# Combinatorial Commutative Algebra

due dates.

o Wednesday, April 8: rough outline. (1-2 pages)

o Monday, May 18: final project. (10
pages in LaTeX, 11pt, single space)

For the final project of the course, you will develop a solid understanding
of a particular aspect

of combinatorial commutative algebra (CCA) that interests you. You may, for
instance:

• Understand the background and significance of an open problem in CCA, and
solve it, or

achieve some partial progress.

• Understand the current state of the art in a branch of CCA, and present it in
a clear,

concise, and useful survey.

• Find a new way of thinking about or proving a known result.

• Write a computer program that will be useful to researchers in CCA.

Below are some possible topics for a final project, in no particular order.
The list is not

comprehensive, and probably biased towards topics that I like. You may choose
your own topic,

but you’ll need me to approve it before you start working on it. I am flexible
about the topics

that you choose, but you must prove to me that you learned a lot of mathematics
related to CCA!

In your proposal you will describe your concrete plan of action and I will offer
feedback.

**some suggested sources.**

The existing literature on CCA is large, deep, and broad; there are many
topics within your reach.

Many papers and books contain interesting open problems which you can understand
and think

about. Search on Google Scholar, the math arXiv, and the American Math.
Society’s mathscinet.

Many of the suggested texts for the course have exercises and comments which
provide good

project directions. Some concrete project suggestions and open problems are in:

1. Cox, Little, O’Shea. Ideals, varieties, and algorithms. Appendix D.

2. D. Eisenbud. Commutative algebra with a view towards algebraic geometry,
particularly

Sections 15.10.9 (open algorithmic questions), 15.12 (computer algebra
projects).

3. R. Stanley. Combinatorics and commutative algebra, p. 135-143.

4. R. Stanley. Positivity problems and conjectures in algebraic combinatorics.

## federico ardila

**some suggested general topics.**

You should be able to find the references below on google, the arxiv, or
mathscinet. You may

need to be creative if you need access to a book that your library doesn’t have.
Let me know if

you have looked carefully and still can’t find some of the references I mention.

**1. The g-theorem and the Upper Bound Conjecture.** The f-vector of a polytope
keeps track of

the number f_{i} of i-dimensional faces. The g-theorem characterizes all possible
f-vectors of

simplicial polytopes. Stanley proved this conjecture using tools from
commutative algebra.

Several extensions, related results, and conjectures that followed.

T.Hibi. Algebraic combinatorics on convex polytopes.

R. Stanley. Combinatorics and commutative algebra.

**2. The cd-index.** A poset is Eulerian if it satisfies a condition that
makes it look like the face

poset of a polytope. Some of the structure of an Eulerian poset is elegantly
encoded in its

cd-index, which has nice properties.

R. Stanley. Combinatorics and commutative algebra.

Billera, L. J., R. Ehrenborg and M. Readdy, The cd-index of zonotopes and
arrangements.

**3. Shellability.** Shellability is a combinatorial condition on a
simplicial complex which implies

many nice algebraic properties. Many nice families of combinatorial simplicial
complexes

are known or conjectured to be shellable.

R. Stanley. Combinatorics and commutative algebra.

**4. Characterizations of Hilbert functions. **What are the possible
Hilbert functions of a graded

ring? Macaulay gave a beautiful characterization for rings satisfying certain
mild conditions.

There are many subsequent variants, strengthenings, and related conjectures.

Stanley. Combinatorics and commutative algebra.

**5. Ehrhart theory.** Given a lattice polytope P, the number E_{P}(n)
of integer points in the scaled

polytope nP is given by a polynomial in n called the Ehrhart polynomial. This
polynomial

has close ties to CCA.

M. Hochster. Rings of Invariants of Tori, Cohen-Macaulay Rings Generated by
Monomials,

and Polytopes.

Stanley. Combinatorics and commutative algebra.

6. Splines on simplicial complexes. A spline on a simplicial complex
is a continuous function

on which is polynomial on each face, and is differentiable to a specified order.
Applications

include numerical analysis and computer graphics.

L. Billera. Homology of smooth splines: generic triangulations and a
conjecture of Strang.

L. Billera and L. Rose. Gr¨obner basis methods for multivariate splines.

R. Stanley. Combinatorics and commutative algebra.

7. Box splines and systems of linear equations. Dahmen and Michelli, among
many others,

showed how the theory of box splines in approximation theory can be applied to
study the

space of nonnegative integer solutions to a system of linear equations.

Dahmen-Michelli, On the number of solutions to systems of linear diophantine
equations

and multivariate splines.

C. De Concini, C. Procesi. The algebra of the box spline. arXiv:0602.5019

O. Holtz and A. Ron. Zonotopal Algebra. arXiv:0708.2632

8. Magic squares. Let H_{n}(r) be the number of n × n N-matrices
whose row sums and column

sums are equal to r. Stanley and Jia used the CCA approach to Ehrhart theory and
box

spline theory to study this function, and offer some related open problems.

Rong-Qing Jia. Multivariate discrete splines and linear diophantine equations.

R. Stanley. Combinatorics and commutative algebra

9. Box splines and index calculations. De Concini, Procesi, and Vergne
generalized aspects of

box spline theory in order to perform computations in the index theory of
elliptic operators.

C. De Concini, C. Procesi, M. Vergne. Vector partition function and generalized
Dahmen-

Micchelli spaces. arXiv:0805.2907

C. De Concini, C. Procesi, M. Vergne. Vector partition functions and index of
transversally

elliptic operators. arXiv:0808.2545

10. Power ideals, fat point ideals, Cox rings. A point configuration in a
vector space determines

several algebraic objects with beautiful combinatorial structure.

F. Ardila and A. Postnikov. Combinatorics and geometry of power ideals.
arXiv:0809.2143

Geramita and Schenck. Fat Points, Inverse Systems, and Piecewise Polynomial
Functions.

B. Harbourne. Problems and Progress: A survey on fat points in P2.

**11. Topology of hyperplane arrangements.** Many topological and
algebraic properties of hyperplane

arrangements can be understood in terms of their combinatorics.

Orlik, Terao. Arrangements of hyperplanes.

**12. Schubert calculus.** The Grassmannian variety, which is the space of
k-dimensional subspaces

of an n-dimensional space, can be stratified into “Schubert varieties”. This
construction is

useful in topology, representation theory, enumerative algebraic geometry, and
symmetric

functions, among others.

Miller and Sturmfels. Combinatorial commutative algebra.

Fulton. Young tableaux.

Manivel. Symmetric functions, schubert polynomials and degeneracy loci.

**13. Gr¨obner bases and polytopes.** An ideal I has different initial
ideals with respect to different

term orders. Study the Grobner fan of an ideal I, a geometric object which
controls these

initial ideals.

Sturmfels. Grobner bases and convex polytopes.

**14. Triangulations of polytopes and toric ideals. **There is a
correspondence between initial ideals

of a toric ideal and the subdivisions of a polytope. The secondary polytope of a
polytope has

faces corresponding to its (regular) subdivisions. The toric Hilbert scheme is
the parameter

space of ideals with the same Hilbert function as a given toric ideal, and it
can be analyzed

in terms of the triangulations of a polytope.

**15. Systems of polynomial equations.** There are nice connections
between a system of polynomial

equations and the combinatorics of the corresponding Newton polytope, such as
Bernstein’s

theorem and Khovanskii’s theorem on systems of equations with few monomials.

B. Sturmfels. Solving systems of polynomial equations.

**16. Applications of polynomial equations.** One can use the tools we’ve
learned to study several

polynomial systems of equations arising in economics, statistics, and
phylogenetics.

B. Sturmfels. Solving systems of polynomial equations.

M. Drton, B. Sturmfels, S. Sullivant. Lectures on Algebraic Statistics

L. Pachter and B. Sturmfels. Algebraic Statistics for Computational Biology,

**17. More applications of polynomial equations.** Other applications
include motion planning for

robots, and algorithms for automatically proving theorems in Euclidean geometry.

Cox, Little, O’Shea. Ideals, varieties, and algorithms.

**18. Tropical geometry.** Tropical geometry studies algebraic varieties
by “tropicalizing” them

into polyhedral complexes that retain some of their structure.

J. Richter-Gebert, B. Sturmfels, T. Theobald. First steps in tropical geometry.

D. Maclagan and B. Sturmfels. Introduction to Tropical Geometry. (draft,
online.)

**19. Invariant theory.** When a group acts on a polynomial ring, it is of
interest to understand

the subring of polynomials invariant under the action. Many results in algebraic
geometry

and commutative algebra were driven by the goal to understand this setup.

Cox, Little, O’Shea. Ideals, varieties, and algorithms.

Kane. Reflection groups and invariant theory.

**20. Cluster algebras.** A cluster algebra is a commutative ring with a
set of generators grouped

into clusters which satisfy certain properties. They are defined in an
elementary way and

have deep connections to many fields.

**21. Topological combinatorics of posets.** Explore the topological
approach to poset combinatorics,

focusing for example on the family of Cohen-Macaulay posets.

A. Bjrner, A.M. Garsia, and R.P. Stanley, An introduction to Cohen-Macaulay
partially

ordered sets. M. Wachs. Poset topology: Tools and applications. arXiv:0602226

22. Resolutions of edge ideals. The edge ideal of a graph G has a generator
x_{i}x_{j} for each edge

ij of the graph. From the invariants of its minimal resolution one can recover
information

about G.

R. Villarreal. Monomial algebras.

23. A. Tchernev. Representations of matroids and free resolutions for multigraded modules.

24. Floystad. The colorful Helly theorem and colorful resolutions of ideals.

25. Suyoung Choi, Jang Soo Kim, A combinatorial proof of a
formula for Betti numbers of a

stacked polytope

26. Ezra Miller, Topological Cohen-Macaulay criteria for monomial ideals

27. Alin Stefan, Classifications of Cohen-Macaulay modules
- The base ring associated to a

transversal polymatroid.

28. T. Kyle Petersen, Pavlo Pylyavskyy, David E Speyer, A
non-crossing standard monomial

theory.

29. Uwe Nagel, Sonja Petrovic, Properties of cut ideals associated to ring graphs.

30. Alicia Dickenstein, Laura Felicia Matusevich, Ezra
Miller, Combinatorics of binomial primary

decomposition.

31. Harm Derksen, Symmetric and Quasi-Symmetric Functions associated to Polymatroids.

32. Francesco Brenti, Volkmar Welker, The Veronese
Construction for Formal Power Series and

Graded Algebras.

33. Uwe Nagel, Victor Reiner, Betti numbers of monomial ideals and shifted skew shapes.

34. Gunnar Floystad, Cellular resolutions of Cohen-Macaulay monomial quotient ring.