arXiv research

Clusters on simple manifolds have connected boundaries.

This paper computes the obstruction to the existence of equivariant extensions of basic gerbes over non-simply connected compact simple Lie groups. By modifying a (finite dimensional) construction of Gawȩdzki-Reis [J. Geom. Phys. 50(1):28-55, 2004], we exhibit basic equivariant bundle gerbes over non-simply connected c…

Study on simplicity of Lie skew braces, proving new results for compact cases.

We give another proof for a result of Brick stating that the simple connectivity at infinity is a geometric property of finitely presented groups. This allows us to define the rate of vanishing of $\p1i$ for those groups which are simply connected at infinity. Further we show that this rate is linear for cocompact latt…

Smooth surfaces in simply connected 4-manifolds yield groups with non-trivial homology.

The paper proves eigenvalues are simple for specific operators on bundles.

In this paper are given explicit calculations of Laplace operator spectrum for smooth real/complex-valued functions on all connected compact simple rank three Lie groups with biinvariant Riemannian metric and established a connection of obtained formulas with the number theory and integer ternary and binary quadratic f…

Simple closed curves in ε-boundaries separate sets in the plane.

In this paper are given explicit calculations of Laplace operator spectrum for smooth real/complex-valued functions on all connected compact simple rank four Lie groups with biinvariant Riemannian metric, corresponding to root systems $B_4$, $C_4$, $D_4$ and established a connection of obtained formulas with the number…

The boundary of hyperbolic groups is locally simply connected.

The study connects conic connections and torsion-free principal connections on G-structures.

We prove that transversal non-simplicity is preserved under taking connect sum, generalizing Vertesi's result.

Spheres in curve complexes are almost simply connected.

By proving a connected sum formula for the Legendrian invariant $λ_+$ in knot Floer homology we exhibit infinitely many transversely non simple knots.

Let $G$ be a compact, simply connected simple Lie group. We give a construction of an equivariant gerbe with connection on $G$, with equivariant 3-curvature representing a generator of $H^3_G(G,\Z)$. Technical tools developed in this context include a gluing construction for gerbes and a theory of equivariant bundle ge…

New definition of Born geometry connects to known geometries.

From the point of view of index theory, we give a simple proof of a Gauss-Bonnet-Chern formula for all Finsler manifolds by the Cartan connection. Based on this, we establish a Gauss-Bonnet-Chern formula for any metric-compatible connection and also derive the Gauss-Bonnet-Chern formula of Lackey.

The study proves stability of a flow on specific Lie groups.

The paper encloses computation of simple centralizer of simple braids and their connection with Fibonacci numbers. Planarity of some commuting graphs is also discussed in the last section.

In this paper, we provide the necessary and sufficient conditions for the connected sum of knots in $S^3$ to be Legendrian simple.

In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of generalized flag manifolds. We prove that all these left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.

Study solves Ricci curvature problem for specific noncompact spaces.

A simple thresholding technique improves graph selection in neural connectivity studies.

Study Hom-Lie algebroid connections on complex manifolds.

We consider collections of disjoint simple closed curves in a compact orientable surface which decompose the surface into pairs of pants. The isotopy classes of such curve systems form the vertices of a 2-complex, whose edges correspond to certain simple moves in which only one curve changes, and whose 2-cells correspo…

We present an explicit construction of the basic bundle gerbes with connection over all connected compact simple Lie groups. These are geometric objects that appear naturally in the Lagrangian approach to the WZW conformal field theories. Our work extends the recent construction of E. Meinrenken \cite{Meinr} restricted…

The paper studies special crystallizations of 4-manifolds to minimize certain PL-invariants.

The paper classifies geodesic orbit spaces with simple isotropy groups.

Bi-invariant metrics on Lie groups and homogeneous spaces are extremal and rigid.

Study of real and quaternionic Lie algebroid connections on manifolds.

Let $G$ be a connected, simply-connected, compact simple Lie group. In this paper, we show that the isometry group of $G$ with a left-invariant pseudo-Riemannan metric is compact. Furthermore, the identity component of the isometry group is compact if $G$ is not simply-connected.

Connected graph for twice-punctured torus curves.

Proves existence of flat connection on theta functions for G-bundles.

Study flat connections on Courant algebroids using Lie groups.

For $d\geq 2$, the regular genus of a closed connected PL $d$-manifold $M$ is the least genus (resp., half of the genus) of an orientable (resp., a non-orientable) surface into which a crystallization of $M$ imbeds regularly. The regular genus of every orientable surface equals its genus, and the regular genus of every…

We propose a unified approach to the theory of connections in the geometry of sprays and Finsler metrics which, in particular, gives a simple explanation of the well-known fact that all the classical Finslerian connections provide exactly the same formulas appearing in the calculus of variations.

We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections on reflexive sheaves at isolated singularities modelled on $μ$-polystable holomorphic bundles over $\mathbf{P}^{n-1}$.

We prove that there does not exist any connected topological proper loop homeomorphic to a quasi-simple Lie group and having a compact Lie group as the group topologically generated by its left translations. Moreover, any connected topological loop homeomorphic to the 7-sphere and having a compact Lie group as the grou…

Let D be a link diagram with n crossings, s_A and s_B its extreme states and |s_AD| (resp. |s_BD|) the number of simple closed curves that appear when smoothing D according to s_A (resp. s_B). We give a general formula for the sum |s_AD|+|s_BD| for a k-almost alternating diagram D, for any k, characterising this sum as…

Neural networks can learn optimal auction mechanisms and satisfy mode connectivity.

The study proves a unique perimeter minimizer for area in simply connected space forms and proves the isoperimetric inequality.

It is proved that the orbit space of an irreducible representation of a simple connected compact Lie group of type B, C, or D can be a smooth manifold only in two cases.

Simple branched coverings established between certain 4-manifolds.

Trivalent $2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed curves where three sheets meet. We obtain a classification of $1$-connected $2$-stratifolds in terms of their associated labeled graphs and develop operations that will construct from a single vertex all graphs that…

In this paper we consider the problem of identifying a connection $\nabla$ on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian $\nabla^*\nabla$ over conformally transversally anisotropic (CTA) manifolds. This was proved in \cite{LCW} for line bundles in the case of t…

All closed geodesics are simple and non-intersecting in dimensions 3 and above.

We describe the range of the attenuated ray transform of a unitary connection on a simple surface acting on functions and 1-forms. We use this to determine the range of the ray transform acting on symmetric tensor fields.

This article discusses the twisted adjoint action $\mathrm{Ad}_{g}^κ:G\rightarrow G$, $x\mapsto gxκ(g^{-1})$ given by a Dynkin diagram automorphism $κ\in\mathrm{Aut}(G)$, where $G$ is compact, connected, simply connected and simple. The first aim is to recover the classification of $κ$-twisted conjugacy classes by elem…