Math 996 (Topics in Algebraic Combinatorics and Commutative Algebra)
Fall 2013

General Information | Final Projects | Links

• Meeting times: MWF, 9:00-9:50 AM, 408 Snow
• KU course line number: 26481
• Syllabus (8/19/13)
• Instructor: Prof. Jeremy Martin
  
  623 Snow, 864-7114
  (The best way to contact me: send me e-mail!)
• Textbook: Combinatorial Commutative Algebra by Ezra Miller and Bernd Sturmfels. Full-text access is available to all KU students from the KU Libraries website.
  Respect the publisher's copyright and do not redistribute the PDF. Also, do not print out the book on the department copier.
• Office hours: MTW 1:00-2:00 PM, 623 Snow Hall
  
• Prerequisites: Math 791 or permission of the instructor.

• Mon 8/26: Monomial ideals
• Wed 8/28: Stanley-Reisner rings; multigraded Hilbert series
• Fri 8/30: Computing Hilbert series; K-polynomial and h-vector; shellability
• Mon 9/2: No class (Labor Day)
• Wed 9/4: Shellability; consequences for h-vector and topology.
  Note: The first construction of a nonshellable Euclidean ball appeared is due to Mary Ellen Rudin (Bull. Amer. Math. Soc. 64 (1958), 90-91). It is of dimension 3 and has 14 vertices and 11 tetrahedral facets.
• Fri 9/6: A whirlwind tour of simplicial homology
• Mon 9/9: More simplicial homology
• Wed 9/11: (Minimal) free resolutions of Stanley-Reisner rings (Example 1.25 in Miller-Sturmfels)
• Fri 9/13: Koszul complex, Betti numbers, Tor
• Mon 9/16: Shellability, in particular the Bruggesser-Mani line shelling (Prof. Marge Bayer)
• Wed 9/18: Free resolutions of arbitrary monomial ideals; interpreting betti numbers
• Fri 9/20: Specializing to the squarefree case; Hochster's formula (dual version)
• Mon 9/23: Alexander duality, links, and an example of Hochster's formula
• Wed 9/25: Simplicial cohomology, the coKoszul complex, and the Alexander duality theorem
• Fri 9/27: Hochster's formula (the original version)
• Mon 9/30: Polyhedra, polytopal complexes
• Wed 10/2: Cellular resolutions I
• Fri 10/4: Cellular resolutions II
• Mon 10/7: Cellular resolutions III
• Wed 10/9: Semigroup rings
• Fri 10/11: Motivation for semigroup rings: chip-firing games and toppling ideals
• Mon 10/14: No class (Fall Break)
• Wed 10/16: Crash course in algbraic geometry
• Fri 10/18: No class (AMS sectional meeting in St. Louis)
• Mon 10/21: Toric varieties (see Mircea Mustata's excellent lecture notes)
• Wed 10/23: Lattice ideals
• Fri 10/25: Rational polyhedral cones
• Mon 10/28: Computing Hilbert bases
• Wed 10/30: Computing Hilbert bases (including Macaulay2 demonstration)
• Fri 11/1: Applications to solving integer equations; saturations and integral closures
• Mon 11/4: Polytopes and toric varieties (Prof. Marge Bayer)
• Wed 11/6: Gröbner bases I: term orders and initial ideals
• Fri 11/8: Gröbner bases II: reduction, Buchberger's criterion
• Mon 11/11: Universal Gröbner bases
• Wed 11/13: Universal Gröbner bases: geometry
• Fri 11/15: Universal Gröbner bases: linear and determinantal ideals
• Mon 11/18: Slope polynomials I (based on this paper)
• Wed 11/20: Slope polynomials II
• Fri 11/22: Minimal and reduced Gröbner bases; Gröbner bases of toric ideals
• Mon 11/25: Graver bases and universal Gröbner bases of toric ideals
• Wed 11/27: No class (Thanksgiving)
• Fri 11/29: No class (Thanksgiving)
• Mon 12/2: Multigraded polynomial rings
• Wed 12/4:
• Fri 12/6:
• Fri 12/6: Final project presentations (3-4 PM)
• Mon 12/9: Final project presentations
• Wed 12/11: Final project presentations
• Fri 12/13 (Stop Day): Final project presentations (3-4 PM)

The final project is to read a current research article in combinatorics, give a short talk on it (20 minutes, like an AMS special session talk) to an audience of fellow graduate students, and provide constructive criticism on another student's talk. Here are the full details. You should choose a paper to present no later than November 15.

Schedule of talks

All talks are in Snow 408.

Date/Time	Presenter	Paper	Reviewer
Friday 12/6 3-4 PM	Alex Lazar	D. Bayer and B. Sturmfels Cellular resolutions of monomial modules J. Reine Angew. Math. 502 (1998), 123-140	Nitin
	Ken Duna	A. Björner and M. Wachs On lexicographically shellable posets Trans. Amer. Math. Soc. 277 (1983), no. 1, 323-341	Logan
Monday 12/9 9-10 AM	Nitin Aggarwal	C. Irving and H. Schenck Geometry of Wachspress surfaces arXiv:1306.1445	Zach
	Tony Se	V. Reiner and V. Welker Linear syzygies of Stanley-Reisner ideals Math. Scand. 89 (2001), no. 1, 117-132	Rob
Wednesday 12/11 9-10 AM	Logan Godkin	R. Cori, D. Rossin and B. Salvy Polynomial ideals for sandpiles and their Gröbner bases Theoret. Comput. Sci. 276 (2002), no. 1-2, 1-15	Alex
	Rob Bradford	A. Conca Reduction numbers and initial ideals Proc. Amer. Math. Soc. 131 (2003), no. 4, 1015-1020	Billy
Friday 12/13 3-4 PM	Billy Sanders	I. Peeva Hyperplane arrangements and linear strands in resolutions Trans. Amer. Math. Soc. 355 (2003), no. 2, 609-618	Ken
	Zach Flores	T. Becker Standard bases in power series rings: ... J. Symbolic Comput. 15 (1993), no. 3, 251-265	Tony

KU links

• Jeremy Martin's home page
• KU Mathematics Department
• KU Registrar
• Fall 2013 academic calendar
• KU policies on academic honesty
• Academic Achievement and Access Center (resources for students with disabilities)

Additional resources

Here are some excellent textbooks and lecture notes that are freely available online and may be useful for this course. Get them!

• Allen Hatcher, Algebraic Topology (Cambridge, 2001)
• Mircea Mustata's lecture notes on toric varieties
• Alexander Schrijver, A Course in Combinatorial Optimization
• Richard P. Stanley, Enumerative Combinatorics, volume 1, 2nd ed. (Cambridge, 2012)
• Richard P. Stanley, Hyperplane Arrangements