Related papers: Double star arrangement and the pointed multinet
We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincare polynomial, and Tutte polynomial. We consider basic algebraic…
A hyperplane arrangement $\cA$ is said to be free if the corresponding Jacobian ideal $J_\cA$ is Cohen-Macaulay. If $\cA$ is free then $J_\cA$ is unmixed (i.e. equidimensional). Freeness is an important property, yet its presence is not…
An arrangement of hyperplanes is a finite collection of hyperplanes in a real Euclidean space. To such a collection one associates the characteristic polynomial that encodes the combinatorics of intersections of the hyperplanes. Finding the…
In this PhD thesis, we give a new geometric approach to higher Teichm\"uller theory. In particular we construct a geometric structure on surfaces, generalizing the complex structure, and we explore its link to Hitchin components. The…
Spatially embedded networks are important in several disciplines. The prototypical spatial net- work we assume is the Random Geometric Graph of which many properties are known. Here we present new results for the two-point degree…
This paper continues the investigation of the configuration space of two distinct points on a graph. We analyze the process of adding an additional edge to the graph and the resulting changes in the topology of the configuration space. We…
We extend the equality-type results of Ito--Takimura and Kindred for the non-orientable genera of alternating knots to the setting of two-component alternating links. We show that, for such links, a unified quantity capturing both…
We show that the linear map defined by multiplication with a general bi-homogeneous form between two bi-graduated pieces of the first cohomology of a nonsingular quadric in the projective space is of maximal rank. This is the first non…
It is known that there exist hyperplane arrangements with same underlying matroid that admit non-homotopy equivalent complement manifolds. In this work we show that, in any rank, complex central hyperplane arrangements with up to 7…
We associate a quotient of superspace to any hyperplane arrangement by considering the differential closure of an ideal generated by powers of certain homogeneous linear forms. This quotient is a superspace analogue of the external…
We developed a modification to the calculation of the two-point correlation function commonly used in the analysis of large scale structure in cosmology. An estimator of the two-point correlation function is constructed by contrasting the…
One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erd\H{o}s: Every set of points $E$ in a projective plane determines at least $|E|$ lines, unless all the points are contained in a…
In this article we prove in main Theorem A that any infinity type real hyperplane arrangement $\mathcal{H}_n^m$ (Definition 2.11) with the associated normal system $\mathcal{N}$ (Definitions [2.2,2.4] can be represented isomorphically…
We show that the number of lines in an $m$--homogeneous supersolvable line arrangement is upper bounded by $3m-3$ and we classify the $m$--homogeneous supersolvable line arrangements with two modular points up-to lattice-isotopy. A lower…
Higher-order exceptional points in the spectrum of non-Hermitian Hamiltonians describing open quantum or wave systems have a variety of potential applications in particular in optics and photonics. However, the experimental realization is…
Two arrangements with the same combinatorial intersection lattice but whose complements have different fundamental groups are called a Zariski pair. This work finds that there are at most nine such pairs amongst all ten line arrangements…
This paper begins by extending the notion of a combinatorial configuration of points and lines to a combinatorial configuration of points and planes that we refer to as configurations of order $2$. We then proceed to investigate a further…
We consider the scheme $X_{r,d,n}$ parametrizing $n$ ordered points in projective space $\mathbb{P}^r$ that lie on a common hypersurface of degree $d$. We show that this scheme has a determinantal structure and we prove that it is…
We produce a criterion for open sets in projective $n$-space over a separably closed field to have \'etale cohomological dimension bounded by $2n-3$. We use the criterion to exhibit a scheme for which \'etale cohomological dimension is…
A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A corresponding framework has been developed by Woronowicz, more…