Hyperplanes, matroids & broken circuits
Thursdays 12-2 at the Villa, Arnimallee 2
[American Mathematical Monthly 1943, Problem E554]:
Show that $n$ cuts can divide a cheese into as many as
Proposed by J.~L.~Woodbridge, Philadelphia
Given a set of hyperplanes (cuts through our cheese ...), we can ask ourselves topological questions (like, how many components does the complement have?). As it turns out, the answer does not really depend on the exact position of the hyperplanes. It only depends on the combinatorial information how the hyperplanes intersect -- the so-called matroid of the arrangement.
In the seminar we will learn what matroids are, what the characteristic polynomial of a matroid is and what it tells us about hyperplane arrangements. The methods to study these questions will touch on topological combinatorics, algebra, geometry and, of course, polytopes.
By the end of the seminar we will not only understand the statement of an old conjecture in this context, but also quite a few of the ingredients of its recent proof by FU-Villa-alumnus Karim Adiprasito and coauthors.
The literature on the subject is vast. A good point to start are Richard Stanley's 2004 PCMI lecture notes; available at www.cis.upenn.edu/~cis610/sp06stanley.pdf.
Rules of the Game
- Presentation 60min., followed by a discussion about the subject at hand and about the presentation.
- You will assign homework, and you will do the homework assigned by the other participants.
- Written draft outline (4 pages) to be handed in 2 weeks before the actual presentation; contains structure (at what point which definitions/results/examples), as well as proof ideas.
- Final version (no more than 8 pages) after the talk.
- It goes without saying that you attend and actively participate during the other presentations.
|5-4||Matrois: graphical, realizable, matroid polytope||Carlos|
|5-11||Intersection poset, Möbius inversion, (geometric) lattices, characteristic polynomial||Evgenia|
|5-18||Deletion/contraction, Tutte polynomial, sign of characteristic coefficients||Marie|
|5-25||--- holiday ---|
|6-1||Simplicial complexes, broken circuits, NBC||Arno|
|6-8||Shelling a poset||Felix|
|6-15||NBC and (reduced) characteristic polynomial||Johanna|
|6-22||Bergman fan and matroid polytope||Lena|
|6-29||Gröbner bases and Stanley-Reisner rings||Florian|
|7-6||The reciprocal plane and its tropicalization||Jorge|
|7-13||Overview & homework makeup||Christian|
|7-20||Gröbner basis for the graph of the reciprocal plane||Giulia|