arXiv research

Improves bounds on surface decompositions.

The Bers-Greenberg theorem tells that the Teichmüller space of a Riemann surface with branch points (orbifold) depends only on the genus and the number of special points, but not on the particular ramification values. On the other hand, the Maskit embedding provides a mapping from the Teichmüller space of an orbifold, …

This article is dedicated to prove Buser's conjecture about Bers' constants for spheres with cusps (or marked points) and for hyperelliptic surfaces. More specifically, our main theorem states that any hyperbolic sphere with $n$ cusps has a pants decomposition with all of its geodesics of length bounded by a constant r…

Develops analogs of character varieties for algebraic correspondences, proving boundedness and compactifications.

Using Lipschitz distance on Outer space we give another proof of the train track theorem.

For compact Riemann surfaces, the collar theorem and Bers' partition theorem are major tools for working with simple closed geodesics. The main goal of this paper is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic two-dimensional orbifolds are a particular case of such surfaces. We consider all cone …

We consider a quotient space of the Bers boundary of Teichmüller space, which we call the reduced Bers boundary, by collapsing each quasi-conformal deformation space into a point. This reduced Bers boundary turns out to be independent of the basepoint, and the action of the mapping class group on the Teichmüller space …

Unified proof of Nielsen-Thurston classification via Teichmüller's theorem.

The paper proves a mapping from a space of holonomy varieties to Teichmüller spaces, with a non-empty discrete intersection.

The article approximates solutions to the Beltrami equation using similarity surfaces.

We prove that every Bers slice of quasi-Fuchsian space is Zariski dense in the character variety.

We prove the Bers' density conjecture for singly degenerate Kleinian surfaces groups without parabolics.

Develops complex harmonic maps for Teichmüller theory, proving new theorems.

The paper discovers new ways Riemann surfaces can degenerate.

In this paper we parametrize the Teichmüller spaces of constructible Koebe groups, that is Kleinian group that arise as covering of $2-$orbifolds determined by certain normal subgroups of their fundamental groups. We also study the covering spaces of the Teichmüller spaces of those Koebe groups. Finally we prove an iso…

Generalizes uniformization to algebraic correspondences.

It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in ge…

Holomorphic solutions vary in Sobolev spaces for Beltrami equations.

Geometric data uniquely determines convex subsets in hyperbolic manifolds.

Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.

A new metric model for quasi-Fuchsian space defined by Bers metrics.

Let $S$ be a closed Riemann surface of genus $g(\geqq 2)$ and set $\dot{S}=S \setminus \{\hat{z}_0 \}$. Then we have the composed map $\varphi\circ r$ of a map $r: T(S) \times U \rightarrow F(S)$ and the Bers isomorphism $\varphi: F(S) \rightarrow T(\dot{S})$, where $F(S)$ is the Bers fiber space of $S$, $T(X)$ is the …

This is the second paper in a series of investigations of the pluripotential theory on Teichmüller space. The main purpose of this paper is to establish the Poisson integral formula for pluriharmonic functions on Teichmüller space which are continuous on the Bers compactification. We also observe that the Schwarz type …

Study bends 2D surfaces in 3D space using special equations.

The paper explores properties of functions on Teichmüller space, proving theorems about limits and non-ergodicity.

Maximal cusps are not dense on Teichmüller space for infinite-type surfaces.

We show that an orientable 3-dimensional manifold M admits a complete riemannian metric of bounded geometry and uniformly pos- itive scalar curvature if and only if there exists a finite collection F of spherical space-forms such that M is a (possibly infinite) connected sum where each summand is diffeomorphic to S2xS1…

The Bers embebbing realizes the Teichmüller space of a Fuchsian group $G$ as a open, bounded and contractible subset of the complex Banach space of bounded quadratic differentials for $G$. It utilizes the schlicht model of Teichmüller space, where each point is represented by an injective holomorphic function on the di…

We shall show that for a given homeomorphism type and a set of end invariants (including the parabolic locus) with necessary topological conditions which a topologically tame Kleinian group with that homeomorphism type must satisfy, there is an algebraic limit of minimally parabolic, geometrically finite Kleinian group…

Unified and generalized mating frameworks for Kleinian groups and rational maps.

We consider the weighted belief-propagation (WBP) decoder recently proposed by Nachmani et al. where different weights are introduced for each Tanner graph edge and optimized using machine learning techniques. Our focus is on simple-scaling models that use the same weights across certain edges to reduce the storage and…

Metric graphs have subgraphs with entropy at least λ.

Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first $3g-3$ geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short ($< \log (g)$) geodesics on a …

The Basilica Julia set is universally equivalent to other complex dynamics sets.

Develops a real-analytic embedding for diffeomorphisms of the line, linking to Fisher-Rao geometry.

Holomorphic map connects Hitchin components to character varieties.

In emerging Internet-of-Nano-Thing (IoNT), information will be embedded and conveyed in the form of molecules through complex and diffusive medias. One main challenge lies in the long-tail nature of the channel response causing inter-symbol-interference (ISI), which deteriorates the detection performance. If the channe…

Thurston's ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free t…

We trained three Binarized Convolutional Neural Network architectures (LeNet-4, Network-In-Network, AlexNet) on a variety of datasets (MNIST, CIFAR-10, CIFAR-100, extended SVHN, ImageNet) using error-prone activations and tested them without errors to study the resilience of the training process. With the exception of …

We prove that a Bers slice is never algebraic, meaning that its Zariski closure in the character variety has strictly larger dimension. A corollary is that skinning maps are never constant. The proof uses grafting and the theory of complex projective structures.

Uniformizes compact complex manifolds via Anosov representations.

We survey some major contributions to Riemann's moduli space and Teichm{ü}ller space. Our report has a historical character, but the stress is on the chain of mathematical ideas. We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space a…

We consider time-domain digital backpropagation with chromatic dispersion filters jointly optimized and quantized using machine-learning techniques. Compared to the baseline implementations, we show improved BER performance and >40% power dissipation reductions in 28-nm CMOS.

The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3-manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We prove that the conjecture obtains for each complete hyperbolic 3-manifold with no cusps and incompr…

Study immersions of surfaces into SL(2,C) and geodesics space.

Bayes Error Rate estimators are evaluated for accuracy and sample requirements.

We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar resu…

In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular, this applies to complements of tubes of radius at least $R_0 = \arctanh(1/\sqrt{3}) \approx 0.65848$ around the singular set of hyperbolic cone manifolds, removing th…