Study geodesics in constrained curve spaces, including elastic curves and concentric circles.
problem Geodesics in constrained curve spaces with Sobolev metrics.
method Intrinsic and constructive approaches.
result Construct geodesics in elastic curve and concentric circle spaces.
In earlier work of NK new closed embedded smooth minimal surfaces in the round three-sphere S3(1) were constructed, each resembling two parallel copies of the equatorial two-sphere Seq2 joined by small catenoidal bridges, with the catenoidal bridges concentrating along two parallel circles, o…
The paper generalizes envelope constructions for chords in circles, revealing complex singularities.
problem Understanding envelopes of chords in circles with varying parameters and configurations.
method Generalizing the embroidery method to rational and concentric circles, breaking symmetry to reveal higher singularities.
result Higher singularities like swallowtails and butterflies can be unfolded, revealing their structure.
The paper examines circle graphs of Gauss diagrams and finds counterexamples to previous descriptions.
problem Problems with previous descriptions of realizable Gauss diagrams.
method Experimental checking and formulation of new descriptions of realizable circle graphs.
result New descriptions of realizable circle graphs and an algorithm for checking realizability.
Study differentiable maps on hypersurface links, finding fold maps with circle singular value sets.
problem Understanding differentiable maps on hypersurface links.
method Restricting holomorphic functions to hypersurface links and analyzing the resulting maps.
result Found fold maps with concentric circle singular value sets.
New patterns on spheres and hyperbolic planes described by integrable systems.
problem Integrable systems and variational principles for spherical and hyperbolic ring patterns.
method Discrete integrable system, variational principles, elliptic dilogarithm function.
result Existence and uniqueness of ring patterns for Dirichlet and Neumann problems.
Normalizing flows are a powerful tool for building expressive distributions in high dimensions. So far, most of the literature has concentrated on learning flows on Euclidean spaces. Some problems however, such as those involving angles, are defined on spaces with more complex geometries, such as tori or spheres. In th…
New discrete cmc surfaces defined from sphere packings and combinatorics.
problem Creating constant mean curvature surfaces from discrete data.
method Discrete cmc surfaces defined via sphere packings and combinatorial patterns.
result Construction of discrete cmc surfaces from orthogonal ring patterns.
Pedal curves derived from ellipses are invariant in area.
problem Finding invariant areas of pedal curves derived from ellipses.
method Analytical proof and explicit area expressions.
result Pedal curves derived from ellipses are invariant in area.
{\it Fold maps} are fundamental tools in generalizing the theory of Morse functions and its application to studies of geometric properties of manifolds. One of the fundamental and important problems in the theory of fold maps is to construct explicit fold maps, which are often difficult. In this paper, we construct new…
Doodles link to commutator identities in a 2-sphere.
problem Understanding commutator identities in free groups via doodles.
method Analyzing doodles with proper noose systems and establishing bijections.
result A bijection between doodles and commutator identities.
In this paper we solve the Plateau problem for spacelike surfaces with constant mean curvature in Lorentz-Minkowski three-space ł3 and spanning two circular (axially symmetric) contours in parallel planes. We prove that rotational symmetric surfaces are the only compact spacelike surfaces in ł3 of constant mean c…
We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. Generalizing Andrews' theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung-Min…
3D manifolds can map to a plane with specific curve patterns.
problem Characterizing 3D manifolds that can map to the plane with certain curve patterns.
method Analyzing fold maps and their critical value sets.
result Closed orientable 3-manifolds admit round fold maps into the plane if and only if they are graph manifolds.
Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Gamma be the fundamental group of a compact orientable surface of genus at least 2. We survey the study of maximal representations, that is the subset of Hom(Gamma,G) which is a union of components characterized by…
Study cohomology rings of 3D manifolds with round fold maps into the plane.
problem Understanding cohomology rings of 3D manifolds with round fold maps.
method Analyzing cohomology rings of 3D manifolds admitting round fold maps into the plane.
result Explicit new study showing relation between coefficient rings and topological types of round fold maps.
Invariants count singularities and vertices of plane curves.
problem Counting singularities and vertices of plane curves.
method Defining and analyzing geometric invariants If and Vf. result For almost all plane curves, these invariants are finite and bounded.
The paper finds circle packings with specific curvatures in hyperbolic geometry.
problem Finding circle packings with prescribed total geodesic curvatures and discrete Gaussian curvatures.
method Established existence and rigidity via variational principle, introduced combinatorial p-th Calabi flows.
result Introduced combinatorial p-th Calabi flows to find circle packings with prescribed curvatures.
The study connects group structure to smooth actions on one-manifolds.
problem Understanding how group actions affect the smoothness of manifolds.
method Analyzes the relationship between group algebraic structure and smoothness of group actions on one-dimensional manifolds.
result Uniform construction of groups acting on compact interval and circle with prescribed regularity.
Study generates infinite circle packings with a specific property.
problem Generating infinite circle packings with a unique property.
method Investigates an infinite family of circle packings and uses them to create Apollonian packings.
result Created an infinite set of circle packings with the Apollonian property.
The paper extends Descartes' circle theorem to n-flower configurations using hyperbolic geometry.
problem Extending Descartes' circle theorem to n-flower configurations.
method Spinorial description of horospheres in hyperbolic geometry.
result An explicit equation satisfied by the curvatures of n-flower configurations.
Solves Apollonius' problem using oriented circles and inversive geometry.
problem Constructing a circle tangent to three given circles.
method Using oriented circles and inversive invariants, reversing each given circle to find solutions.
result The problem has 0, 1, or 2 solutions, depending on the configuration of given circles.
The paper explores universal circles for Anosov foliations and their uniqueness.
problem Exploring the uniqueness of universal circles for Anosov foliations.
method Using the flow space of an Anosov flow to parameterize the circle bundle at infinity of the foliations.
result Several constructions of a universal circle are typically distinct and not conjugate.
Link projections with the same circle arrangement can be transformed by specific moves.
problem Characterizing link projections based on their circle arrangements.
method Local moves to transform link projections and analyze their circle arrangements.
result Two link projections have the same circle arrangement if and only if they can be transformed into each other by certain local moves.
A Steiner chain of length k consists of k circles, tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that once a chain of k circles exists, there exists a 1-parameter family of such chains with the same parent circles that can be …
Proves rigidity of circle packings in the plane, generalizing previous work.
problem Rigidity of infinite inversive distance circle packings in the plane.
method Maximal principle for generic weighted Delaunay inversive distance circle packings and ring lemma for inversive distance circle packings in hexagonal triangulated plane.
result Proves Bowers-Stephenson's conjecture for inversive distance circle packings.
Theory for algebraic data on categories via concentration structures.
problem Defining algebraic structures on categories.
method Introducing concentration structures and concentration monoids.
result Every group can be represented as a concentration monoid of a trivial category.
We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex projective structure. Motivated by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending …
Classifies surfaces with great and small circles through each point.
problem Identifying surfaces with specific circle properties.
method Topological classification of surfaces in 3D unit sphere.
result Surfaces are homeomorphic to five normal forms.
Study local diffeomorphisms of conformal circles in pseudo-Riemannian manifolds.
problem Understanding local diffeomorphisms of conformal circles.
method Variations of conformal circles and pseudo-Riemannian manifolds.
result Local diffeomorphisms of conformal circles are conformal local diffeomorphisms.
Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.
problem Comparing geometric quantities of circle packings with different boundary values.
method Combinatorial Calabi flows and maximum principle.
result Discrete Schwarz-Pick lemma proven for generalized circle packings.
A ``hyperideal circle pattern'' in S2 is a finite family of oriented circles, similar to the ``usual'' circle patterns but such that the closed disks bounded by the circles do not cover the whole sphere. Hyperideal circle patterns are directly related to hyperideal hyperbolic polyhedra, and also to circle packings. …
Circle graph automorphisms match circle's and are strongly universal.
problem Identifying the automorphism group of the circle.
method Proving the circle graph's automorphism group coincides with the circle's and showing the circle graph's rational chords form a strongly universal element.
result The circle graph's automorphism group is strongly universal.
The paper studies circle packings using renormalization and subdivision rules.
problem Characterizing and proving properties of circle packings with specific subdivision rules.
method Iterations of skinning maps on Teichmüller spaces, renormalization theory, subdivision rules.
result Uniformly contracting renormalization operator and geometric inflexibility of circle packings.
Paper proves circle packings converge to Riemann mapping for Jordan domains.
problem Proving discrete conformal maps converge to Riemann mapping.
method Establishing solvability theorem for inversive distance circle packings.
result Bowers-Stephenson's conjecture for Jordan domains is proven.
Paper introduces new flows to find circle packings with specific curvature.
problem Finding circle packings with prescribed total geodesic curvatures.
method Introduces combinatorial Calabi flow, fractional combinatorial Calabi flow, and combinatorial p-th Calabi flow.
result Establishes conditions for the longtime behaviors of these flows.
Paper addresses concentration of distances for fractional quasi p-norms, identifying conditions for concentration and anti-concentration.
problem Understanding concentration of distances for fractional quasi p-norms in high dimensions.
method Analyzes conditions for concentration and anti-concentration of distances for fractional quasi p-norms.
result Identifies conditions for concentration and anti-concentration of fractional quasi p-norms, ruling out some approaches and specifying conditions for control.
Study of combinatorial Calabi flow on ideal circle patterns.
problem Finding ideal circle patterns with prescribed curvatures.
method Combinatorial Calabi flow in hyperbolic and Euclidean geometry.
result Flow converges exponentially to ideal circle patterns.
Proves existence of circle patterns on surfaces with cusps.
problem Existence of circle patterns with prescribed angles on surfaces with cusps.
method Introduced combinatorial Ricci and Calabi flows to prove longtime existence and convergence.
result Existence of generalized circle patterns with prescribed angles on surfaces with cusps.
Study Finsler metric measure manifolds' concentration properties.
problem Understanding concentration properties in Finsler metric measure manifolds.
method Established relationships with observable diameter, isoperimetric inequalities, and first eigenvalue.
result Derived a Cheng type upper bound estimate for the first closed eigenvalue.
Unique circle patterns on spheres found for spherical conical metrics.
problem Non-uniqueness in circle packing for spherical metrics.
method Prescribed geodesic total curvature instead of cone angles.
result Unique existence of circle patterns for spherical conical metrics.
Paper introduces 'zippers' for constructing universal circles.
problem Constructing universal circles for hyperbolic 3-manifolds.
method Introduces zippers to directly construct universal circles.
result New and direct way to construct universal circles.
Paper proves existence and uniqueness of circle patterns on surfaces with assigned geodesic curvatures.
problem Existence and uniqueness of circle patterns on surfaces with prescribed geodesic curvatures.
method Applied Perron's method and Thurston's algorithm to prove existence and convergence.
result Existence and uniqueness of circle patterns on surfaces with prescribed geodesic curvatures.
Study circle patterns on tori, linking symplectic forms and homeomorphisms.
problem Understanding circle patterns on tori and their symplectic properties.
method Investigates the space of circle patterns on closed tori with complex projective structures, embedding it into Teichmüller spaces and analyzing symplectic forms.
result Non-degeneracy of the pulled-back Weil-Petersson symplectic form and homeomorphism between circle patterns and Teichmüller spaces.
Study of circle arrangements related to Morse-Bott functions.
problem Understanding the geometry and singularity theory of Morse-Bott functions.
method Systematic construction of circle arrangements centered at existing circles, studying local changes in Reeb graphs.
result Reeb graphs of Morse-Bott functions are spaces of all components of preimages of single points.
The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.
problem Existence and rigidity of circle packings with conical singularities.
method Variational principle and combinatorial Ricci flow.
result Existence and rigidity of circle packings with prescribed total geodesic curvature.
Asymptotics for equidistribution of circles on hyperbolic surfaces.
problem Equidistribution of circles on hyperbolic surfaces.
method Spectral method and statistical limit theorems.
result Precise asymptotics for the rate of equidistribution of circles.
New method improves missing mass concentration bounds.
problem Missing mass concentration problem
method New method of estimating concentration of heterogenic sums
result Slightly improved state-of-the-art bounds