sn. We will participate in the MAA MathFest in Albuquerque, August 4-6, 2005.
References by Frank Morgan:
"Double Bubble No More Trouble," Math Horizons, November, 2000.
"Proof of the Double Bubble Conjecture," Amer. Math. Monthly, March,
2001.
Geometric Measure Theory, Academic Press, 3rd ed, 2000, chapters 13
and 14.
With Hutchings, Ritore, and Ros,
"Proof of the Double Bubble Conjecture,"
ERA AMS 6 (2000) 45-49; Ann. Math. 155 (2002) 459-489.
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 with an emphasis on the function field situation.
Some knowledge of abstract algebra is required.
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 space. An interesting class of examples is given by polynomial maps defined on some compact and open subsets of the p-adic numbers. We will study properties such as ergodicity and mixing for these maps. In particular we will consider d-dimensional versions of maps such as those discussed in "Measurable dynamics of simple p-adic polynomials" by Bryk and Silva.