Related papers: Grassmannians and Pseudosphere Arrangements
We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial…
The cohomology of the configuration space of n points in R^3 admits a symmetric group action and has been shown to be isomorphic to the regular representation. One way to prove this is by defining an S^1-action whose fixed point set is the…
We analyze combinatorial structures which play a central role in determining spectral properties of the volume operator in loop quantum gravity (LQG). These structures encode geometrical information of the embedding of arbitrary valence…
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 a…
The Folkman-Lawrence topological representation theorem, which states that every (loop-free) oriented matroid of rank $r$ can be represented as a pseudosphere arrangement on the $(r-1)$-dimensional sphere $S^{r-1}$, is one of the most…
In the recent past, nested structures in Riemannian manifolds has been studied in the context of dimensionality reduction as an alternative to the popular principal geodesic analysis (PGA) technique, for example, the principal nested…
We provide two alternate settings for a family of varieties modeling the uniserial representations with fixed sequence of composition factors over a finite dimensional algebra. The first is a quasi-projective subvariety of a Grassmannian…
Many classes of projective algebraic varieties can be studied in terms of graded rings. Gorenstein graded rings in small codimension have been studied recently from an algebraic point of view, but the geometric meaning of the resulting…
We prove rigidity results describing contextually-constrained maps defined on Grassmannians and manifolds of ordered independent line tuples in finite-dimensional vector or Hilbert spaces. One statement in the spirit of the Fundamental…
This article is a survey of matroid theory aimed at algebraic geometers. Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be…
The nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its study involves interesting combinatorial objects, such as positroids and plabic graphs. Remarkably, the same combinatorial…
By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all…
The $g$-vector of a simplicial complex contains a lot of information about the combinatorial and topological structure of that complex. Several classification results regarding the structure of normal pseudomanifolds and homology manifolds…
We study realization spaces of matroids over hyperfields (in the sense of Baker and Bowler). More precisely, given a matroid M and a hyperfield H we determine the space of all H-matroids over M. This can be seen as the matroid stratum of…
A positroid is the matroid of a real matrix with nonnegative maximal minors, a positroid variety is the closure of the locus of points in a complex Grassmannian whose matroid is a fixed positroid, and a positroid class is the cohomology…
In this paper we study the mod $2$ cohomology ring of the Grasmannian $\widetilde{G}_{n,3}$ of oriented $3$-planes in $\mathbb{R}^n$. We determine the degrees of the indecomposable elements in the cohomology ring. We also obtain an almost…
It is known that the complex Grassmannian of $k$-dimensional subspaces can be identified with the set of projection matrices of rank $k$. It is also classically known that the convex hull of this set is the set of Hermitian matrices with…
A phased matroid is a matroid with additional structure which plays the same role for complex vector arrangements that oriented matroids play for real vector arrangements. The realization space of an oriented (resp., phased) matroid is the…
\def\mon{S^3\stackrel{S^1}{\rightarrow}S^2} \def\inst{S^7\stackrel{S^3}{\rightarrow}S^4} \def\octo{S^{15}\stackrel{S^7}{\rightarrow}S^8} In semilocal theories, the vacuum manifold is fibered in a non-trivial way by the action of the gauge…
The general construction of self-adjoint configuration space representations of the Heisenberg algebra over an arbitrary manifold is considered. All such inequivalent representations are parametrised in terms of the topology classes of flat…