ACE Summer School lectures 2005- Vic Reiner

ACE Summer School 2005 in Geometric Combinatorics

Lectures by Vic Reiner

Topic: matroids and oriented matroids

Lectures (tentative titles/topics )

Lecture 1: Motivation, axioms, examples
Lecture 2: Constructions, representations, and realizations
Lecture 3: Invariants: enumerative, topological, algebraic
Lecture 4: New directions: ??

Here are the lecture notes in PostScript or PDF for Lectures 1,2,3, containing all the homework exercises at the end.

Let me know about typos/errors (there will be plenty!) when you see me in Vienna, or otherwise, send me an e-mail.

Reading

Matroid surveys
- R. J. Wilson, "An introduction to matroid theory" Amer. Math. Monthly 80 (1973), 500--525.
- D.J.A. Welsh "Matroids: fundamental concepts", in the Handbook of Combinatorics Vol. 1, (eds. R. L. Graham, M. Grötschel and L. Lovász., Elsevier; MIT Press, Cambridge, MA, 1995)
Oriented matroid surveys
- Chapter 1 of Björner, Las Vergnas, Sturmfels, White, and Ziegler, "Oriented matroids" (see full reference below).
- Lecture 6 of G.M. Ziegler's "Lectures on polytopes", Graduate Texts in Mathematics, 152. Springer-Verlag, New York, 1995.
- J. Richter-Gebert and G.M. Ziegler, "Oriented matroids" in CRC Handbook of Discrete and Computational Geometry
Here are some of the standard reference texts on matroids and oriented matroids.
It is probably not necessary to look through any of these before the summer school;
they are only here in case the reader wants to know more about the topics mentioned in the surveys.
Standard matroid texts
- D.J.A. Welsh, "Matroid theory", L. M. S. Monographs, No. 8. Academic Press, London-New York, 1976.
- J.G. Oxley, "Matroid theory", Oxford Science Publications. Oxford University Press, New York, 1992
- Three volumes (N. White, editor) from the Encyclopedia of Mathematics and Its Applications series by Cambridge Univ. Press:
  - Theory of matroids, Vol. 26, 1986.
  - Combinatorial geometries, Vol. 29, 1987.
  - Matroid applications, Vol. 40, 1992.
- H.H. Crapo and G.-C. Rota, "On the foundations of combinatorial theory: Combinatorial geometries. Preliminary edition", The M.I.T. Press, Cambridge, Mass.-London, 1970.
- R. von Randow, "Introduction to the theory of matroids", Lecture Notes in Economics and Mathematical Systems, Vol. 109. Springer-Verlag, Berlin-New York, 1975.
- A. Recski, "Matroid theory and its applications in electric network theory and in statics", Algorithms and Combinatorics, 6. Springer-Verlag, Berlin; Akadémiai Kiadó, Budapest, 1989
Standard oriented matroid texts
- A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G.M. Ziegler, "Oriented matroids", Encyclopedia of Mathematics and its Applications, 46. Cambridge University Press, Cambridge, 1999. Also see G. M. Ziegler's accompanying dynamic survey on Oriented Matroids (available as DS4 at www.combinatorics.org)
- A. Bachem and W. Kern "Linear programming duality. An introduction to oriented matroids", Universitext. Springer-Verlag, Berlin, 1992
Here are some (relatively) recent papers that represent a few trends/threads/developments of interest (to me).
In most cases only one paper about a topic is listed, but the interested reader can find more related references within it.
Recent threads/developments
- D. Biss, "The homotopy type of the matroid grassmannian", Annals Math. 158 no. 3 (2003), 929-952.
- T. Hausel "Quaternionic Geometry of Matroids",math.AG/0308146.
- E. Swartz, "Topological representations of matroids", math.CO/0208157.
- I. Novik, A. Postnikov and B. Sturmfels, "Syzygies of oriented matroids", math.CO/0009241
- J. Herzog and T. Hibi, "Discrete Polymatroids", math.AC/0307236.
- H. Crapo and W. Schmitt,"A unique factorization theorem for matroids", math.CO/0409099.
- F. Vallentin,"A Note on Space Tiling Zonotopes", math.MG/0402053.
- E.M. Feichtner and B. Sturmfels, "Matroid polytopes, nested sets and Bergman fans", math.CO/0411260
- A. Borovik, I.M. Gelfand, and N. White, "Coxeter matroids", Progress in Mathematics, 216. Birkhäuser Boston, Inc., Boston, MA, 2003.
- B. Bollobás and O. Riordan, "A polynomial of graphs on surfaces", Math. Ann. 323 (2002), no. 1, 81--96.
- M. Korn and I. Pak, "Combinatorial evaluations of the Tutte polynomial", preprint, 2003, available at www-math.mit.edu/~pak/research.html.
- Y.-B. Choe, J.G. Oxley, A.D. Sokal, and D.G. Wagner, "Homogeneous multivariate polynomials with the half-plane property", math.CO/0202034 and Adv. Appl. Math. 32 (2004), 88-187.
- J.E. Bonin, A. de Mier, and M. Noy, "Lattice path matroids: enumerative aspects and Tutte polynomials", math.CO/0211188.
- C. Merino López, "Chip-firing and the Tutte polynomial", Ann. Comb. 1 (1997), no. 3, 253--259.
- A.D. Sokal,"The multivariate Tutte polynomial (alias Potts model) for graphs and matroids", math.CO/0503607.
- M. Develin, J.L. Martin and V. Reiner, "Rigidity theory for matroids", math.CO/0503050.