Upper bounds on geodesics in hyperbolic manifolds.
problem Bounding the number of shortest closed geodesics in hyperbolic manifolds.
method Using the Selberg trace formula, we derive upper bounds on the number of shortest and primitive geodesics.
result Generalized Parlier's theorem to higher dimensions and uniform bounds for manifolds with bounded geometry.
In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsific…
Mirzakhani's thesis counts geodesics on hyperbolic surfaces, finding a specific asymptotic formula.
problem Counting simple closed geodesics on hyperbolic surfaces.
method Inspired by lattice point counting, uses principles of homogeneous dynamics.
result The number of simple closed geodesics of length ≤ L is asymptotic to L^(6g-6) times a constant.
Two Riemannian manifolds are called eigenvalue equivalent when their sets of eigenvalues of the Laplace-Beltrami operator are equal (ignoring multiplicities). They are (primitive) length equivalent when the sets of lengths of their (primitive) closed geodesics are equal. We give a general construction of eigenvalue equ…
Motivated by the results of Scott and Patel about "untangling" closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back …
For hyperbolic surfaces, primitive lengths are bounded below by a specific formula.
problem Understanding the distribution of primitive closed-geodesic lengths on hyperbolic surfaces.
method Analyzing Teichmüller space and proving a lower bound on the number of distinct primitive lengths.
result There exists a lower bound on the number of distinct primitive closed-geodesic lengths for hyperbolic surfaces.
Study counts geodesics on hyperbolic 3-manifolds, proving prime theorems.
problem Counting primitive closed geodesics on compact hyperbolic 3-manifolds.
method Proves prime geodesic theorems with symmetric error terms in length and holonomy.
result Effective equidistribution of holonomy and symmetric error terms.
We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of minimal configurations not realized by geodesics in any hyperbolic metric.
Geometric techniques reveal new insights into Gromov-Witten invariants.
problem Formulating Gromov-Witten invariants for complete intersections in projective space.
method Combining geometric group theory and geometric topology, focusing on geodesic laminations.
result Primitive cohomologies unify mathematical formulations of Gromov-Witten invariants.
The paper shows how to uniquely determine a connection up to gauge.
problem Determining a connection up to gauge from its holonomies.
method Using hyperbolic dynamical systems and transport operators.
result The primitive trace map is locally injective near generic points.
New method calculates winding of geodesics on surfaces.
problem Understanding the distribution of geodesics on surfaces.
method Introducing a new construction of winding numbers for geodesics on cusped hyperbolic orbifolds.
result Winding numbers can be expressed by Rademacher symbols for various arithmetic families of surfaces.
We consider non-elementary representations of two generator free groups in PSL(2,C), not necessarily discrete or free, G=<A,B>. A word in A and B, W(A,B), is a palindrome if it reads the same forwards and backwards. A word in a free group is {\sl primitive} if it is part of a minimal generating …
The study counts geodesics on curved surfaces with specific intersections.
problem Counting geodesics with exact intersection numbers on curved surfaces.
method Introduced a dynamical scattering operator and used Pollicott-Ruelle resonances.
result Asymptotic growth of geodesics with prescribed intersections.
New representation theory for closed geodesic subflows.
problem Classifying representations with good geometric properties.
method Restricting to invariant closed geodesic subflows.
result Equivalent characterizations and properties of new representations.
The paper finds lower bounds for volumes of complex geometric structures.
problem Estimating the volume of complex geometric structures.
method Reduction to a counting problem in the unit tangent bundle, solved using exponential multiple mixing for the geodesic flow.
result First known lower bound for the volume of these manifolds in terms of curve length.
New results on geodesic flows using curve shortening flow.
problem Stability and existence of geodesics in Riemannian surfaces.
method Curve shortening flow approach.
result Existence of infinitely many geodesics intersecting a given one in every primitive class.
Uniform counting formulas for orthogeodesics in Kleinian groups converge.
problem Counting orthogeodesics in Kleinian groups converging to a limit.
method Spectral gap of the limit manifold and geodesic flow mixing property.
result Asymptotically uniform counting formulas for orthogeodesics.
Using the Selberg trace formula, we show that for a hyperbolic 2-orbifold, the spectrum of the Laplacian acting on functions determines, and is determined by, the following data: the volume; the total length of the mirror boundary; the number of conepoints of each order, counting a mirror corner as half a conepoint; an…
Researchers found new functions for spherical clothoids using special functions.
problem Developing new mathematical functions for spherical clothoids.
method Used confluent hypergeometric functions and Meixner-Pollaczek polynomials.
result Presented Cartesian coordinate functions and stereographic projections.
The study calculates the growth rate of reciprocal hyperbolic elements in Hecke groups.
problem Counting reciprocal hyperbolic elements in Hecke groups.
method Analyzes conjugacy classes of hyperbolic elements associated with reciprocal geodesics.
result Determines the asymptotic growth rate and limiting constant of primitive conjugacy classes of reciprocal hyperbolic elements.
Paper proves stability for recovering connections from holonomy traces.
problem Recovering a connection from holonomy traces on Riemannian manifolds.
method Combination of microlocal analysis and non-Abelian approximate Livsic Theorem.
result Hölder type stability estimates for holonomy inverse problem.
The paper extends geometric results from negatively-curved spaces to strictly convex Hilbert geometry.
problem Extending geometric results from negatively-curved spaces to strictly convex Hilbert geometry.
method Demonstrates dynamical and counting results for geometrically-finite strictly convex projective structures with Hilbert metric.
result Hilbert geodesic flow is strongly mixing and orbits and primitive closed geodesics equidistribute.
Let X be a globally symmetric space of noncompact type, and $Γ\subset\Isom(X)$ a Schottky group of axial isometries. Then M:=X/Γ is a locally symmetric Riemannian manifold of infinite volume. The goal of this note is to give an asymptotic estimate for the number of primitive closed geodesics in M modulo free homo…
New findings on geometric flows and equidistribution in Hilbert geometry.
problem Characterizing dynamical and counting results in Hilbert geometry.
method Study of dynamical and counting results in rank-one properly convex projective structures with Hilbert metrics.
result Hilbert geodesic flow is strongly mixing and orbits and primitive closed geodesics equidistribute.
Anosov subgroup equidistributes geodesics and holonomies on homogeneous spaces.
problem Equidistribution of geodesics and holonomies in Anosov homogeneous spaces.
method Analyzes maximal flat cylinders and their holonomies for Anosov subgroups.
result Joint equidistribution of maximal flat cylinders and holonomies as circumference tends to infinity.
Berge introduced knots that are primitive/primitive with respect to the genus 2 Heegaard surface, F, in S3; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to F; surgery on these knots at the surface slope yield…
The twisted torus knots lie on the standard genus 2 Heegaard surface for S3, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted tor…
Primitive curves in handlebodies form a connected complex.
problem Understanding the structure of curves in handlebodies.
method Defining and analyzing primitive curves and constructing sequences between them.
result The primitive curve complex for a handlebody is connected.
The study describes maximal Fuchsian subgroups of a specific Bianchi group and computes their covolumes.
problem Characterizing maximal Fuchsian subgroups of a Bianchi group and computing their covolumes.
method Explicit description of conjugacy classes of maximal Fuchsian subgroups using quaternion algebras.
result Explicit description and covolumes of maximal Fuchsian subgroups.
Let G=⟨A,B⟩ be a non-elementary two generator subgroup of the isometry group of H2, the hyperbolic plane. If G is discrete and free and geometrically finite, its quotient is a pair of pants and in prior work we produced a formula for the number of essential self intersections (ESIs) of a…
Note on connectedness of primitive disk complex.
problem Whether primitive disk complex is connected for genus > 3 Heegaard splittings.
method Defined and quotiented primitive disk complex to prove connectedness.
result Homotopy primitive disk complex is connected.
Study primitive cohomology in symplectic manifolds.
problem Understanding primitive cohomology in symplectic geometry.
method Reviewing superbundle-valued forms and proving a transgression formula.
result Introduced primitive characteristic classes and proved a transgression formula.
Disk surgery on primitive disks of genus-3 Heegaard splittings of 3-sphere yields no primitive disks.
problem Characterizing primitive disks in genus-3 Heegaard splittings of 3-sphere.
method Analyzing the effect of disk surgery on primitive disks in genus-3 Heegaard splittings of 3-sphere.
result Primitive disks in genus-3 Heegaard splittings of 3-sphere are not weakly closed under disk surgery.
No primitive Teichmüller curves found in Prym(2,2).
problem Existence of primitive Teichmüller curves in Prym(2,2).
method Completed work by Lanneau and Möller.
result No primitive Teichmüller curves in Prym(2,2).
Study local Weyl law on hyperbolic surfaces, identifying geodesic loops.
problem Understanding the variance of a local Weyl law on hyperbolic surfaces.
method Explicit integration of test functions, stationary phase arguments, and geometric analysis of geodesic loops.
result Identifies length-minimizing geodesic loops and sequences, proving they are simple.
The paper analyzes vehicle encounters using driving primitives.
problem Understanding complex vehicle encounters for autonomous driving.
method Decompose driving data into primitives using nonparametric Bayesian learning.
result More than 4000 driving primitives identified from 976 encounters.
A symplectic form has a primitive with nowhere vanishing β.
problem Existence of nowhere vanishing primitives for symplectic forms.
method Constructing a primitive β for an exact symplectic form ω. result There exists a nowhere vanishing primitive β for the symplectic form ω. We show that lens space surgeries on knots in S3 which arise from the primitive/Seifert type construction also arise from the primitive/primitive construction. This is the first step of a three step program to prove the Berge conjecture for tunnel number one knots.
For a genus two Heegaard splitting of a lens space, the primitive disk complex is defined to be the full subcomplex of the disk complex for one of the handlebodies of the splitting spanned by all vertices of primitive disks. In this work, we describe the complete combinatorial structure of the primitive disk complex fo…
We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into primitive forms and therefore lead directly to the construction of primitive cohom…
Study primitive decompositions for harmonic forms on almost Kähler manifolds.
problem Decomposing harmonic forms on almost Kähler manifolds.
method Proved primitive decompositions for Bott-Chern and Aeppli harmonic forms in specific bidegrees.
result Optimal bidegrees for primitive decompositions of harmonic forms.
In this article, we prove that every arithmetic locally symmetric orbifold of classical type without Euclidean or compact factors has arbitrarily long arithmetic progressions in its primitive length spectrum. Moreover, we show the stronger property that every primitive length occurs in arbitrarily long arithmetic progr…
A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in F, the genus 2 Heegaard surface for S3. Primitive/primitive and primitive/Seifert knots lie in F in a particular way. Dean gives sufficient conditions for the parameters of the tw…
Two conditions on primitive elements are shown to be equivalent.
problem Equivalence of primitive stability and Bowditch's BQ-condition. method Proof of equivalence between two conditions on primitive elements.
result Primitive stability and Bowditch's BQ-condition are equivalent. A new reinforcement learning approach using competitive primitives that specialize and specialize based on information needs.
problem Complex environments require efficient and specialized decision-making.
method Decomposes policy into competitive primitives that decide based on information needs, regularized to use minimal information.
result Improves generalization over flat and hierarchical policies.
Algorithm finds smooth primitives for exact forms.
problem Computing smooth primitives for exact forms on manifolds.
method Diagram chasing in Čech-de Rham complex, explicit formulas.
result Explicit formulas for primitive families.
Lengths of simple closed geodesics on hyperbolic surfaces in prescribed homology classes
problem Estimating the number of simple closed geodesics of a fixed length and homology class on a hyperbolic surface
method Using asymptotic formulas and numerical evidence
result Proving an asymptotic lower bound for the number of such geodesics
In this paper, we give a complete criterion for a discrete faithful representation $ρ:F_n \ra \pslc$ to be primitive stable. This will answer Minsky's conjectures about geometric conditions on $\H^3/ρ(F_n)$ regarding the primitive stability of ρ.