What are some open problems in hyperplane/subspace arrangements, preferably of the combinatorial algebraic topology kind, and where can one read about them? That is, where are they discussed, and where can one learn the necessary background knowledge to understand them properly and in context?

I have been working one one such problem for a while, but I'm ignorant about the big questions, and more generally, interesting problems in this field, and also current trends.

(I'm aware of a few unresolved questions in Richard P. Stanley's An Introduction to Hyperplane Arrangements (available online, the unresolved problems are signaled by [5]))

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    $\begingroup$ It's possible that some (new) open problems on hyperplane arrangements will be available later this year at: www-math.mit.edu/~rstan/217 $\endgroup$ – Sam Hopkins Jun 14 '15 at 0:47

The classic book ``Hyperplane Arrangements'' by Orlik and Terao gives many open problems (some listed there are solved). A more recent, but not published, book which gives many open problems is at http://www.math.uiuc.edu/~schenck/cxarr.pdf.

There are many other surveys that list open problems. Alex Suciu wrote two survery's, http://arxiv.org/pdf/math/0010105.pdf and http://arxiv.org/pdf/1301.4851.pdf each with many problems. On the more algebraic side, Masahiko Yoshinaga wrote the survey http://arxiv.org/abs/1212.3523 which has many open problems. The survey by Sergey Yuzvinsky ``Resonance varieties of arrangement complements'' also has some open problems. The survey http://arxiv.org/abs/1101.0356 by Hal Schenck also has many open problems. One of the oldest open problems is Terao's conjecture: freeness depends on the intersection lattice (which is in the two books and in Yoshinaga's survey).


Here is a problem a came to know about from Ryan Kinser's talk at the 2015 Midwest Combinatorics Conference. Slides from this talk can be found here.

Given a subspace arrangement $\{V_i : i \in I\}$ one gets a rank function rank function $rk: 2^I \to \mathbb{Z}$ by $rk(A) = \dim \sum_{i \in A} V_i$. This function satisfies:

  1. $rk(\emptyset) = 0$
  2. $rk(A) \leq rk(B)$ if $A \subseteq B$
  3. $rk(A \cup B) + rk(A \cap B) \leq rk(A) + rk(B)$

This makes $(I, rk)$ a polymatroid. In general a polymatroid is any pair $(I, rk)$ satisfying the conditions above. If a polymatroid arises from a subspace arrangement it is called realizable (over which ever field the subspace arrangement is over).

If one identifies $\{f: 2^I \to \mathbb{R}\}$ with $\mathbb{R}^{2^{|I|}}$ one can talk about the cone of polymatroids which is defined by the inequalities above. One can also talk about the cone of realizable polymatroids. Not all polymatroids are realizable. Some open questions are:

  • What additional inequalities hold for realizable polymatroids? That is, what inequalities of the rank function must hold if the rank function comes from a subspace arrangement? Some results in this direction are from Kinser and Dougherty-Freiling-Zeger.
  • Is the cone of realizable polymatroids polyhedral? That is, are there finitely many inequalities defining it?

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