Here are two problems about hyperplane arrangements.

*Problem 1:* If you take a collection of planes in $\mathbb{R}^3$,
then the number of lines you get by intersecting the planes is at
least the number of planes. This is an example of a more general
statement, called the “Top-Heavy Conjecture”, that Dowling and
Wilson conjectured in 1974.

*Problem 2:* Given a hyperplane arrangement, I will explain how
to uniquely associate a certain polynomial (called its
Kazhdan–Lusztig polynomial) to it. These polynomials should have
nonnegative coefficients.

Both of these problems were formulated for all matroids, and in the case of hyperplane arrangements they are controlled by the Hodge theory of a certain singular projective variety, called the Schubert variety of the arrangement. For arbitrary matroids, no such variety exists; nonetheless, I will discuss a solution to both problems for all matroids, which proceeds by finding combinatorial stand-ins for the cohomology and intersection cohomology of these Schubert varieties and by studying their Hodge theory. This is joint work with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang.

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Last updated: 30 May 2023