Summer Research in Mathematics & Computation

The online application is now live on MathPrograms.org.

Summer 2012 dates: June 4, 2012 - July 27, 2012

The Bard College REU in Mathematics & Computation is an eight week summer program in which undergraduates work in groups with individual faculty members on active and current research projects in pure and applied mathematics, mathematical physics, and mathematical computation. Throughout the summer, students will be exposed to theoretical constructs, computational techniques, and real world applications, and they will develop the background and skills to prepare them for further study or employment in a mathematics-related career. The REU will also provide workshops on LaTeX, Sage, Matlab, and other mathematical software.

Bard College is situated on over 500 acres along the Hudson River, 90 miles north of New York City (Campus Map and Tour). Workshops, seminars, and lectures will held in the state-of-the-art Gabrielle H. Reem and Herbert J. Kayden Center for Science and Computation, opened Fall 2007, and students will be provided space in a dedicated computer lab. REU participants will be part of a summer research community of dozens of students, with regular social events.

Participating students will receive a stipend of $3200, free double-occupancy on-campus housing, an allowance for food, and up to $800 in travel expenses. This financial support and free housing is available only to U.S. citizens or permanent residents.

Please email questions to mathreu@bard.edu. (This address is NOT for submitting application materials or letters of recommendation. We will NOT read any such materials or letters sent by email or postal mail. All application materials and letters of recommendation MUST be submitted via MathPrograms.org.)

Eligibility

We welcome applications from both current sophomores and juniors, including students who have decided to major in mathematics as late as their junior year and students who are undecided about a career in mathematics. Applicants must not yet have graduated with an undergraduate degree at the time of the REU.

Women and underrepresented groups are particularly encouraged to apply.

Due to National Science Foundation restrictions, financial support is available only to U.S. citizens or permanent residents. Students who are not eligible for financial support are still welcome to apply, provided they can obtain alternative funding from their home institution or another source.

How to Apply

Summer 2012 application information.

For full consideration, please submit your application by February 15, 2012. Applications will continue to be accepted after that date until all position have been filled.

Applications must be submitted online via MathPrograms.org, and include:

• A curriculum vitae.
• A short essay, less than one page, addressing the following points:
  • How did you become interested in mathematics research?
  • Describe your future career goals.
• Two letters of reference from professors who can discuss your research potential.
  (Letters of recommendation MUST be submitted online via MathPrograms.org.
  Once you specify your letter writers on your online application, they will
  be sent emails explaining how to submit their letters online via MathPrograms.org.
  We will NOT read any letters submitted by email or postal mail.)
• A transcript of grades (does not need to be official).

The topic for this project will be chosen from the following:

Voronoi Diagrams and Delaunay Tessellations
The goal of this research is to explore geometric and combinatorial properties of Voronoi diagrams, their Delaunay tessellations, and their underlying graphs.

Derived Graphs of Voltage Graphs
Voltage Graphs are graphs whose edges are labeled with elements of a group. For each labeling, there is an associated graph called the derived graph. We will explore various properties of these pairs of graphs.

Combinatorial Aspects of Bivariate Splines
A bivariate spline is a piecewise polynomial function whose domain is a region of the plane broken up into triangles. What types of splines can occur depends on the number of triangle and how they fit together.

Prerequisites: Multivariable Calculus, Linear Algebra, an introductory proofs-based course, and at least one upper level proofs-based course. Computer programming experience and a course in Abstract Algebra, or Combinatorics, or Graph Theory is helpful but not required.

Arithmetic in Iterated Towers of Number Fields

One of the most fruitful recent developments in Number Theory is the subject of Arithmetic Dynamics. This area of mathematics typically centers around the number-theoretic properties of iterated rational maps. Some well-studied examples are: the prime divisors of recursively-defined sequences; the geometric properties of iterated isogenies of elliptic curves; and p-adic fractals.

Gauss' law of quadratic reciprocity tells us how primes decompose in quadratic extensions of the rational numbers Q. Given an irreducible polynomial f(x) with integral coefficients, one can construct an iterated tower of number fields by looking at the splitting fields of f(x), f(f(x)), f(f(f(x))), etc. In this project we will work on "iterated reciprocity laws", where we study how primes decompose in these extensions of Q.

Prerequisites: A strong background in Abstract Algebra. This can include: a good knowledge of groups, rings, and fields; some exposure to Galois theory; or coursework in algebraic geometry.

Using Machine Learning to Simplify Text

How many words are needed to accurately express the content of a typical Wikipedia article? Can words, sentences, and even ideas be simplified automatically by a computer to make them more easily understood by children, second language learners, robots, and others who have only mastered a portion of English? This project explores how statistically based machine learning can be modified and thereby used to translate between a complex text and a simplified version.

Applicants to this project should be interested in both applied mathematics, computational linguistics, and artificial intelligence. In the course of this project they will develop a deeper understanding of Markov models, statistical learning, system optimization and human language.

Prerequisites: Applicants for this project should have completed one year of calculus, a least one course in probability and/or statistics, and also have intermediate programming skills.

Ecological Stability Analysis

Transient stability of natural and managed ecosystems

Modern ecology relies heavily on models, and emphasizes strongly linear stability as a theoretical and practical conservation tool. Most often, stability is interpreted in the context of traditional, eigenvalue based, analysis. Yet actual ecosystems and agroecosystems are extremely rarely quiescent and docile enough for the asymptotically valid eigenvalue based state to be realized. Instead, a much more useful view of such systems is as stochastically forced, transiently responding ones in which variance is maintained in the presence of asymptotic stability by the constant energy input into the system by the stochastic forcing. In this project, we will address the transient stability of semi-realistic and nearly realistic ecosystems. The emphasis is somewhat flexible and can be agroecosystems or natural ones. In the former, the main question is this: can diverse agroecosystems exhibit larger resilience, all else being equal, then their more homogeneous counterparts? In the latter case, the emphasis may be placed on Mediterranean oak forests and their response to climate change. Either way, the main tool will be generalized stability theory.

Interactions of north American agricultural land with the midlatitude westerlies, and their effects on potential cyclogenesis

The North American continent is a main conduit through which the global midlatitude westerlies complete their circumnavigational path. Since the 1700s, just under half of that surface area has been farmland. This land use change has profound effect on the surface properties as seen by the overlaying westerlies, and thus affects the available potential energy budget of the traversing westerlies, and consequently their potential to support explosive variance growth. In this project, we will use data and theory to better quantify some of the main effects of the grand North American land use change.

Prerequisites: Linear Algebra, Introductory Physics, and either Differential Equations or Dynamical Systems. Some background with statistics is recommended. Intimate familiarity with unix/linux machines and coding experience is a must.

For more information, see: