18.318 M.I.T. Spring 2010
Topics in Combinatorics:
Splines and Combinatorics
Class meets: Tuesday, Thursday 2:304
room 2136 
Instructor:
Alexander Postnikov
apost at math
room 2389
Course webpage:
http://wwwmath.mit.edu/~apost/courses/18.318/
Synopsis:
Splines are smooth piecewise polynomial functions. They were introduced by
Schoenberg in the 1940's. They are widely used in numerical analysis,
computeraided 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, OrlikSolomon algebra, convex polytopes, their volumes and integer
points, Ehrhart polynomials, zonotopes, f and hvectors, multivariate splines, boxsplines,
DahmenMicchelli 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
helpful.
Course Level: Graduate
Problem Sets:
Lectures:
 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.
Tutte polynomial.
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.
OrlikSolomon algebra.
 T 02/23/2010. Lecture 6. Kostant's partition function. Flow polytope.
Reduction rules for graphs.
 R 02/25/2010. Lecture 7. NBCreduction. Piecewise polynomiality
of Kostant's partition function.
 T 03/02/2010. Lecture 8. NBCtrees & alternating trees.
Linial arrangement. ChanRobbinsYuen polytope.
 R 03/04/2010. Lecture 9.
Volume of CRYpolytope (cont'd). Perfect matching polytope.
Path polytope.
 T 03/09/2010. Lecture 10.
Multivariate spline and Boxspline.
 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.
Mixed subdivisions.
Cayley trick.
T 03/23/2010. no class: Spring vacation
R 03/25/2010. no class: Spring vacation
 T 03/30/2010. Lecture 14.
DahmenMicchelli space.
 R 04/01/2010. Lecture 15.
Discussion of problem set.
 T 04/06/2010. Lecture 16.
DahmenMicchelli space (cont'd) and related spaces.
Power ideals.
 R 04/08/2010. Lecture 17.
Parking functions and the inversion polynomial.
Alternating permutations.
 T 04/13/2010. Lecture 18.
Proofs of theorems on DahmenMichelli spaces.
 R 04/15/2010. Lecture 19.
Proofs (cont'd). Graphical case.
Gparking functions.
T 04/20/2010. no class: Patriots Day
 R 04/22/2010. Lecture 20.
Gparking functions (cont'd). Abelian sandpile model.
 T 04/27/2010. Lecture 21.
rhoparking functions. PitmanStanley polytope.
 R 04/29/2010. Lecture 22.
Discrete DahmenMicchelli theory. Vector partition
functions.
 T 05/04/2010. Lecture 23. Proof of DahmenMicchelli's theorem
on vector partition function. Cocircuit recurrences. Reciprocity.
 R 05/06/2010. Lecture 24. Quasipolynomiality.
Relation between discrete and continuous DahmenMicchelli spaces.
 T 05/11/2010. Lecture 25. Discussion of problem set.
 R 05/13/2010. Lecture 26. Power ideals.
Texts:

C. De Concini, C. Procesi: "Topics in Hyperplane Arrangements, Polytopes,
and BoxSplines," forthcoming book, available at
http://www.mat.uniroma1.it/~procesi/dida.html
See also "Polytopes, Partition Functions
and BoxSplines" by C. Procesi, UCSD seminar notes,
available at the same place.
 R. P. Stanley: "An Introduction to Hyperplane Arrangements,"
available at
http://wwwmath.mit.edu/~rstan/arrangements/arr.html
 W. Dahmen, C. A. Micchelli: The number of solutions to linear Diophantine
equations and multivariate splines,
Transactions of the AMS, 308 (1988), 509532,
available at
http://www.ams.org/journals/tran/198830802/S0002994719880951619X/home.html
 F. Ardila, A. Postnikov: Combinatorics and geometry of power ideals,
to appear in Transactions of the AMS, available at
http://arxiv.org/abs/0809.2143
last updated: May 6, 2010