arXiv research

Paper defines a new braid invariant and shows it's commutative.

Mathematical framework for brane quantization using SYZ mirror symmetry.

Constructs continuous families of minimal surfaces and holomorphic immersions.

Associated to each material body $\mathcal{B}$ there exists a groupoid $Ω\left( \mathcal{B} \right)$ consisting of all the material isomorphisms connecting the points of $\mathcal{B}$. The uniformity character of $\mathcal{B}$ is reflected in the properties of $Ω\left( \mathcal{B} \right)$: $\mathcal{B}$ is uniform if,…

For every $\;b>1\;$ fixed, we explicitly construct $1$-dimensional families of embedded constrained Willmore tori parametrized by their conformal class $\;(a,b)$\; with $\; a \sim_b 0^+\;$ deforming the homogenous torus \;$f^b$ of conformal class \;$(0,b).$ The variational vector field at $f^b$ is hereby given by a non…

Optimized strategies for graph-structured bandits reduce regret efficiently.

Let $Q$ be a smooth compact orientable 3--manifold with smooth boundary $\partial Q$. Let $\mathcal{B}$ be the set of exact 2--forms $B\inΩ^2(Q)$ such that $j_{\partial Q}^*B=0$, where $j_{\partial Q}:{\partial Q}\to Q$ is the inclusion map. The group $\mathcal{D}=\mathrm{Diff}_0(Q)$ of self-diffeomorphisms of $Q$ isot…

We study the behaviour of analytic torsion under smooth fibrations. Namely, let F \to E \to^{f} B be a smooth fiber bundle of connected closed oriented smooth manifolds and let $V$ be a flat vector bundle over $E$. Assume that $E$ and $B$ come with Riemannian metrics and $V$ comes with a unimodular (not necessarily fla…

Suppose a and b are distinct isotopy classes of essential simple closed curves in an orientable surface S. Let T_a and T_b represent the respective Dehn twists along a and b. In this paper, we study the subgroups of Mod(S) generated by X and Y, where X belongs to {(T_aT_b)^k,(T_bT_a)^k}, k an integer, and Y belongs to …

Suppose subgroups $A,B < MCG(S)$ in the mapping class group of a closed orientable surface $S$ are given and let $\langle A, B \rangle$ be the subgroup they generate. We discuss a question by Minsky asking when $\langle A, B \rangle \simeq A*_{A \cap B} B$ for handlebody subgroups $A,B$.

Let $G$ be a group given by the presentation [<a_1,...,a_k,b_1,... b_k\,| a_i=u_i(\bar b), b_i=v_i(\bar a) \hbox{for} 1\le i\le k>,] where $k\ge 2$ and where the $u_i\in F(b_1,..., b_k)$ and $w_i\in F(a_1,..., a_k)$ are random words. Generically such a group is a small cancellation group and it is clear that $(a_1,...,…

Study b-divisors on Kähler manifolds linking them to currents.

Conditions found for linearizing divergence-free fields on invariant tori.

We consider two eigenvalue problems for Laplacian on some specific doubly connected domain. In particular, we study the following two eigenvalue problems. Let $B_1$ be an open ball in $\mathbb{R}^n$ and $B_0$ be a ball contained in $B_1$. Let $ν$ be the outward unit normal on $\partial B_1$. Then the first eigenvalue o…

The study classifies $b$-contact structures on $b$-manifolds, especially on $S^3$.

This paper classifies all admissible quotients of a surface braid group up to order 127.

The study bounds exceptional surgeries for hyperbolic knots.

Let $Γ$ be the fundamental group of a surface of finite type and Comm$(Γ)$ be its abstract commensurator. Then Comm$(Γ)$ contains the solvable Baumslag--Solitar groups $\langle a ,b : a b a^{-1} = b^n \rangle$ for any $n > 1$. Moreover, the Baumslag--Solitar group $\langle a ,b : a b^2 a^{-1} = b^3 \rangle$ has an imag…

In this paper biharmonic maps between doubly warped product manifolds are studied. We show that the inclusion maps of Riemannian manifolds $B$ and $F$ into the doubly warped product $_{f}B\times_{b}F$ can not be proper biharmonic maps. Also we analyze the conditions for the biharmonicity of projections $_{f}B\times_{b}…

The paper constructs Darboux transforms for a specific hierarchy and its related flows.

We study immersed tori in $3$-space minimizing the Willmore energy in their respective conformal class. Within the rectangular conformal classes $\;(0,b)\;$ with $\;b \sim 1\;$ the homogenous tori $\;f^b\;$ are known to be the unique constrained Willmore minimizers (up to invariance). In this paper we generalize this r…

Quantization and reduction for coisotropic A-branes on Hamiltonian manifolds.

A {\em blink} is a plane graph with a bipartition (black, gray) of its edges. Subtle classes of blinks are in 1-1 correspondence with closed, oriented and connected 3-manifolds up to orientation preserving homeomorphisms \cite{lins2013B}. Switching black and gray in a blink $B$, giving $-B$, reverses the manifold orien…

Let $π\colon (M,ω)\to B$ be a non-singular Lagrangian torus fibration on a complete base $B$ with prequantum line bundle $\bigl(L,\nabla^L\bigr)\to (M,ω)$. Compactness on $M$ is not assumed. For a positive integer $N$ and a compatible almost complex structure $J$ on $(M,ω)$ invariant along the fiber of $π$, let $D$ be …

In this paper we present two new bases, $B^{\prime}_{H_2}$ and $\mathcal{B}_{H_2}$, for the Kauffman bracket skein module of the handlebody of genus 2 $H_2$, KBSM($H_2$). We start from the well-known Przytycki-basis of KBSM($H_2$), $B_{H_2}$, and using the technique of parting we present elements in $B_{H_2}$ in open b…

Study on automorphisms of complex $b^k$-manifolds, extending previous work.

For a closed topological $n$--manifold $K$ and a map $p:K\to B$ inducing an isomorphism $π_1(K)\toπ_1(B)$, there is a canonicaly defined morphism $b:H_{n+1}(B,K,\mathbb{L})\to \mathbb{S} (K)$, where $\mathbb{L}$ is the periodic simply-connected surgery spectrum and $\mathbb{S} (K)$ is the topological structure set. We …

Smooth manifold structure on Möbius transformations of quaternionic ball identified.

Defines new structures on Courant algebroids, focusing on B_n types.

This paper covers non-orientable Euclidean manifolds B3 and B4, detailing their n-fold coverings.

Let $Σ_{g,b}$ denote a closed orientable surface of genus $g$ with $b$ punctures and let $\rm Mod(Σ_{\textit{g,b}})$ denote its mapping class group. In [Luo] Luo proved that if the genus is at least 3, $\rm Mod(Σ_{\textit{g,b}})$ is generated by involutions. He also asked if there exists a universal upper bound, indepe…

A torti-rational knot, denoted by K(2a,b|r), is a knot obtained from the 2-bridge link B(2a,b) by applying Dehn twists an arbitrary number of times, r, along one component of B(2a,b). We determine the genus of K(2a,b|r) and solve a question of when K(2a,b|r) is fibred. In most cases, the Alexander polynomials determine…

New algebra connects braid group actions to disc curves.

The abstract discusses families of holomorphic maps to Oka manifolds with approximation theorems.

Geometrically represents L-homology classes using normal maps.

Newlander-Nirenberg theorem extended to complex b-manifolds.

This paper connects symplectic and Kähler manifolds via brane quantization.

It is shown that for given positive integers g and b, there is a number C(g,b), such that any orientable compact irreducible 3-manifold of Heegaard genus g has at most C(g,b) disjoint, nonparallel incompressible surfaces with first Betti number b_1 < b.

Introduces $b$-Lie groups and studies their symplectic structures and reductions.

Constructs CMC hypersurfaces in S^4 from piecewise-smooth unions of spheres.

Given a knot $K$ we may construct a group $G_n(K)$ from the fundamental group of $K$ by adjoining an $n$th root of the meridian that commutes with the corresponding longitude. For $n\geq2$ these "generalised knot groups" determine $K$ up to reflection (Nelson and Neumann, 2008; arXiv:0804.0807). The second author has s…

We consider a vector space V over K=R or C, equipped with a skew symmetric bracket [.,.]: V x V --> V and a 2-form omega:V x V --> K. A simple change of the Jacobi identity to the form [A,[B,C]]+[C,[A,B]]+[B,[C,A]]=omega(B,C)A+omega(A,B)C+omega(C,A)B opens new possibilities, which shed new light on the Bianchi classifi…

Let $π{:}\,(M,\mathcal{H})\to (B,b)$ be a submersion equipped with a horizontal connection $\cal H$ over a Riemannian manifold $(B,b)$. We present an intrinsic curvature condition that only depends on the pair $(\cal H,b)$. By studying a set of relative flat planes, we prove that a certain class of pairs $(\cal H,b)$ a…

Let $a>b>0$ and $f$ be a conformal map from $B_a\setminus B_b\subseteq R^2$ into $\R^n$, with $|\nabla f|^2=2e^{2u}$. Then $(e_1, e_2)$ with $e_1=e^{-u}\frac{\partial f}{\partial r},$ and $e_2=r^{-1}e^{-u}\frac{\partial f}{\partialθ}$ is a moving frame on $f(B_a\setminus B_b)$. It satisfies the following equation $$d\s…

This classification is found by analyzing the action of a normal subgroup of $B_3$ as hyperbolic isometries. This paper gives an example of an unfaithful specialization of the Burau representation on $B_4$ that is faithful when restricted to $B_3$, as well as examples of unfaithful specializations of $B_3$.

We prove a $\mathsf{C}^1$-elliptic estimate of the form $\sup_{B(x,r/2)} |\mathrm{grad} (ψ) | \leq C \left\{ \sup_{B(x,r)} |Δψ| + \sup_{B(x,r)} |ψ| \right\},$ valid on any complete Riemannian manifold $M$ and for any smooth function $ψ$ which is defined in a nighbourhood of $B(x,r)$, with an explicit quantitative con…

In this paper we introduce $B_{α,β}^{k}$-manifolds as generalizations of the notion of smooth manifolds with $G$-structure or with $k$-bounded geometry. These are $C^{k}$-manifolds whose transition functions $\varphi_{ji}=\varphi_{j}\circ\varphi_{i}^{-1}$ are such that $\partial^μ\varphi_{ji}\in B_{α(r)}\cap C^{k-β(r)}…

Let p: M -> B be a family of compact manifolds equipped with a unitarily flat vector bundle F -> M. We generalize Igusa's higher Franz-Reidemeister torsion τ(M/B;F) to the case that the fibre-wise cohomology H^*(M/B;F) -> B carries a parallel metric. If moreover M admits a fibre-wise Morse function, we compute the diff…