REU/Math 685: Algorithmic Algebraic Geometry and Applications (Summer 2009)

REU/Math 685: Algorithmic Algebraic Geometry and Applications

MAIN LECTURES on M-Th: 11:00-12:00 & 14:00-15:00
in Blocker 111.

Instructor: Prof. J. Maurice Rojas
E-mail: rojas@math.tamu.edu, Office Phone: (979) 845-2083
Web Page: http://www.math.tamu.edu/~rojas
Office Hours: By appointment, in Milner 206

Assistant Instructor: Prof. Scott Zrebiec
E-mail: zrebiec@math.tamu.edu, Office Phone: (979) 845-0261
Web Page: http://www.math.tamu.edu/~zrebiec
Office Hours: By appointment, in Milner 224

Assistant Instructor: Ashraf Ibrahim
E-mail: aibrahim@math.tamu.edu, Office Phone: (979) 845-1897
Web Page: http://www.math.tamu.edu/~aibrahim
Office Hours: By appointment, in Blocker 611B

HANDOUTS, HOMEWORK, AND OTHER ANNOUNCEMENTS
POLICIES AND SYLLABUS

MAIN REFERENCES:

  • [PS04] Algebraic Statistics for Computational Biology, by Lior Pachter and Bernd Sturmfels, Oxford University Press, 2004.
  • [Roj09] Algorithmic Complexity in Algebraic Geometry, by J. Maurice Rojas, in preparation.
  • Course Reader (a constantly growing collection of excerpts from recent books, expository papers, and research papers).

    SUPPLEMENTAL REFERENCES:

  • [Ahl78] Complex Analysis, Lars V. Ahlfors, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., 1978.
  • [BS96] Algorithmic Number Theory, Eric Bach and Jeff Shallit, MIT Press, 1996.
  • [BCSS98] Complexity and Real Computation, by L. Blum, F. Cucker, M. Shub, and S. Smale, Springer-Verlag 1998.
  • [CLO97] Ideals, Varieties, and Algorithms, by David A. Cox, John B. Little, and Donal O'Shea, Springer-Verlag, 1997.
  • [CLO98] Using Algebraic Geometry, by David A. Cox, John B. Little, and Donal O'Shea, Springer-Verlag, 1998.
  • [DSS08] Lectures on Algebraic Statistics (Oberwolfach Seminars), M. Drton, B. Sturmfels, and S. Sullivant, Birkhauser Basel, 2008.
  • [Ewa96] Combinatorial Convexity and Algebraic Geometry, by Gunter Ewald, Springer-Verlag, 1996.
  • [GKZ94] Discriminants, Resultants, and Multidimensional Determinants, by Israel M. Gel'fand, Misha M. Kapranov, and Andrei V. Zelevinsky, Mathematics: Theory & Applications, Birkhauser Boston, Inc., Boston, MA, 1994.
  • [IMS09] Tropical Algebraic Geometry, by Ilia Itenberg, Grigory Mikhalkin, and Evgenii Shustin; Oberwolfach Seminars, Birkhauser Basel, 2nd ed., 2009.
  • [Pap94] Computational Complexity, by Christos H. Papadimitriou, Addison-Wesley, Reading, MA, 1994.
  • [Sch86] Theory of Linear and Integer Programming, by Alexander Schrijver, Wiley-Interscience Series in Discrete Mathematics, 1986.
  • [Stu02] Solving Systems of Polynomial Equations, by Bernd Sturmfels, CBMS Lecture Series, AMS Press, 2002.
  • [Zie95] Lectures on Polytopes, Gunter M. Ziegler, Springer-Verlag, 1995.

































    Detailed Course Information
    Prerequisites:
    Linear algebra and an open mind. On occasion, we'll use some basic notions from topology, such as compactness and denseness.
    Syllabus:
    We'll cover topics in numerical analysis (NA) not usually covered in any algebraic geometry class, topics in algebraic geometry (AG) not usually covered in any numerical analysis class, and topics in algorithmic complexity not usually covered in any course on NA or AG.
    Topics include: Applications of polynomial system solving, Homotopy inspired proof of the Fundamental Theorem of Algebra, Descartes' Rule of Signs and its Generalizations, Algebraic Statistical Models, the Gelfond-Kazarnovski-Khovanski-Voorhoeve Theorem on Complex Fewnomials, Binomial Systems, $A$-discriminants, Connections to Tropical Geometry, Smale's 17th Problem, the P=NP? Question.
    Grading Policy: Weekly Homeworks (60%) + Participation (40%)
    Students With Disabilities:
    Every effort will be made to accomodate your specific needs --- just discuss the matter with me at the beginning of the semester. Please contact the Office of Services for Students with Disabilities (845-1637) if you need any additional assistance.
  • Please see http://www.math.tamu.edu/~rojas/hqfreu09.html for further details on homeworks, handouts, projects, and other announcements.

  • Copyright Information: Please note that all written and web materials for this course have an implied copyright. In particular, you can xerox (or download) ONE copy for your own use, but you may not reproduce them for others. SELLING THIS MATERIAL IS EXPRESSLY FORBIDDEN, VIOLATES COPYRIGHT LAWS, AND WILL BE VIGOUROUSLY PROSECUTED.