Textbook : Ideals, Varieties and Algorithms by
David Cox, John Little, and Donal O'Shea, 2nd. edition, Springer-Verlag.
E-mail: Use: my lastname at math dot umass dot edu
Office: Lederle GRT 1235E
Office Hours:
Monday: 11 to 12
Wednesday: 1 to 2
Thursday: 1 to 2
Course Description
This will be a non-traditional course on Affine and Projective
Geometry. Rather than following an axiomatic approach to
Euclidean and non-Euclidean geometries, we will concentrate on the
study of the basic objects of affine and projective geometries, namely
the varieties defined as the
zeroes of polynomials. In this setting, lines and planes
correspond to varieties defined by linear polynomials. Of course,
it is much easier to deal with linear equations thanks to the theory of
linear algebra, where tools such as Gaussian elimination allow for
great computational simplification. Thus, one of our goals will
be to develop similar algorithms to
deal with non-linear polynomial equations. We will accomplish
this through the study of Gröbner bases and the use of Mathematica
to carry out most of the computations and to develop geometric
intuition through the graphing of curves and surfaces. The
main algebraic object in this course will be the study of ideals in the ring of
polynomials. No previous acquaintance with these concepts is
assumed. Solving polynomial equations is a topic of great
interest in applications, we will discuss some of this if time permits.
The goal of the course is to cover Chapters 1-4, 6, and 8 of the Cox,
Little, and O'Shea textbook.
Course Structure and Grading Policies:
There will be regular homework assignments. Some of the problems
will
be graded. No late homework
will be accepted. However, only the best 80% of your
homework grades will be used to determine your final grade.
Students will be expected to acquire a working knowledge of
Mathematica, sufficient for using the Gröbner bases
algorithms. Some Mathematica notebooks will be provided. No
previous familiarity with Mathematica is assumed.
There will be a midterm and a final exam. The final
exam will be comprehensive but will emphasize the material not covered
in the midterm. The date of the midterm exam will be announced at
least a week in advance. It is the policy of the University of
Massachusetts that final exams must be taken at the date and time
scheduled by the University.
Your homework scores will count for 25% of the grade. The
lowest exam score will count for 35% of the grade and the highest exam
score will count for the remaining 40% of the grade.
Using Mathematica:
The current version of Mathematica is installed in many campus
computers including those at the Math & Stat Resource Center (LGRT
110), at UMASTR Lab (Du
Bois Library 1667), and at all OIT-managed public labs. It includes
complete on-line documentation. You can also buy your own copy of
"Mathematica for Students" at the Campus Center store. This is
not however necessary given the wide availability of Mathematica on
campus.
The following introductory notebooks, which were prepared by Professor
Murray
Eisenberg for his Math. 421 course, may be useful to you.