Mathematics Program at Bard College
Greg Landweber
Mathematics Chair
Phone: (845) 758-7093
Email: gregland@bard.edu

Mathematics Program
Bard College
P.O. Box 5000
Annandale, NY 12504

Summer Research in Mathematics & Computation

Our summer 2013 REU is canceled. Thank you for your applications.

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 $4000, free double-occupancy on-campus housing, 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.

Projects

Supersymmetry, Graphs, and Codes

Greg Landweber, Assistant Professor of Mathematics

In 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.

Topics in Discrete Geometry

Lauren Rose, Associate Professor of Mathematics

Students will work on one of two different projects, depending on their backgrounds and interests.

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.

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.

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.

Gerrymandering: Theory and Practice

Jeff Suzuki, Associate Professor of Mathematics, Brooklyn College

In the United States and other modern democracies, voters are partitioned into legislative districts, and the voters within a district elect someone to represent them in the legislature. Gerrymandering is the practice of dividing the districts in such a fashion as to give one political party an unfair advantage.

The ultimate goal of this research is to derive measures of district shape that can be used to assess districting plans, with an eye towards practical utility (e.g., incorporating such measures into evaluations of districting plans by both legislatures and the courts). There are several directions this research might take:

  • Investigation into whether gerrymandering is an inevitable result of representative democracies. Researcg suggests that gerrymander-proof systems will, under certain seemingly weak conditions, disenfranchise groups of voters. What additional conditions must we impose to ensure that such disenfranchisement does not occur in a gerrymander-proof system?

  • Defining what an "ideal" partition of a geographic region into districts should be then, given the ideal partition, measuring a particular districting plan's deviation from ideal and using hypothesis testing to evaluate if the deviation is too great to be attributed to chance.

  • Investigating the correlation between existing measures of district size/shape and currently permitted districts; also the correlation between the measures and (court-determined) illegal partisan and/or racial gerrymanders.

  • Using some assumptions on voter behavior, deriving limits for optimal gerrymandering. Due to long-term partisan shifts, a gerrymander that gives one party a political advantage in the short term could place the party at a significant disadvantage over the long term (producing what pundits call a "dummymander").

    Prerequisites: Applicants for this project should have completed multivariable calculus and taken at least one upper-level applied mathematics course, such as differential equations, numerical analysis, or mathematical modeling. Coursework or experience with probability, statistics, geometry, GIS, and programming will be helpful, but is not required.

    Using Machine Learning to Simplify Text

    Sven Anderson, Associate Professor of Computer Science

    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.

    Self-Distributivity and Computation

    Bob McGrail, Assistant Professor of Computer Science and Mathematics

    We will consider the implications of a general characterization of the P = NP problem into finite algebras with self-distributive operations. Algebras with self distributive binary operations arise in many different areas of mathematics. For example, quandles constitute a strong invariant for the classification of three-dimensional knots and also arise from group conjugation. Racks are employed in a similar fashion for braids. Certain left distributive algebras, such as Laver tables, are derived from elementary embeddings in large cardinal set theory.

    It is expected that many key questions will be settled via counterexample using computational discovery. That is, the project team will create computer programs in systems such as Mathematica, Mace4, and other systems to find counterexamples that refute claims.

    References:
  • D. Hobby and R. N. McKenzie. The Structure of Finite Algebras, volume 76 of Comtemporary Mathematics. American Mathematical Society, Providence, R. I., 1988.
  • Golbus, McGrail, Przytycki, Sharac, and Chakarov. Tricolorable torus knots are NP-complete. Proceedings of the 47th Annual ACM Southeast Regional Conference. April, 2009.

    Autonomous Mobile Projector-Camera Systems

    Keith O'Hara, Assistant Professor of Computer Science

    Mixed Reality (MR) and Augmented Reality (AR) are terms used to describe technologies that visually blend the physical and virtual worlds. Compared to the interest in wearable mobile projector-camera systems for AR/MR, there has been relatively little research on using mobile projector-camera systems on mobile robots. The Intelligent Mobile Projector (IMP) system which combines recent advances in mobile robot and projector-camera system research. Coupling a projector with a camera, creates a projector-camera system which can provide a large interaction surface available to multiple users at the same time. Our overarching research question is whether mobility can be used to create expansive mixed-reality interfaces.

    Placing a projector-camera system on a mobile robot has a variety of uses. First, the robot can use the projector-camera system as an inexpensive range scanner to detect obstacles or build 3D maps - for which the Microsoft Kinect infrared projector-camera sensor has found widespread use. Second, the projector-camera system can provide new ways of interacting with a robot. Third, a mobile robot can actively reorient the projector to follow a moving person or find suitable projection surfaces. Finally, by exploiting the autonomy, a new type of mixed reality lets the robot control not only what is projected, but also where the scene is projected.

    References:
  • Aaaron Strauss, Anis Zaman, and Keith O'Hara. The IMP: An Intelligent Mobile Projector. In AAAI Exhibition and Robot Workshop, 2010.
  • I. Garcia Dorado and J. Cooperstock. Fully automatic multi-projector calibration with an uncalibrated camera. In PROCAMS11, pages 29-36, 2011.

    Prerequisites: Multivariable Calculus, Linear Algebra, at least one course in probability and/or statistics, and intermediate programming skills.