Projects

The term paper is to be a ten-page essay on a topic related to the course. The goal is for you to learn something new, and to explain it clearly to others in the class, or better, to other upper-class math majors. The paper must be written in a professional style, and formatted in AMS-LaTeX, like the papers in MIT's Undergraduate Journal of Mathematics. Some helpful resources are found in the study materials section. If you do a good job on your paper, then, possibly after further editing, it can be published in the next volume. (MIT students only.)

Term Paper Key Dates

• Ses #16: Outline due
• Ses #21: First draft due
• Ses #26: Final draft due

Below is a list papers written for 18.704 in previous terms, to give you some idea of possible topics. These papers were published in MIT's Undergraduate Journal of Mathematics (not available to OCW users).

Volume 1, 1999

• Paul Grayson, Robotic Motion Planning, pp. 57-67.

Volume 2, 2000

• Ted Allison, Complexity of Computations of Ideal Membership, pp. 1-9.

Volume 3, 2001

• Ethan Cotterill, Syzygies over Polynomial Rings, pp. 29-41.
• Geoffrey L. Goodell, Algebraic Coding Theory, pp. 71-80.
• Matt Menke, Running time of Groebner Basis Algorithms, pp. 145-151.
• Brian D. Smithling, A Proof of Hilbert's Syzygy Theorem, pp. 199-207.

Volume 4, 2002

• Peter Ahumada, An Algorithm for Integer Programming Problems, pp. 1-11.
• Nicholas Cohen, Automatic Geometric-Theorem Proving, pp. 29-38.
• Leah Schmelzer, Implicitization via Resultants, pp. 179-188.

Volume 5, 2003

• Eric Schwerdtfeger, An Introduction to Symmetric Polynomials, pp. 265-273.

Volume 6, 2004

• Paul Gorbow, Ideals from Graphs, pp. 69-84.

Volume 9, 2007

• Hyeyoun Chung, Computing Invariants of Finite Groups, pp. 11-29.
• Anand Deopurkar, Normalization of Algebraic Varieties, pp. 43-63.
• Pablo Solis, Splines on a Finer Subdivision, pp. 133-142.

Volume 10, 2008

• Alessandro Chiesa, Companion Matrices for Systems of Polynomial Equations, pp. 31-41.
• Philip Engel, Colorings and Cycles of Graphs, pp. 43-52.

Our text, Ideals, Varieties, and Algorithms, describes a number of possible topics in Appendix D.

More topic possibilities are found in the following books:

Adams, William W., and Philippe Loustaunau. An Introduction to Gröbner Bases. Providence, RI: American Mathematical Society, 1994. ISBN: 9780821838044.

Cox, David A., John B. Little, and Donal O'Shea. Using Algebraic Geometry. Graduate texts in mathematics, 185. New York, NY: Springer, 2005. ISBN: 9780387207063.

Cox, D., and B. Sturmfels. "Applications of Computational Algebraic Geometry, Lectures Presented at the AMS Short Course held in San Diego, CA, January 6-7, 1997." Proceedings Symposia Applied Math, 53, AMS Short Course Lecture Notes, Amer. Math. Soc., 1998.

Dickenstein, Alicia, and Ioannis Z. Emiris. Solving Polynomial Equations: Foundations, Algorithms, and Applications. Algorithms and computation in mathematics, vol. 14. Berlin, Germany: Springer, 2005. ISBN: 9783540243267.

Eisenbud, David. Commutative Algebra with a View Toward Algebraic Geometry. Graduate texts in mathematics, 150. New York, NY: Springer-Verlag, 1995. ISBN: 9783540942696.

Greuel, G.-M., and Gerhard Pfister. A Singular Introduction to Commutative Algebra. Berlin, Germany: Springer, 2002. ISBN: 9783540428978.

Schenck, Hal. Computational Algebraic Geometry. London Mathematical Society student texts, 58. Cambridge, UK: Cambridge University Press, 2003. ISBN: 9780521536509.

CBMS Conference on Solving Polynomial Equations, and Bernd Sturmfels. Solving Systems of Polynomial Equations. Providence, RI: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 2002. A preliminary edition is available. (PDF)

Vasconcelos, Wolmer V., and David Eisenbud. Computational Methods in Commutative Algebra and Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 2. Berlin, Germany: Springer, 1997. ISBN: 9783540605201.