18.318 M.I.T. Spring 2010
Topics in Combinatorics:
Splines and Combinatorics
Class meets: Tuesday, Thursday 2:30-4
apost at math
Splines are smooth piecewise polynomial functions. They were introduced by
Schoenberg in the 1940's. They are widely used in numerical analysis,
computer-aided design, and computer graphics for interpolation of curves and
surfaces. For example, they are used to model airplane and automobile bodies.
During the last 30 years various researchers discovered that the multivariate
spline theory has deep links with combinatorics, in particular with
hyperplane arrangements, convex polytopes, matroid theory, and with
other areas of mathematics. An important
example of a piecewise polynomial function is the Kostant partition function,
which plays a central role in Lie theory.
The course will focus on the links between combinatorics and splines.
The course will include the following topics:
Hyperplane arrangements, graphical arrangements, matroids, Tutte polynomial,
broken circuits, Orlik-Solomon algebra, convex polytopes, their volumes and integer
points, Ehrhart polynomials, zonotopes, f- and h-vectors, multivariate splines, box-splines,
Dahmen-Micchelli space, partition functions, contingency tables, flow polytopes, root systems,
power ideals, parking functions, toric arrangements, wonderful models, splines on simplicial complexes, etc.
The course should be accessible to first year graduate students. There are no
any special prerequisites though some background in combinatorics would be
Course Level: Graduate
- T 02/02/2010. Lecture 1. Course overview.
- R 02/04/2010. Lecture 2. Matroids and hyperplane arrangements.
- T 02/09/2010. Lecture 3. Characteristic polynomial. Zonotopes.
- R 02/11/2011. Lecture 4. Zonotopes (cont'd). Unimodularity.
T 02/16/2010. no class: Monday class schedule
- R 02/18/2010. Lecture 5.
Tutte polynomial (cont'd). Internal and external activity. Broken circuits.
- T 02/23/2010. Lecture 6. Kostant's partition function. Flow polytope.
Reduction rules for graphs.
- R 02/25/2010. Lecture 7. NBC-reduction. Piecewise polynomiality
of Kostant's partition function.
- T 03/02/2010. Lecture 8. NBC-trees & alternating trees.
Linial arrangement. Chan-Robbins-Yuen polytope.
- R 03/04/2010. Lecture 9.
Volume of CRY-polytope (cont'd). Perfect matching polytope.
- T 03/09/2010. Lecture 10.
Multivariate spline and Box-spline.
- R 03/11/2010. Lecture 11.
Piecewise polynomiality. Decomposition into big cells
(aka chamber complex).
- T 03/16/2010. Lecture 12.
Chamber complex (cont'd).
- R 03/18/2010. Lecture 13.
Chamber complex vs diagonal sections.
Triangulations of products of simplices.
T 03/23/2010. no class: Spring vacation
R 03/25/2010. no class: Spring vacation
- T 03/30/2010. Lecture 14.
- R 04/01/2010. Lecture 15.
Discussion of problem set.
- T 04/06/2010. Lecture 16.
Dahmen-Micchelli space (cont'd) and related spaces.
- R 04/08/2010. Lecture 17.
Parking functions and the inversion polynomial.
- T 04/13/2010. Lecture 18.
Proofs of theorems on Dahmen-Michelli spaces.
- R 04/15/2010. Lecture 19.
Proofs (cont'd). Graphical case.
T 04/20/2010. no class: Patriots Day
- R 04/22/2010. Lecture 20.
G-parking functions (cont'd). Abelian sandpile model.
- T 04/27/2010. Lecture 21.
rho-parking functions. Pitman-Stanley polytope.
- R 04/29/2010. Lecture 22.
Discrete Dahmen-Micchelli theory. Vector partition
- T 05/04/2010. Lecture 23. Proof of Dahmen-Micchelli's theorem
on vector partition function. Cocircuit recurrences. Reciprocity.
- R 05/06/2010. Lecture 24. Quasi-polynomiality.
Relation between discrete and continuous Dahmen-Micchelli spaces.
- T 05/11/2010. Lecture 25. Discussion of problem set.
- R 05/13/2010. Lecture 26. Power ideals.
C. De Concini, C. Procesi: "Topics in Hyperplane Arrangements, Polytopes,
and Box-Splines," forthcoming book, available at
See also "Polytopes, Partition Functions
and Box-Splines" by C. Procesi, UCSD seminar notes,
available at the same place.
- R. P. Stanley: "An Introduction to Hyperplane Arrangements,"
- W. Dahmen, C. A. Micchelli: The number of solutions to linear Diophantine
equations and multivariate splines,
Transactions of the AMS, 308 (1988), 509-532,
- F. Ardila, A. Postnikov: Combinatorics and geometry of power ideals,
to appear in Transactions of the AMS, available at
last updated: May 6, 2010