Math 99r Fall 2000 - Computational Algebraic
The interplay between algebra and geometry is a beautiful (and fun!)
area of mathematical investigation. Advances in computing and algorithms
over the last quarter century have revolutionized the area, making
many (formerly inaccessable) problems tractable, and providing a fertile
ground for experimentation and conjecture. We'll begin by studying
some basic commutative algebra and connections to geometry; i.e. rings,
and ideals, varieties, the Hilbert basis theorem and nullstellensatz.
Then we'll discuss graded objects and varieties in projective space;
the Hilbert syzygy theorem says (basically) we can approximate a graded
module with a finite sequence of free modules (a finite free resolution).
The Grobner basis algorithm (which we'll study) actually lets us
compute these things; and so we have a great source of examples.
The real objective of the course is to bring whatever we choose to study
next to life by doing lots of examples. There are many paths we can take;
a homological direction (free resolutions, Tor and Ext, Hilbert syzygy
theorem), a combinatorial direction (Stanley-Reisner rings, upper
bound theorem, applications to polytopes and discrete geometry); or
an applied direction (coding theory, integer programming, mathematical
modelling, and robot control). Or we can take a tapas approach, and
study some of everything.
421h Science Center
Cox, Little, O'Shea, ``Ideals, varieties, and algorithms", Springer UTM,
Cox, Little, O'Shea, ``Using algebraic geometry", Springer GTM, 1998.
Eisenbud, ``Commutative Algebra with a view toward algebraic geometry", Springer GTM, 1995.
Stanley, ``Commutative algebra and combinatorics", Birkhauser, 1995.
Links to lots of online notes on Basic Algebraic Geometry.
Syllabus and Notes
The class notes turned into the book
Cambridge University Press in
the London Mathematical Society Student Text Series. The book has been
I've found a