Math 99r Fall 2000 - Computational Algebraic Geometry

### Course Description

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.

### Instructor

Hal Schenck
421h Science Center
schenck@math.harvard.edu

### Tapas Texts

Cox, Little, O'Shea, ``Ideals, varieties, and algorithms", Springer UTM, 1997.
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 published by Cambridge University Press in the London Mathematical Society Student Text Series. The book has been reviewed in I've found a couple typos, pdf, ps.
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