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Greg Landweber Mathematics Chair Phone: (845) 758-7093 Email: gregland@bard.edu
Mathematics Program | ||
Summer Research in Mathematics & Computation
Summer 2010 dates: June 1, 2010 - July 23, 2010
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.
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 2010 application deadline: April 1, 2010
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.) - A transcript of grades (does not need to be official).
Projects
Voronoi Diagrams and Delaunay Tessellations
Lauren Rose, Associate Professor of MathematicsThe concept of a Voronoi Diagram has been independently discovered many times, and in many different fields over the past 350-400 years. Descartes used Voronoi tessellations in 1644 in a model of the universe. The definition of a Voronoi diagram was formalized in two and three dimensions in 1858 by Dirichlet, and rediscovered in 1908 by Voronoi who defined them in n-dimensional space. Applications of Voronoi diagrams can be found in a broad spectrum of fields such as computer science, biology, cartography, physiology, crystallography, engineering, and computer imaging.
The goal of this research is to determine geometric and combinatorial properties of Voronoi diagrams, their Delaunay tessellations, and their underlying graphs. This is closely linked to the inscribability and circumscribability of polyhedra. There are several different projects that are likely to lead to some interesting results and a better understanding of Voronoi diagrams and related Combinatorial and Geometric Objects.
Prerequisites: Multivariable Calculus, Linear Algebra, an introductory proofs-based course, and at least one upper level proofs-based course. Computer programming experience and a background in Abstract Algebra or Combinatorics is helpful but not required.Supersymmetry, Graphs, and Codes
Greg Landweber, Assistant Professor of MathematicsIn physics, supersymmetry is a pairing between bosons and fermions appearing in theories of subatomic particles. To study supersymmetry, Faux and Gates recently introduced Adinkras, graphs with vertices representing the particles in a supersymmetric theory and edges corresponding to the supersymmetry pairings.
Although Adinkras arise in the study of supersymmetric physics, participants will study them from a pure mathematics point of view, starting with their basic axioms. In graph-theoretic terms, Adinkras are N-regular, edge N-colored bipartite graphs with heights assigned to the vertices and signs assigned to the edges, satisfying various conditions. Adinkras are closely related to Clifford algebras, and we have recently discovered that the underlying graph of an Adinkra is actually the Schreier coset graph corresponding to a doubly-even linear binary error correcting code. Participants will study Adinkras from various points of view, including graph theory, linear algebra, abstract algebra, and coding theory. Also, participants will be enlisted in the ongoing project of classifying the simplest examples of Adinkra graphs, using computational software such as Sage and Mathematica as necessary.
Prerequisites: Linear Algebra and an introductory proofs-based course. Students must also have at least one of the following: Abstract Algebra, Combinatorics, Graph Theory, Coding Theory, or computer programming experience. No background in physics is required.
For more information, see:
Neural and Biomechanical Modeling
Sven Anderson, Associate Professor of Computer ScienceHow do the brains and vocal apparatus of humans and passerine birds come to produce speech and song? Can mathematics help explain neural processing and thought? The students in this group project will build and study dynamic systems models that explore how birdsong is produced at both the neural and biomechanical levels. Students will extend models that the faculty mentor has developed in collaboration with undergraduate students. Members of the group will work on mathematical analysis and computational simulation of realistic spiking neural networks and related sound production mechanisms. Our long-term goal is to combine these two modeling projects and evaluate their accuracy with respect to the bigger problem of understanding song learning in passerine birds.
Applicants to this project should be interested in both applied mathematics and biological modeling. They will be mentored to develop a solid grounding in model building, analysis, and simulation within the context of understanding a complex system.
Prerequisites: Differential Equations and the equivalent of one semester of computer programming experience. Linear Algebra, Probability, and Mechanics are also beneficial but not required.
For more information, see:
Ecological Stability Analysis
Gidon Eshel, Bard Center Fellow in Environmental Physics and Applied MathematicsTransient 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: