Connor Simpson
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View article: Combinatorial flats and Schubert varieties of subspace arrangements
Combinatorial flats and Schubert varieties of subspace arrangements Open
The lattice of flats of a matroid is combinatorially well-behaved and, when is realizable, admits a geometric model in the form of a “Schubert variety of hyperplane arrangement”. In contrast, the lattice of flats of a polymatroid e…
View article: Toric varieties modulo reflections
Toric varieties modulo reflections Open
Let $W$ be a finite group generated by reflections of a lattice $M$. If a lattice polytope $P \subset M \otimes_{\mathbb Z}\mathbb R$ is preserved by $W$, then we show that the quotient of the projective toric variety $X_P$ by $W$ is isomo…
View article: Combinatorial flats and Schubert varieties of subspace arrangements
Combinatorial flats and Schubert varieties of subspace arrangements Open
The lattice of flats $\mathcal L_M$ of a matroid $M$ is combinatorially well-behaved and, when $M$ is realizable, admits a geometric model in the form of a "Schubert variety of hyperplane arrangement". In contrast, the lattice of flats of …
View article: Total positivity for matroid Schubert varieties
Total positivity for matroid Schubert varieties Open
We define the totally nonnegative matroid Schubert variety $\mathcal Y_V$ of a linear subspace $V \subset \mathbb R^n$. We show that $\mathcal Y_V$ is a regular CW complex homeomorphic to a closed ball, with strata indexed by pairs of acyc…
View article: Simplicial generation of Chow rings of matroids
Simplicial generation of Chow rings of matroids Open
We introduce a presentation of the Chow ring of a matroid by a new set of generators, called “simplicial generators.” These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation vi…
View article: A new generic vanishing theorem on homogeneous varieties and the positivity conjecture for triple intersections of Schubert cells
A new generic vanishing theorem on homogeneous varieties and the positivity conjecture for triple intersections of Schubert cells Open
In this paper we prove a new generic vanishing theorem for $X$ a complete homogeneous variety with respect to an action of a connected algebraic group. Let $A, B_0\subset X$ be locally closed affine subvarieties, and assume that $B_0$ is s…
View article: The Bergman fan of a polymatroid
The Bergman fan of a polymatroid Open
We introduce the Bergman fan of a polymatroid and prove that the Chow ring of the Bergman fan is isomorphic to the Chow ring of the polymatroid. Using the Bergman fan, we establish the Kähler package for the Chow ring of the polymatroid, r…
View article: Chow rings of vector space matroids
Chow rings of vector space matroids Open
The Chow ring of a matroid (or more generally, atomic latice) is an invariant whose importance was demonstrated by Adiprasito, Huh and Katz, who used it to resolve the long-standing Heron-Rota-Welsh conjecture. Here, we make a detailed stu…
View article: Simplicial generation of Chow rings of matroids
Simplicial generation of Chow rings of matroids Open
We introduce a presentation of the Chow ring of a matroid by a new set of generators, called "simplicial generators." These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation vi…
View article: Flow Polytopes of Partitions
Flow Polytopes of Partitions Open
Recent progress on flow polytopes indicates many interesting families with product formulas for their volume. These product formulas are all proved using analytic techniques. Our work breaks from this pattern. We define a family of closely…
View article: The Set Splittablity Problem
The Set Splittablity Problem Open
A collection of sets is called splittable if there is a set S such that for each set B in the collection, the intersection of S and B is half the size of B. Splittability is a generalization of graph colorability, which is an active area o…
View article: The set splittability problem
The set splittability problem Open
The set splittability problem is the following: given a finite collection of finite sets, does there exits a single set that contains exactly half the elements from each set in the collection? (If a set has odd size, we allow the floor or …