Related papers: Topology of spaces of hyperbolic polynomials and c…
This note is a survey on the topology of hyperplane arrangements. We mainly focus on the relationship between topology and the real structure, such as adjacent relations of chambers and stratifications related to real structures.
We prove that the rational cohomology of the space of non-singular complex homogeneous polynomials of degree d in a fixed number of variables stabilizes to the cohomology of the general linear group for d sufficiently large.
We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the "splitting basis" for the homology of the partition lattice given in [Wa96],…
We study framed translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces, for which a horizontal separatrix is marked for each pole or zero. Such geometric structures naturally appear when studying flat…
We study the moduli space of logarithmic connections of rank $2$ on $\mathbb{P}^1 \setminus \{ t_1, \dots, t_5 \}$ with fixed spectral data. The aim of this paper is to compute the cohomology of such space, and this computation will be used…
Using a hyperbolic complex plane, we study the realization of the underlying hyperbolic symmetry as an internal symmetry that enables the unification of scalar fields of cosmological and particle physics interest. Such an unification is…
On a 4-dimensional compact symplectic manifold, we study how suitable perturbations of a toric system to a family of completely integrable systems with $\mathbb{S}^1$-symmetry lead to various hyperbolic-regular singularities. We compute and…
For a manifold $W$ and an $E_d$-algebra $A$, the factorisation homology $\int_W A$ can be seen as a generalisation of the classical configuration space of labelled particles in $W$. It carries an action by the diffeomorphism group…
In this paper we study the moduli spaces $Simp^m_n$ of degree $n+1$ morphisms $ \mathbb{A}^1_{K} \to \mathbb{A}^1_{K}$ with "ramification length $<m$" over an algebraically closed field $K$. For each $m$, the moduli space $Simp^m_n$ is a…
Self-assembling novel ordered structures with nanoparticles has recently received much attention. Here we use computer simulations to study a two-dimensional model system characterized by a simple isotropic interaction that could be…
We provide a naturally isomorphic description of the persistence map from merge trees to barcodes in terms of a monotone map from the partition lattice to the subset lattice. Our description is local, which offers the potential to speed up…
In this paper, we introduce a birationally admissible stratification on the Deligne-Mumford stack of stable minimal models (e.g., the KSBA moduli stack), such that the universal family over each stratum admits a simple normal crossing log…
We study the parameter space structure of degree $d \ge 3$ one complex variable polynomials as dynamical systems acting on $\C$. We introduce and study {\it straightening maps}. These maps are a natural higher degree generalization of the…
We show that strata of oriented toric hyperplane arrangements are in bijection with a collection of lattice points in a zonotope. Moreover, we relate the dimension of the stratum and the dimension of the minimal face of the zonotope…
We give an expository overview over recent results on the global structure and geometry of the Newton stratification of the reduction modulo p of Shimura varieties of Hodge type with hyperspecial level structure. More precisely, we discuss…
The set of homogeneous polynomials of degree $D$ is a topological space that contains the subspace $Hyp(D)$ constituted only by hyperbolic polynomials. In 2002, V. I. Arnold conjectured that the number of connected components of $Hyp (D)$…
Complex networks, which are the abstractions of many real-world systems, present a persistent challenge across disciplines for people to decipher their underlying information. Recently, hyperbolic geometry of latent spaces has gained…
Robustness against small perturbations is a crucial feature of topological properties. This robustness is both a source of theoretical interest and a drive for technological applications, but presents a challenge when looking for new…
This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of $n$-variate polynomial equations is specified through $n$ monomial bases. The natural locus for the…
This paper investigates the independence polynomials arising from iterated strong products of cycle graphs, examining their algebraic symmetries and combinatorial structures. Leveraging modular arithmetic and Galois theory, we establish…