arXiv research

The paper studies Majorana Fermions and their braid group representations.

New formula connects braid dilatation with fixed points.

The paper explores normal subgroups of braid groups and mapping class groups.

The mixed braid groups are the subgroups of Artin braid groups whose elements preserve a given partition of the base points. We prove that the centralizer of any braid can be expressed in terms of semidirect and direct products of mixed braid groups. Then we construct a generating set of the centralizer of any braid on…

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…

Let $PB_n(S_{g,p})$ be the pure braid group of a genus $g>1$ surface with $p$ punctures. In this paper we prove that any surjective homomorphism $PB_n(S_{g,p})\to PB_m(S_{g,p})$ factors through one of the forgetful homomorphisms. We then compute the automorphism group of $PB_n(S_{g,p})$, extending Irmak, Ivanov and McC…

Let B_n be the braid group on n > 3 strands. We prove that B_n modulo its center is co-Hopfian. We then show that any injective endomorphism of B_n is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from B_n to B_n+1 is geometric. Additionally, we ob…

In this paper, we prove that each automorphism of the Torelli group of a surface is induced by a diffeomorphism of the surface, provided that the surface is a closed, connected, orientable surface of genus at least 3. This result was previously announced by Benson Farb for genus at least 4 and has recently been announc…

We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a sphere with at least 5 punctures. In particular, we show that any injective homomorphism between these groups is parameterized by a homeomorphism of a punctured sphere together with a map to the integers.…

We exhibit pseudo Riemannian manifolds which are Szabó nilpotent of arbitrary order, or which are Osserman nilpotent of arbitrary order, or which are Ivanov-Petrova nilpotent of order 3.

Study automorphism groups of geodesic currents and measured laminations on surfaces.

Let $Γ_{g,b}$ denote the orientation-preserving Mapping Class Group of a closed orientable surface of genus $g$ with $b$ punctures. For a group $G$ let $Φ_f(G)$ denote the intersection of all maximal subgroups of finite index in $G$. Motivated by a question of Ivanov as to whether $Φ_f(G)$ is nilpotent when $G$ is a fi…

We present a new test for studying asphericity and diagrammatic reducibility of group presentations. Our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails. We use this criterion to generalize results of J. Howie and S.M. Gersten on asphericity of LOTs and of Adian pre…

Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and $\varepsilon$-spaces exhaust the class of $n$-dimensional Lorentzian manifolds admitting a group of isometries of dimension at least ${1/2} n(n-1)+1$, for almost all val…

We present some examples of curvature homogeneous pseudo-Riemannian manifolds which are k-spacelike Jordan Stanilov; their higher order curvature operator has constant Jordan normal form on the Grassmannian of unoriented k-dimensional spacelike subspaces of the tangent plane.

We study geometric realization questions of curvature in the affine, Riemannian, almost Hermitian, almost para Hermitian, almost hyper Hermitian, almost hyper para Hermitian, Hermitian, and para Hermitian settings. We also express questions in Ivanov-Petrova geometry, Osserman geometry, and curvature homogeneity in ter…

We exhibit 3 families of complete curvature homogeneous pseudo-Riemannian manifolds which are modeled on irreducible symmetric spaces and which are not locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some of the manifolds are, in addition, Jordan Osserman and Jordan Ivanov-Petrova.

Optimal inequalities for metric surfaces derived from filling minimality.

For any k which is at least 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not k+1-affine curvature homogeneous, and hence not locally homogeneous. All the local scalar Weyl invariants of these manifolds vanish. These manifolds are Ricci flat, Osserman, and Ivanov…

Notes on continuity of discrete-spectrum Fredholm operators.

Anomaly flow studied on flat and non-flat nilmanifolds.

Study on functional ellipsoids to decompose the identity.

The study examines the rigidity of mapping class groups under large powers of twists.

This is our second paper in a series to study gravitational instantons, i.e. complete hyperkäler 4-manifolds with faster than quadratic curvature decay. We prove two main theorems: 1.The asymptotic rate of gravitational instantons to the standard models can be improved automatically. 2.Any ALF-D_k gravitational instant…

Probabilistic model for exhaustion in infinite-genus curve complexes.

We introduce a new volume definition on normed vector spaces. We show that the induced $k$-area functionals are convex for all $k$. In the particular case $k=2$, our theorem implies that Busemann's 2-volume density is convex, which was recently shown by Burago-Ivanov. We also show how the new volume definition is relat…

Classic braids embed in virtual braids.

The paper finds minimal generating sets and abelianizes the quasitoric braid group.

We study the motion of a charge on a conformally flat Riemannian torus in the presence of magnetic field. We prove that for any non-zero magnetic field there always exist orbits of this motion which have conjugate points. We conjecture that the restriction of conformal flatness of the metric is not essential for this r…

We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…

Satellite links of fully positive braids are characterized.

Formula found for braid index of $n$-bridge braids.

We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven dimensional solvable Lie groups having an integra…

Most simple braids have positive topological entropy.

This paper is concerned with detecting when a closed braid and its axis are 'mutually braided' in the sense of Rudolph. It deals with closed braids which are fibred links, the simplest case being closed braids which present the unknot. The geometric condition for mutual braiding refers to the existence of a close contr…

Proves normal subgroups of mapping class groups have specific properties.

We show that 3-braid links with given (non-zero) Alexander or Jones polynomial are finitely many, and can be effectively determined. We classify among closed 3-braids strongly quasipositive and fibered ones, and show that 3-braid links have a unique incompressible Seifert surface. We also classify the positive braid wo…

The paper examines subgroup separability for surface and virtual braid groups.

Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility, stability and Reidemeister moves. We show that virtual braids are in a bijective…

Study on deformation cohomology for braided commutative structures.

Magic braids can be made with leatherworking technique.

This paper identifies braided 3-belts that can be written in a braid-only form.

Study virtual braid groups, proving a key subgroup result.

Machine learning classifies braids and discovers new invariants.

In the present paper we give a new method for converting virtual knots and links to virtual braids. Indeed the braiding method given in this paper is quite general, and applies to all the categories in which braiding can be accomplished. We give a unifying topological interpretation of virtuals and flats (virtual strin…

We exhibit a family of generalized plane wave manifolds of signature (2,2). The geodesics in these manifolds extend for infinite time (i.e. they are complete), they are spacelike and timelike Jordan Osserman, and they are spacelike and timelike Jordan Ivanov-Petrova. Some are irreducible symmetric spaces. Some are homo…

This paper extends braid lifting to coloured braid groupoids for all simple disc covers.

Paper studies special braids from chromatic configuration spaces.