arXiv research

When $S$ is a closed, orientable surface with genus $g(S) \geq 2$, we show that the automorphism group of the compression body graph $\mathcal{CB}(S)$ is the mapping class group. Here, vertices are compression bodies with exterior boundary $S$, and edges connect pairs of compression bodies where one contains the other.

New method connects compression bodies through cone manifolds.

We show that the compression body graph has infinite diameter.

Detects handlebodies and mapping class extensions using bordered Floer homology.

A recent paper (\cite{BJM}) by Biringer, Johnson, and Minsky prove that any pseudo-Anosov whose stable lamination is the limit of disks in a compression body has a power which extends over some non-trivial minimal compression body. This paper presents an alternative proof of their theorem, using techniques of Long and …

Let M be a nontrivial compression body without toroidal boundary components. We study the dynamics of the group of outer automorphisms of the fundamental group of M on the PSL(2,C)-character variety of M.

Given a 3-manifold that can be written as the double of a compression body, we compute the Chern-Simons critical values for arbitrary compact connected structure groups. We also show that the moduli space of flat connections is connected when there are no reducibles.

New proof of Laudenbach and Poénaru's theorem on 4D 1-handlebodies.

We show that a pseudo-Anosov map on a boundary component of an irreducible 3-manifold has a power that partially extends to the interior if and only if its (un)stable lamination is a projective limit of meridians. The proof is through 3-dimensional hyperbolic geometry, and involves an investigation of algebraic limits …

We analyze the mapping class group of extendible automorphisms of the exterior boundary W of a compression body of dimension 3 or 4, which extend over the compression body (Q,V), where V is the interior boundary. Those that extend as automorphisms of (Q,V) rel V are called discrepant automorphisms, forming the mapping …

Let M be a hyperbolizable, nontrivial compression body without toroidal boundary components. In this paper, we characterize which discrete and faithful representations of the fundamental group of M into PSL(2,C) are separable-stable. The set of separable-stable representations forms a domain of discontinuity for the ac…

The (isothermic) compressibility of lattice knots can be examined as a model of the effects of topology and geometry on the compressibility of ring polymers. In this paper, the compressibility of minimal length lattice knots in the simple cubic, face centered cubic and body centered cubic lattices are determined. Our r…

We consider hyperbolic structures on the compression body C with genus 2 positive boundary and genus 1 negative boundary. Note that C deformation retracts to the union of the torus boundary and a single arc with its endpoints on the torus. We call this arc the core tunnel of C. We conjecture that, in any geometrically …

If M is a manifold with compressible boundary, we analyze essential disks in M, as well as incompressible, but not necessarily boundary incompressible, surfaces in M. We are most interested in the case where M is a handlebody or compression body. The analysis depends on a new normal surface theory. We hope the normal s…

Telemonitoring of electroencephalogram (EEG) through wireless body-area networks is an evolving direction in personalized medicine. Among various constraints in designing such a system, three important constraints are energy consumption, data compression, and device cost. Conventional data compression methodologies, al…

Fetal ECG (FECG) telemonitoring is an important branch in telemedicine. The design of a telemonitoring system via a wireless body-area network with low energy consumption for ambulatory use is highly desirable. As an emerging technique, compressed sensing (CS) shows great promise in compressing/reconstructing data with…

Efficient point cloud compression is fundamental to enable the deployment of virtual and mixed reality applications, since the number of points to code can range in the order of millions. In this paper, we present a novel data-driven geometry compression method for static point clouds based on learned convolutional tra…

We prove that an irreducible 3-manifold whose fundamental group satisfies a certain group-theoretic property called RFRS is virtually fibered. As a corollary, we show that 3-dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. These include the Seifert Weber …

We give the rectangle condition for strong irreducibility of Heegaard splittings of $3$-manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of a $2$-fold covering of $S^3$ branched along a link. The condition implies that any thin meridional level surface in the link complement is incom…

Study quotients of curve complex actions by mapping class group.

New framework for sutured manifolds using handleslides and Heegaard invariants.

New model-independent compact representations of imaginary-time data are presented in terms of the intermediate representation (IR) of analytical continuation. This is motivated by a recent numerical finding by the authors [J. Otsuki et al., arXiv:1702.03056]. We demonstrate the efficiency of the IR through continuous-…

Let (V,W;F) be a weakly reducible, unstabilized, genus three Heegaard splitting in an orientable, irreducible 3-manifold M and DVW(F) the subset of the disk complex D(F) consisting of simplices having at least one vertex from V and at least one vertex from W. In this article, we describe the shape of DVW(F) and prove t…

We propose a new embedding method which is particularly well-suited for settings where the sample size greatly exceeds the ambient dimension. Our technique consists of partitioning the space into simplices and then embedding the data points into features corresponding to the simplices' barycentric coordinates. We then …

Sparsity-constrained optimization has wide applicability in machine learning, statistics, and signal processing problems such as feature selection and compressive Sensing. A vast body of work has studied the sparsity-constrained optimization from theoretical, algorithmic, and application aspects in the context of spars…

New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.

Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds: (1) N has non-empty conformal boundary, (2) N is not homotopy equivalent to a compres…

It is unknown whether an unknotting tunnel is always isotopic to a geodesic in a finite volume hyperbolic 3-manifold. In this paper, we address the generalization of this problem to hyperbolic 3-manifolds admitting tunnel systems. We show that there exist finite volume hyperbolic 3-manifolds with a single cusp, with a …

The paper proves a conjecture about the shape of floating bodies.

New index characterizes non-smooth Zoll convex bodies.

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

A groupoid $Ω\left( \mathcal{B} \right)$ called material groupoid is naturally associated to any simple body $\mathcal{B}$. The material distribution is introduced due to the (possible) lack of differentiability of the material groupoid. Thus, the inclusion of these new objects in the theory of material bodies opens th…

Study shows volumes of complex classes can be represented by convex bodies.

The paper finds the unique minimizer of area for hyperbolic bodies with curvature constraints.

Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.

Convolutional Neural Networks (CNNs) are extremely computationally demanding, presenting a large barrier to their deployment on resource-constrained devices. Since such systems are where some of their most useful applications lie (e.g. obstacle detection for mobile robots, vision-based medical assistive technology), si…

We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, an…

Study on discrete Okounkov bodies and their applications.

Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of resu…

We consider the motion of small bodies in general relativity. The key result captures a sense in which such bodies follow timelike geodesics (or, in the case of charged bodies, Lorentz-force curves). This result clarifies the relationship between approaches that model such bodies as distributions supported on a curve, …

Prove a generalization of Werner's formula for the volume of illumination bodies on Riemannian manifolds.

This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric inequalities. We also prove that the homogeneous geominimal surface areas are continuous, …

For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area…

The paper proves no multiple equichordal points exist in convex bodies.

We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This…

The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Alexandrov's problem consists in finding a convex body with given curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and sufficient condit…

For a knot $K\subset S^3$, its exterior $E(K) = S^3\backslashη(K)$ has a singular foliation by Seifert surfaces of $K$ derived from a circle-valued Morse function $f\colon E(K)\to S^1$. When $f$ is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critica…

Let $Δ$ be a $d$-dimensional normal pseudomanifold, $d \ge 3.$ A relative lower bound for the number of edges in $Δ$ is that $g_2$ of $Δ$ is at least $g_2$ of the link of any vertex. When this inequality is sharp $Δ$ has relatively minimal $g_2$. For example, whenever the one-skeleton of $Δ$ equals the one-skeleton of …