Topics in Combinatorics:   Splines and Combinatorics

Class meets: Tuesday, Thursday   2:30-4   room 2-136

Instructor: Alexander Postnikov   apost at math   room 2-389

Course webpage: http://www-math.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, 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 helpful.

Course Level: Graduate

Problem Sets:

• Problem Set 1
• Problem Set 2

Lectures:

1. T 02/02/2010. Lecture 1. Course overview.

2. R 02/04/2010. Lecture 2. Matroids and hyperplane arrangements.

3. T 02/09/2010. Lecture 3. Characteristic polynomial. Zonotopes.

4. R 02/11/2011. Lecture 4. Zonotopes (cont'd). Unimodularity. Tutte polynomial.

   T 02/16/2010. no class: Monday class schedule

5. R 02/18/2010. Lecture 5. Tutte polynomial (cont'd). Internal and external activity. Broken circuits. Orlik-Solomon algebra.

6. T 02/23/2010. Lecture 6. Kostant's partition function. Flow polytope. Reduction rules for graphs.

7. R 02/25/2010. Lecture 7. NBC-reduction. Piecewise polynomiality of Kostant's partition function.

8. T 03/02/2010. Lecture 8. NBC-trees & alternating trees. Linial arrangement. Chan-Robbins-Yuen polytope.

9. R 03/04/2010. Lecture 9. Volume of CRY-polytope (cont'd). Perfect matching polytope. Path polytope.

10. T 03/09/2010. Lecture 10. Multivariate spline and Box-spline.

11. R 03/11/2010. Lecture 11. Piecewise polynomiality. Decomposition into big cells (aka chamber complex).

12. T 03/16/2010. Lecture 12. Chamber complex (cont'd).

13. 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

14. T 03/30/2010. Lecture 14. Dahmen-Micchelli space.

15. R 04/01/2010. Lecture 15. Discussion of problem set.

16. T 04/06/2010. Lecture 16. Dahmen-Micchelli space (cont'd) and related spaces. Power ideals.

17. R 04/08/2010. Lecture 17. Parking functions and the inversion polynomial. Alternating permutations.

18. T 04/13/2010. Lecture 18. Proofs of theorems on Dahmen-Michelli spaces.

19. R 04/15/2010. Lecture 19. Proofs (cont'd). Graphical case. G-parking functions.

    T 04/20/2010. no class: Patriots Day

20. R 04/22/2010. Lecture 20. G-parking functions (cont'd). Abelian sandpile model.

21. T 04/27/2010. Lecture 21. rho-parking functions. Pitman-Stanley polytope.

22. R 04/29/2010. Lecture 22. Discrete Dahmen-Micchelli theory. Vector partition functions.

23. T 05/04/2010. Lecture 23. Proof of Dahmen-Micchelli's theorem on vector partition function. Cocircuit recurrences. Reciprocity.

24. R 05/06/2010. Lecture 24. Quasi-polynomiality. Relation between discrete and continuous Dahmen-Micchelli spaces.

25. T 05/11/2010. Lecture 25. Discussion of problem set.

26. R 05/13/2010. Lecture 26. Power ideals.

Texts:

• C. De Concini, C. Procesi: "Topics in Hyperplane Arrangements, Polytopes, and Box-Splines," forthcoming book, available at http://www.mat.uniroma1.it/~procesi/dida.html

  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," available at http://www-math.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), 509-532, available at http://www.ams.org/journals/tran/1988-308-02/S0002-9947-1988-0951619-X/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