Math 662: A tapas of algebra, geometry, and
combinatorics

### Instructor

Hal Schenck

620B Blocker

schenck@math.tamu.edu

http://www.math.tamu.edu/~schenck/VIGRE.html

### Students Enrolled

Jenny Gilmore (undergrad) and Xianjin Chen, Marvin Decker, Scott
Evans, Larry Holifield, Lanvaya Kannan, Youngdeug Ko, Vince
Lemoine, Rober Main, Roel Morales, Jody Wilson and Zhigang Zhang
(graduate students).

### 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. We really will take a
tapas approach, studying homological algebra (free resolutions, Tor
and Ext, Hilbert syzygy theorem), algebraic combinatorics and
algebraic topology (simplicial homology, Stanley-Reisner rings,
upper bound theorem, applications to polytopes and discrete
geometry); and classical algebraic geometry (the geometry of points
and curves in projective space).