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


Hal Schenck
620B Blocker

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).