Fletcher Jones Summer 2011 Research Projects

Distribution of Two-Dimensional Well-Rounded Lattices

 

Supervised by Lenny Fukshansky
Department of Mathematics, Claremont McKenna College

 

Project Description 

Lattices are discrete periodic structures in Euclidean spaces, which are fundamental objects in Number Theory and Discrete Geometry. Their importance in mathematics is emphasized by numerous applications in Discrete Optimization, Digital Communications, Coding Theory, Cryptography, Computer Science, and many other areas of science and engineering. More precisely, let $ N \geq 2 $ be an integer, then a lattice $ \Lambda $ in $ N $-dimensional Euclidean space $ \mathbb R^N $ is a set of integral linear combinations of a collection of basis vectors, say $ \boldsymbol x_1,\dots,\boldsymbol x_N $, so

$$\Lambda = \left\{ \sum_{i=1}^N a_i \boldsymbol x_i : a_i \in \mathbb Z\ \forall\ 1 \leq i \leq N \right\}.$$

Define the minimal norm of $ \Lambda $ to be

$$|\Lambda| = \min_{\boldsymbol x \in \Lambda \setminus \{\boldsymbol 0\}} \|\boldsymbol x\|,$$

and let

$$S(\Lambda) = \{ \boldsymbol x \in \Lambda : \|\boldsymbol x\| = |\Lambda| \}$$

be the set of minimal vectors of $ \Lambda $. This set is always finite, and in each dimension there is a bound on how large it can be. For many applications, it is important that the set $ S(\Lambda) $ is large and contains many linearly independent vectors: this insures that the lattice is especially symmetric. Of course the maximal possible number of linearly independent vectors $ S(\Lambda) $ can contain is $ N $, in which case $ \Lambda $ is called well-rounded. Well-rounded lattices come up frequently in a variety of contexts in Discrete Geometry, Combinatorics, and Optimization. In this project, we will study various geometric, arithmetic, and analytic properties of well-rounded lattices in $ \mathbb R^2 $ with a view towards optimization problems. Our specific interest will be to understand how well-rounded lattices are distributed among all lattices in the plane. The ideal candidates for this project will have some background in Abstract Algebra; some knowledge of Real Analysis is also helpful, although not absolutely necessary.

 

Yang-Baxter Equations and Integrable Systems

 

Supervised by Gizem Karaali
Department of Mathematics, Pomona College

 

Project Description

Here is an innocuous-looking collection of mathematical symbols:

$$[r^{12},r^{13}] + [r^{12},r^{23}] + [r^{13},r^{23}] = 0.$$

This equation, called the classical Yang-Baxter equation (CYBE), will be the main focus of our project this summer. Much active research revolves around the CYBE and its many relatives. Our group will focus on exploring the various meanings of the CYBE and relating it to the objects of classical mechanics. In particular our main goal will be to construct (precise mathematical descriptions for) specific mechanical systems (those nice ones called integrable systems) arising from solutions of the CYBE. Along the way we may even get a glimpse of how modern physics and modern mathematics are inextricably intertwined, and learn about our most current physical theories of the universe, including the standard model and supersymmetries.

This is a well-established path of research, and there are several research articles that we will read through the summer. Students will be exposed to several big ideas of mathematics, in particular in representation theory, Lie theory, mathematical physics, classical mechanics. This list may sound intimidating, but the summer will provide us ample time to learn! The only prerequisites from the students at the beginning are: 

  • A course in abstract algebra (e.g., Math 171) and a solid understanding of linear algebra (e.g., Math 60)
  • Some curiosity about mathematical physics (no previous exposure to physics beyond high school is required)

Students who have some physics training and those who have taken a course in representation theory (e.g., Math 174) may find the background of the project more accessible, but neither of these will be necessary or expected.

 

Linear Rank Tests of Uniformity: Generalizations, Symmetries and Algorithms

 

Supervised by Michael Orrison
Department of Mathematics, Harvey Mudd College

 

Project Description

Suppose a group of judges has been asked to rank a set of alternatives from most to least preferred. For example, the judges might be mathematics majors who have been asked to rank a set of elective courses. How might we use the resulting rankings to determine if the judges perceive any significant differences among the alternatives?

For convenience, we\'ll assume that the judges’ rankings are governed by a single probability distribution $ P $ defined on the collection of all possible rankings of the alternatives.  In this case, the lack of any perceived differences among the alternatives would manifest itself as $ P $ being the uniform distribution.  As such, we might consider using a test of uniformity to test the null hypothesis that $ P $ is the uniform distribution.

The focus of this project will be on a collection of tests known as linear rank tests of uniformity.  These tests use straightforward summary statistics that are the result of applying simple linear transformations to the rank data generated by the judges. Interestingly, different linear rank tests of uniformity can lead to conflicting results when applied to the same rank data.  Furthermore, these tests can be generalized to a multitude of settings that go well beyond rank data.

This summer, our team will explore several generalizations of some of the best known linear rank tests of uniformity.  In particular, we will study situations in which judges are asked to choose items from sets that are highly symmetric, and we will develop tests of uniformity that are then based on the symmetries of those sets. We will also create accompanying efficient algorithms for running our resulting tests of uniformity.

The ideal student researchers for this project will have had solid introductory course work in linear algebra, group theory, statistics, and computing.  Advanced course work in statistics and abstract algebra is highly desirable, but not necessary.