arXiv research

The paper examines conditions for commuting conjugates of finite-order mapping classes.

New displacement technique vanishes bounded cohomology in all degrees.

Study on homeomorphism groups of telescoping 2-manifolds showing strong distortion.

Bavard proved a duality theorem between commutator length and quasimorphisms. Burago, Ivanov and Polterovich introduced the notion of a conjugation-invariant norm which is a generalization of commutator length. Entov and Polterovich proved that Oh-Schwarz spectral invariants are subset-controlled quasimorphisms which a…

We classify algebraic curvature tensors such that the Ricci operator is simple (i.e. the Ricci operator is complex diagonalizable and either the complex spectrum consists of a single real eigenvalue or the complex spectrum consists of a pair of eigenvalues which are complex conjugates of each other) and which are Jacob…

The paper introduces statistical and geometric structures on anti-commutable pre-Leibniz algebroids.

In this paper, we give a proof of the result of Brandenbursky and Kȩdra which says that the commutator subgroup of the infinite braid group admits stably unbounded norms. Moreover, we observe the norms which we constructed are equivalent to the biinvariant word norm studied by Brandenbursky and Kȩdra.

Study natural and conjugate mates of Frenet curves in Lie groups.

We define a quasihomomorphism from braid groups to the concordance group of knots and examine its properties and consequences of its existence. In particular, we provide a relation between the stable four ball genus in the concordance group and the stable commutator length in braid groups, and produce examples of infin…

The purpose of this paper is the study of the roots in the mapping class groups. Let $Σ$ be a compact oriented surface, possibly with boundary, let $\PP$ be a finite set of punctures in the interior of $Σ$, and let $\MM (Σ, \PP)$ denote the mapping class group of $(Σ, \PP)$. We prove that, if $Σ$ is of genus 0, then ea…

The paper proves vanishing bounded cohomology for various groups.

The paper examines uniform perfectness of diffeomorphism groups on open manifolds.

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 establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of com…

The paper examines boundedness of diffeomorphism groups of manifold pairs, focusing on the circle case.

We develop a new criterion to tell if a group $G$ has the maximal gap of $1/2$ in stable commutator length (scl). For amalgamated free products $G = A \star_C B$ we show that every element $g$ in the commutator subgroup of $G$ which does not conjugate into $A$ or $B$ satisfies $scl(g) \geq 1/2$, provided that $C$ embed…

New connections on symmetric spaces with invariant properties.

We prove a generalized version of the Morse index theorem for geodesics endowed with a non positive definite metric tensor (semi-Riemannian manifolds). We apply the result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of manifolds. More specifically, we c…

The energy of any $C^1$ representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant determined by the asymptotic geometry of the target, with equality if and only …

In this paper, we establish a rigorous correspondence between the two tube algebras, that one comes from the Turaev-Viro-Ocneanu TQFT introduced by Ocneanu and another comes from the sector theory introduced by Izumi, and construct a canonical isomorphism between the centers of the two tube algebras, which is a conjuga…

The purpose of this article is two-fold: We first give a more elementary proof of a recent theorem of Korkmaz, Monden, and the author, which states that the commutator length of the n-th power of a Dehn twist along a boundary parallel curve on a surface with boundary S of genus g at least two is the floor of (|n|+3)/2 …

We extend Obata's rigidity theorem to free probability.

Simple construction of Lie 2-groups from loop group extensions.

We continue the investigation of the correspondence between systems of conservation laws and congruences of lines in projective space. Relationship between "additional" conservation laws and hypersurfaces conjugate to a congruence is established. This construction allows us to introduce, in a purely geometric way, the …

The Cayley graph of quandles reveals structural properties and is studied for various classes.

Study on geodesic distances on SE(3)/SO(2) in machine learning.

In a range of fields including the geosciences, molecular biology, robotics and computer vision, one encounters problems that involve random variables on manifolds. Currently, there is a lack of flexible probabilistic models on manifolds that are fast and easy to train. We define an extremely flexible class of exponent…

Let $Σ$ be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that the solution of generalized (higher genus) Kashiwara-Vergne equations for an automorphism $F \in {\rm Aut}(L)$ of a free Lie algebra implies an isomorphism between the Goldman-Turaev Lie bial…

The Novikov-Shubin invariants for a non-compact Riemannian manifold M can be defined in terms of the large time decay of the heat operator of the Laplacian on square integrable p-forms on M. For the (2n+1)-dimensional Heisenberg group H, the Laplacian can be decomposed into operators in the conjugate of the generalised…

New manifolds found without interior conjugate points.

New algebraic structures on manifolds generalize supergeometry concepts.

Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined as the supremum of commutator lengths of elements of its commutator subgroup. We …

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigen…

For a smooth (locally trivial) principal bundle in Ehresmann's sense, the relation between the commuting vertical and horizontal actions of the structural Lie group and the structural Lie groupoid (isomorphisms between vertical fibers) is regarded as a special case of a symmetrical concept of conjugation between "princ…

Doodles link to commutator identities in a 2-sphere.

Study on deformation cohomology for braided commutative structures.

Examining singularities of commuting vector fields on submanifolds.

Two commutators generate mapping class groups of surfaces with genus ≥5.

Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.

A condition for a statistical manifold to have an equiaffine structure is studied. The facts that dual flatness and conjugate symmetry of a statistical manifold are sufficient conditions for a statistical manifold to have an equiaffine structure were obtained in [2] and [3]. In this paper, a fact that a statistical man…

The paper tackles efficient computation of optimal transport by approximating conjugates with amortized optimization.

Estimates geodesics on surfaces without conjugate points.

New proof and description of commutator subgroups for free and surface groups.

New method speeds up inference for non-conjugate Gaussian processes.

We introduce a multiple conjugation biquandle, and show that it is the universal algebra to define a semi-arc coloring invariant for handlebody-links. A multiple conjugation biquandle is a generalization of a multiple conjugation quandle. We extend the notion of $n$-parallel biquandle operations for any integer $n$, an…

The paper studies Jacobi fields and conjugate points in projective sprays.

Paper describes how to extend multiple conjugation quandles using maps.

Euclidean metrics without conjugate points are proven to be flat.