Related papers: Sausages and Butcher Paper

The shift locus is the space of normalized polynomials in one complex variable for which every critical point is in the attracting basin of infinity. The method of sausages gives a (canonical) decomposition of the shift locus in each degree…

This article introduces a finite piecewise Euclidean cell complex homeomorphic to the space of monic centered complex polynomials of degree $d$ whose critical values lie in a fixed closed rectangular region. We call this the branched…

Given a maximal geodesic lamination $\lambda$ on a closed oriented surface $S$ of genus $g$, the space of $d$-pleated surfaces with pleating locus $\lambda$ is an open subset of $\mathrm{Hom}(\pi_1(S),\mathsf{PGL}_d(\mathbb{C}))$ obtained…

Let $M_d$ be the moduli space of one-dimensional complex polynomial dynamical systems. The escape rates of the critical points determine a critical heights map $G: M_d \to \mathbb{R}^{d-1}$. For generic values of $G$, each connected…

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…

Given a $d$-dimensional manifold $M$ and a knotted sphere $s\colon\mathbb{S}^{k-1}\hookrightarrow\partial M$ with $1\leq k\leq d$, for which there exists a framed dual sphere $G\colon\mathbb{S}^{d-k}\hookrightarrow\partial M$, we show that…

We relate generic points in the shift locus $\mathcal{S}_D$ of degree $D\ge 2$ polynomials to metric graphs. Using thermodynamic metrics on the space of metric graphs, we obtain a distance function $\rho_D$ on $\mathcal{S}_D$. We study the…

We prove the connectedness of the following locus: the space of degree-$d$ branched self-coverings of $S^2$ with two critical points of order $d$, one of which is $n$-periodic.

The Schur class, denoted by $\mathcal{S}(\mathbb{D})$, is the set of all functions analytic and bounded by one in modulus in the open unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$, that is \[ \mathcal{S}(\mathbb{D}) = \{\varphi…

We study the Disc-structure space $S^{\rm Disc}_\partial(M)$ of a compact smooth manifold $M$. Informally speaking, this space measures the difference between $M$, together with its diffeomorphisms, and the diagram of ordered framed…

Consider the topologically enriched category of compact smooth manifolds (possibly with corners), with morphisms given by codimension zero smooth embeddings. Now formally identify any object X with its thickening X x [-1,1]. We prove that…

In this paper we define a family of topological spaces, which contains and vastly generalizes the higher-dimensional Dunce hats. Our definition is purely combinatorial, and is phrased in terms of identifications of boundary simplices of…

Let $M^d$ be a simply connected spin manifold of dimension $d \geq 5$ admitting Riemannian metrics of positive scalar curvature. Denote by $\mathcal{R}^+(M^d)$ the space of such metrics on $M^d$. We show that $\mathcal{R}^+(M^d)$ is…

For a fixed $d \ge 5$, the Noether-Lefschetz locus parametrizes smooth degree $d$ surfaces in $\mathbb{P}^3$ with Picard number greater than $1$. This is a countable union of proper algebraic varieties. It is known (due to works of Voisin,…

The program of understanding Shape Theory layer by layer topologically and geometrically -- proposed in Part I -- is now addressed for 4 points in 1-$d$. Topological shape space graphs are far more complex here, whereas metric shape spaces…

Among the simple Lie algebras, $D_4$ is distinguished as the unique one whose group of outer-automorphisms is bigger than $\mathbb{Z}_2$. We study the compactifications of the $D_4$ (2,0) Theory on a punctured Riemann surface, $C$, with…

We classify four-dimensional compact solvmanifolds up to diffeomorphism, while determining which of them have complex analytic structures. In particular, we shall see that a four-dimensional compact solvmanifold S can be written, up to…

The Coulomb branch geometry of a 4d $\mathcal{N}=2$ SCFT is encoded in the data of a complex integrable system. In class-S, this is the Hitchin System (of ADE type) on the punctured curves $C$ on which we compactified from 6d to 4d. As we…

We investigate models of (1+d)-D Lorentzian semi-random lattices with one random (space-like) direction and d regular (time-like) ones. We prove a general inversion formula expressing the partition function of these models as the inverse of…

The observable universe has undergone periods of expansion that are well approximated by de Sitter (dS) space. Still lacking is a quantum mechanical description of dS, both globally and when restricted to the static patch. We develop a…