The paper studies eigenvalues in negatively curved spaces, showing their relationship to the thick-thin decomposition.
problem Investigating the relationship between eigenvalues and the geometric structure of negatively curved spaces.
method Analyzing the eigenvalues of the Laplacian on the thick part of a manifold's thick-thin decomposition.
result Eigenvalues in the thick part of a negatively curved manifold are significantly smaller than those in the entire manifold.
New theory of distributions on spaces with singular submanifolds.
problem Defining distributions on spaces with singular submanifolds.
method Construction of thick distributions, operations, and special distributions.
result Clarified connection between thick and classical distributions.
Path-connectivity of thick laminations on high-genus surfaces.
problem Path-connectivity of thick laminations on high-genus surfaces.
method Teichmüller ray analysis and subshift of finite type construction.
result Path-connectedness of the Morse boundary of the mapping class group.
What length of rope (of given diameter) is required to tie a particular knot? To answer this question, we define some new notions of thickness for a space curve, one based on Gromov's distortion, and another generalizing the thickness of Litherland, Simon et al. We prove a basic inequality between these thickness measu…
New examples show limits of physical link isotopies.
problem Limits of physical link isotopies in thick and thickly embedded links.
method Construction of specific examples of thick and thickly embedded links.
result Explicit examples of links that cannot be split through thick homotopies.
A knot's thickness is measured by its β invariant, a new numerical invariant.
problem Measuring the thickness of knots.
method Introduced a new invariant β(K) and proved an inequality between β(K) and knot Floer thickness.
result All Montesinos knots have thickness at most one.
The first algorithm for sampling the space of thick equilateral knots, as a function of thickness, will be described. This algorithm is based on previous algorithms of applying random reflections. To prove the existence of the algorithm, we describe a method for turning any knot into the regular planar polygon using on…
Groups with certain thickness have empty Floyd boundary.
problem Understanding Floyd boundaries of groups.
method Proving triviality of Floyd boundaries for thick groups.
result Trivial Floyd boundary for thick groups.
Method estimates section thickness and XY anisotropy in ssEM images.
problem Accurate 3D reconstructions require precise section thickness and XY anisotropy estimates.
method Non-parametric Bayesian regression of image statistics.
result Method has lower estimation error compared to existing methods.
Algorithm computes knot Floer complex for knots of thickness one.
problem Computing knot Floer complexes for knots of thickness one.
method Developed and implemented an algorithm for knots of thickness one.
result Algorithm can compute full knot Floer complex for knots of thickness one.
For right-angled Coxeter groups WΓ, we obtain a condition on Γ that is necessary and sufficient to ensure that WΓ is thick and thus not relatively hyperbolic. We show that Coxeter groups which are not thick all admit canonical minimal relatively hyperbolic structures; further, we show that in such a structure, …
We describe some problems, observations, and conjectures concerning thickness and packing density of knots and links in $\sp^3$ and R3. We prove the thickness of a nontrivial knot or link in $\sp^3$ is no more than 4π, the thickness of a Hopf link. We also give arguments and evidence supporting the conject…
Bound skinning map diameter on hyperbolic 3-manifold rays.
problem Bounding skinning map diameter on hyperbolic 3-manifold rays.
method Using acylindrical hyperbolic 3-manifolds and thick Teichmüller geodesic rays.
result Diameter of skinning map bounded by a constant.
New examples of gordian unlinks show different rope geometries.
problem Tackling the existence of non-trivial unknots with prescribed length and thickness.
method Provided the first examples of gordian unlinks with thickness in [1,2).
result Thinner normal tubes lead to different rope geometries.
We show that thick morphisms (or microformal morphisms) between smooth (super)manifolds, introduced by us before, are classical limits of `quantum thick morphisms' defined here as particular oscillatory integral operators on functions.
Thick morphisms link quantum mechanics and supermanifolds.
problem Understanding the relationship between quantum mechanics and supermanifolds.
method Introducing and analyzing thick morphisms of supermanifolds.
result Quantum thick morphisms provide a spinor representation.
The paper explores new algebraic structures and morphisms in graded settings.
problem Understanding new algebraic structures and morphisms in graded settings.
method Introducing and analyzing L∞-, P∞-, and S∞-algebras, and thick morphisms in a Z2imesZ-graded context. result Shifted S∞-thick morphisms induce L∞-morphisms of shifted S∞-structures. The paper introduces boundary thickness as a measure for improving model robustness.
problem Improving the robustness of machine learning models to adversarial and non-adversarial corruptions.
method Introducing boundary thickness as a measure and showing how various procedures can increase it.
result Thicker decision boundaries lead to improved robustness against adversarial and out-of-distribution transforms.
We introduce a new quasi-isometry invariant of 2-dimensional right-angled Coxeter groups, the hypergraph index, that partitions these groups into infinitely many quasi-isometry classes, each containing infinitely many groups. Furthermore, the hypergraph index of any right-angled Coxeter group can be directly computed f…
New proof shows knot Floer thickness limits bad domains in diagrams.
problem Understanding knot thickness and dealternating number.
method Modified Stipsicz-Szabo approach using Kauffman states.
result Knot Floer thickness is a lower bound on dealternating number.
We show that a complete hyperbolic n-manifold has a geodesic triangulation such that the tetrahedra contained in the thick part are L-bilipschitz diffeomorphic to the standard Euclidean n-simplex, for some constant L depending only on the dimension and the constant used to define the thick-thin decomposition of M.
Study on reducing surgeries on knots, developing thickness and genus bounds.
problem Understanding reducible surgeries on knots in S3. method Developed thickness bounds for L-space knots and lower bounds on slice genus; used d-invariants and mapping cone formula from Heegaard Floer homology. result Provided new upper bounds on reducing slopes for fibered, hyperbolic slice knots and on multiple reducing slopes for slice knots; verified the Cabling Conjecture for thin knots.
We show the existence of a thick thin decomposition of the domain of a pseudo holomorphic curve with boundary. The geometry of the thick part is bounded uniformly in the energy. Furthermore, in the thick part, there is a uniform bound on the differential which is exponential in the energy. The thin part consists of ann…
We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or on a tree, connection…
New phenomena in knot thickness revealed by Legendrian large cables.
problem Understanding non-uniformly thick knots and their Legendrian large cables.
method Definition and analysis of Legendrian large cables, and construction of knots.
result Existence of knots that are not uniformly thick and have virtually overtwisted contact structures.
A graphical calculus for microformal morphisms simplifies complex operations in classical and quantum physics.
problem Simplifying operations in classical and quantum microformal morphisms.
method Developed a graphical calculus inspired by Cattaneo-Dherin-Felder's work on formal symplectic groupoids, extended to quantum thick morphisms.
result Infinite series can be written as sums over bipartite trees for both classical and quantum thick morphisms.
Classifies divergence and thickness in right-angled Coxeter groups.
problem Characterizing the divergence and thickness of right-angled Coxeter groups.
method Completely classifies divergence functions and proves conditions for thickness using the hypergraph index.
result Exact divergence functions of RACGs can be computed from their defining graphs.
We show that every closed, virtually fibered hyperbolic 3-manifold contains immersed, quasi-Fuchsian surfaces with convex cores of arbitrarily large thickness.
We investigate the relationship between a discrete version of thickness and its smooth counterpart. These discrete energies are defined on equilateral polygons with n vertices. It will turn out that the smooth ropelength, which is the scale invariant quotient of length divided by thickness, is the Γ-limit of the di…
We describe a natural decomposition of a normal complex surface singularity (X,0) into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the…
Geodesic currents on surfaces have comparable metrics in thick regions.
problem Comparing the geometry of geodesic currents and their minimizing metrics.
method Analyzing the space of geodesic currents on surfaces and comparing metrics on thick components.
result Geometries of geodesic currents and their minimizing metrics are comparable in thick regions.
Constructs geometric decompositions for thick hyperbolic 3-manifolds with bounded rank.
problem Understanding the structure of thick hyperbolic 3-manifolds with bounded rank.
method Geometric decomposition of the convex core of M.
result Upper bounds on Heegaard genus and radius of embedded balls in terms of rank and injectivity radius.
Study on divergence and thickness for Coxeter groups, generalizing previous work.
problem Characterizing and bounding divergence and thickness for Coxeter groups.
method Characterization of linear divergence, introduction of hypergraph index, new construction of Coxeter systems.
result Upper bounds on divergence and thickness for Coxeter groups, conjectured to be equalities.
Infinite volume found in the thick part of PSLn(R)-Hitchin-Riemann moduli space.
problem Proving infinite volume in the thick part of PSLn(R)-Hitchin-Riemann moduli space. method Employing Goldman flows and internal sequences to find an infinite series of subsets of identical volume.
result Infinite total Atiyah--Bott--Goldman volume for n>2. Graphs of multicurves are hyperbolic, relatively hyperbolic, or thick.
problem Characterizing graphs of multicurves based on their geometric properties.
method Proving graphs of multicurves are hyperbolic, relatively hyperbolic, or thick based on subsurface intersections.
result Geometric characterization of graphs of multicurves.
Study differential operators over maps and their applications in supermanifolds.
problem Understanding differential operators over smooth maps and their applications.
method Recall and study differential operators, formal ℏ-differential operators, pullbacks by thick morphisms, and quantization of symplectic micromorphisms. result Developed constructions and examples of differential operators over maps.
Two subset germs of Euclidean spaces are called blow-spherically equivalent, if their spherical modifications are homeomorphic and the homeomorphism induces homeomorphic tangent links. Blow-spherical equivalence is stronger than the topological equivalence but weaker than the Lipschitz equivalence. We introduce the thi…
We study Khovanov homology classes which have state cycle representatives, and examine how they interact with Jacobsson homomorphisms and Lee's map Φ. As an application, we describe a general procedure, quasipositive modification, for constructing H-thick knots in rational Khovanov homology. Moreover, we show that sp…
New link pairs exist that can't be deformed without increasing length.
problem Link isotopy and thick isotopy constraints.
method Constructed isotopic link pairs with preserved total length.
result Link pairs exist that are not thick isotopic.
Study of cable links of uniformly thick knots, revealing new isotopy phenomena.
problem Understanding Legendrian isotopy in cable links of uniformly thick knots.
method Introduced new technique of Legendrian surgeries to classify Legendrian knots in negative cables of twist knots.
result Found new phenomena of stabilized Legendrian links that are smoothly isotopic but not Legendrian isotopic.
New multi-spiral approach improves packaging of thick membranes.
problem Deploying thick membranes on curved surfaces efficiently.
method Multi-spiral folding approach with prismatic folding lines.
result Improved deployment performance of thick membranes on curved surfaces.
We review and organize some results describing the behavior of a Teichmüller geodesic and draw several applications: 1) We show that Teichmüller geodesics do not back track. 2) We show that a Teichmüller geodesic segment whose endpoints are in the thick part has the fellow travelling property. This fails when the endpo…
Study of Milnor invariants and ropelength of spherical links.
problem Understanding the relationship between the thickness of spherical links and their Milnor invariants.
method Generalized Massey products and Milnor invariants to spherical links, finding optimal asymptotic bounds.
result Optimal asymptotic bounds on Milnor invariants in terms of thickness, revealing a polynomial vs exponential regime.
The paper finds the unique minimizer of area for hyperbolic bodies with curvature constraints.
problem Finding the unique minimizer of area for hyperbolic bodies with curvature constraints.
method Introduced the concept of 'thick λ-sausage' bodies and used extra assumption of thickness to handle non-convex inner parallel bodies. result The thick λ-sausage body is the unique minimizer of area among all bodies with a given length and curvature constraints. Nonuniform tubular neighborhoods of curves in Euclidean n-space are studied by using weighted distance functions and generalizing the normal exponential map. Different notions of injectivity radii are introduced to investigate singular but injective exponential maps. A generalization of the thickness formula is obtaine…
Let X be a building of uniform thickness q+1. L^2-Betti numbers of X are reinterpreted as von-Neumann dimensions of weighted L^2-cohomology of the underlying Coxeter group. The dimension is measured with the help of the Hecke algebra. The weight depends on the thickness q. The weighted cohomology makes sense for all re…
In this paper we explore relationships between divergence and thick groups, and with the same techniques we estimate lengths of shortest conjugators. We produce examples, for every positive integer n, of CAT(0) groups which are thick of order n and with polynomial divergence of order n+1, both these phenomena are new. …
The paper defines a condition for the finiteness of mapping class groups of Heegaard splittings.
problem Conditions for the finiteness of mapping class groups of Heegaard splittings.
method Use of the thick isotopy lemma to establish conditions for finiteness.
result Heegaard splittings with topologically minimal disk complex and finitely generated homotopy group have finitely generated mapping class groups.