arXiv research

The authors found geodesics, shortest arcs, cut loci, and conjugate sets for left-invariant sub-Riemannian matric on the Lie group $SL(2)$, which is right-invariant relative to the Lie subgroup $SO(2)\subset SL(2)$ (in other words, for invariant sub-Riemannian metric on weakly symmetric space $(SL(2)\times SO(2))/SO(2)…

In this paper we show that flat (m-1)-dimensional tori give nontrivial rational homology cycles in congruence covers of the locally symmetric space SL(m,Z) \SL(m,R)/SO(m). We also show that the dimension of the subspace of H_{m-1}(Γ\SL(m,R)/SO(m);Q) spanned by flat (m-1)-tori grows as one goes up in congruence covers.

The paper finds a contact form on SL(2p) for p > 1.

Study geodesics on $SL(n, \mathbb{R})$ adjoint orbits.

New complete minimal submanifolds found in specific Riemannian spaces.

Proved dynamical Alekseevskii conjecture in 5D.

Study of a series of Lorentzian structures on SL(2,R) with SO(1,1) symmetry.

We prove that the well-rounded retract of SO_n\SL_n(R) is a minimal SL_n(Z)-invariant spine.

Study on deformations of Bianchi groups into SU(3,1) and SL(4,R).

We prove that when n >= 5, the Dehn function of SL(n;Z) is quadratic. The proof involves decomposing a disc in SL(n;R)/SO(n) into triangles of varying sizes. By mapping these triangles into SL(n;Z) and replacing large elementary matrices by "shortcuts," we obtain words of a particular form, and we use combinatorial tec…

This paper consists of three parts. First, we generalize the Jaeger Formula to express the Kauffman-Vogel graph polynomial as a state sum of the Murakami-Ohtsuki-Yamada graph polynomial. Then, we demonstrate that reversing the orientation and the color of a MOY graph along a simple circuit does not change the sl(N) Mur…

Defines and parametrizes $\mathfrak{sl}(2)$-type singular fibres in symplectic and odd orthogonal Hitchin systems.

Asymptotic symmetries of the five dimensional noncompact symmetric space SL(3)/SO(3) are found to form an infinite dimensional Lie algebra, analogously to the asymptotic symmetries of anti-de Sitter spaces in two and three dimensions. Possible exact solvability of the corresponding Chern-Simons theory and the AdS/CFT c…

Classifies limits of groups of involutions in SL(2,F) over local fields.

Characterizes knotted subgroups of Lie groups and provides examples.

Describes representations of modular group into SL(3,R)/SO(3).

There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichm{ü}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of su…

We introduce an sl(n) homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorifications of enhanced sl(2) foams and categorified quantum gl(m), via classical skew Ho…

Study of unoriented SL(4) foams in 3-manifolds.

New proof of SL(n) skein algebra for twice punctured sphere, showing it's a polynomial algebra.

In the first part we define a "BTZ" black hole in anti de Sitter space in any dimension by defining as "singular" the closed orbits of the Iwasawa component of SO(2,n). In the second part, a strict quantization of the black hole by action of group is performed and its Dirac operator is computed. We introduce, in the ap…

Study classifies totally geodesic surfaces in nearly Kähler space.

One computes the cohomology of the projective embedding of sl(m+1,R) acting on the differential operators on densities on R^m of various weights. This cohomology is non vanishing only for some special critical values of the weights. This allows us first to explain some strange feature pointed out by Gargoubi in his cla…

Study on Selberg's modified metric in symmetric spaces.

In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than one and their geometric perturbations. In our previous work we described the resolvent, and specifically the asymptotic behavior of the Green'…

We show that the resolvent of the Laplacian on SL(3,$\mathbb{R}$)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of $\mathbb{C}$. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue op…

New computations show sl(N) homology is related to SU(N) representations of knots.

We construct $\sorth{p} \times \sorth{q}$-invariant special Lagrangian (SL) cones in $\C^{p+q}$. These SL cones are natural higher-dimensional analogues of the $\sorth{2}$-invariant SL cones constructed previously by MH and used in our gluing constructions of higher genus SL cones in $\C^{3}$. We study in detail the ge…

We introduce a new method for constructing complex-valued $r$-harmonic functions on Riemannian manifolds. We then apply this method for the important semisimple Lie groups $SO(n)$, $SU(n)$, $Sp(n)$, $SL_n(R)$, $Sp(R,n)$, $SU(p,q)$, $SO(p,q)$, $Sp(p,q)$, $SO^*(2n)$ and $SU^*(2n)$.

Sub-Riemannian Selberg trace formulae for compact quotients of SL(2, R)

Spider category comparison proves equivalence to Sikora's quotient category.

Let $q$ be a $2N$th root of unity where $N$ is odd. Let $U_q(sl_2)$ denote the quantum group with large center corresponding to the lie algebra $sl_2$ with generators $E,F,K$, and $K^{-1}$. A semicyclic representation of $U_q(sl_2)$ is an $N$-dimensional irreducible representation $ρ:U_q(sl_2)\rightarrow M_N(\mathbb{C}…

In this note we present an explicit isomorphism between Khovanov-Rozansky $sl_2$-homology and ordinary Khovanov homology. This result was originally stated in Khovanov and Rozansky's paper \cite{KRI}, though the details have yet to appear in the literature. The main missing detail is providing a coherent choice of sign…

The purpose of this paper is twofold. On one hand, we introduce a modification of the dual canonical basis for invariant tensors of the 3-dimensional irreducible representation of $U_q(sl_2)$, given in terms of Jacobi diagrams, a central tool in quantum topology. On the other hand, we use this modified basis to study t…

Formula for $\mathfrak{sl}_2$ weight system on complete bipartite graphs.

This is the fourth in a series of papers math.DG/0008021, math.DG/0008155, math.DG/0010036 constructing explicit examples of special Lagrangian submanifolds (SL m-folds) in C^m. A submanifold of C^m is ruled if it is fibred by a family of real straight lines in C^m. This paper studies ruled special Lagrangian 3-folds i…

We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allo…

In this paper we study the invariant Carnot-Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$ and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory dis…

In this note we describe the recursion relations between two parameter HOMLFY and Kauffman polynomials of framed links These relation correspond to embeddings of quantized universal enveloping algebras. The relation corresponding to embeddings $g_{n}\supset g_{k}\times sl_{n-k}$ where $g_{n}$ is either $so_{2n+1}$, $so…

Study finds lower bounds on flat cycles in congruence covers of symmetric spaces.

The classical Rankin-Cohen brackets are bi-differential operators from $C^\infty(\mathbb R)\times C^\infty(\mathbb R)$ into $C^\infty(\mathbb R)$. They are covariant for the (diagonal) action of ${\rm SL}(2,\mathbb R)$ through principal series representations. We construct generalizations of these operators, replacing…

Following Vinberg, we find the criterions for a subgroup generated by reflections $Γ\subset \SL^{\pm}(n+1,\mathbb{R})$ and its finite-index subgroups to be definable over $\mathbb{A}$ where $\mathbb{A}$ is an integrally closed Noetherian ring in the field $\mathbb{R}$. We apply the criterions for groups generated by re…

Second part of proving linearization theorem for sl2(C).

Study Poisson cohomology and linearize Lie algebra structures.

Using the identification of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ with the Teichmüller space of flat $n$-tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration both from Teichmüller theory and symmetric spaces. We define and study analogs of t…

Study conformal foliations on Lie groups, finding new families and harmonic morphisms.

In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than one and their geometric perturbations. Our goal here is to explain how analysis of the Laplacian on the globally symmetric space $\SL(3,\RR)/\…

Floating geodesic planes in Hitchin manifolds have fractal closures with non-integer dimensions.