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Enumeration of points, lines, planes, etc
One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erdős: Every set of points $E$ in a projective plane determines at least $|E|$ lines,
Cohomology jump loci of quasi-projective varieties
We prove that the cohomology jump loci in the space of rank one local systems over a smooth quasi-projective variety are finite unions of torsion translates of subtori. The main ingredients are a
Torsion points on the cohomology jump loci of compact K\"ahler manifolds
We prove that each irreducible component of the cohomology jump loci of rank one local systems over a compact Kahler manifold contains at least one torsion point. This generalizes a theorem of
Cohomology support loci of local systems
The support S of Sabbah's specialization complex is a simultaneous generalization of the set of eigenvalues of the monodromy on Deligne's nearby cycles complex, of the support of the Alexander
A semi-small decomposition of the Chow ring of a matroid
We give a semi-small orthogonal decomposition of the Chow ring of a matroid M. The decomposition is used to give simple proofs of Poincare duality, the hard Lefschetz theorem, and the Hodge-Riemann
Singular Hodge theory for combinatorial geometries.
We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain
Cohomology jump loci of differential graded Lie algebras
To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local
Cohomology jump loci of quasi-compact Kähler manifolds
We give two applications of the exponential Ax-Lindemann Theorem to local systems. One application is to show that for a connected topological space, the existence of a finite model of the real
Correlation bounds for fields and matroids
Let $G$ be a finite connected graph, and let $T$ be a spanning tree of $G$ chosen uniformly at random. The work of Kirchhoff on electrical networks can be used to show that the events $e_1 \in T$ and