arXiv research

Study flat GL(1|1) connections using fatgraphs and coordinates.

Four-dimensional GL(2) structures are flat with a specific subgroup transformation.

Study cobrackets on flat GL(1, R)-bundles over surfaces, finding none for Turaev cobracket.

New proof classifies orbit closures in Hodge bundle.

We prove the vanishing of a certain characteristic class of flat vector bundles when the structure groups of the bundles are contained in GL(N,Z). We do so by explicitly writing the characteristic class as an exact form on the base of the bundle.

We study the 8 natural GL equivariant geometric realization questions for the space of generalized algebraic curvature tensors. All but one of them is solvable; a non-zero projectively flat Ricci antisymmetric generalized algebraic curvature is not geometrically realizable by a projectively flat Ricci antisymmetric tor…

Stability of flat vector bundles on complex parallelizable manifolds proven.

We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of $\mathbf{GL}_+(2,\R)$ avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surf…

We provide a new topological interpretation of the symplectic properties of gluing equations for triangulations of hyperbolic 3-manifolds, first discovered by Neumann and Zagier. We also extend the symplectic properties to more general gluings of PGL(2,C) flat connections on the boundaries of 3-manifolds with topologic…

New invariants for families of flat connections constructed using fiber integration.

The study classifies flat solvmanifolds and finds $G_2$-structures on them.

The study classifies Veech groups of flat surfaces with poles.

Study partially hyperbolic flows on flat bundles, proving equivalence for complete affine manifolds.

Invariant structures link to algebraic curves with specific properties.

Method constructs quantum holonomies for BPS states on line defects.

Equivalence proven between two torsion invariants for flat vector bundles.

We characterize constant mean curvature surfaces in the three-dimensional Heisenberg group by a family of flat connections on the trivial bundle $\D \times \GL$ over a simply connected domain $\mathbb{D}$ in the complex plane. In particular for minimal surfaces, we give an immersion formula, the so-called Sym-formula, …

Random flat bundles on surfaces have least eigenvalues at least 1/4.

The paper connects GL-racks to knot coloring invariants.

We classify indefinite simply connected hyper-Kaehler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m,4m). We establish a natural 1-1 correspondence between simply connected hyper-Kaehler symmetric spaces of dimension 8m and orbits of the general linear group GL(m,H…

Extends gl(m|k) construction using Hilbert scheme of points.

Study of generalized Legendrian racks and their GL-structures.

The group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus is the action of the affine group $\Aff(2)$ on $\bbR^2$. Since this action has non-compact stabiliser $\GL(2,\bbR)$, the underlying locally homogeneous geometry is highly non-Riemannian. In this articl…

Study the geometry of a Lie group using Hessian and flat affine structures.

Study invariant connections on multivariate Gaussian distributions.

Study $GL(2)$-structures on manifolds leading to complex structures.

We study five dimensional geometries associated with the 5-dimensional irreducible representation of GL(2,R). These are special Weyl geometries in signature (3,2) having the structure group reduced from CO(3,2) to GL(2,R). The reduction is obtained by means of a conformal class of totally symmetric 3-tensors. Among all…

Consider a finite dimensional (generally reducible) polynomial representation ρof GL_n. A projective compactification of GL_n is the closure of ρ(GL_n) in the space of all operators defined up to a factor (this class of spaces can be characterized as equivariant projective normal compactifications of GL_n). We give an …

Researchers compute $\mathfrak{gl}_2$-skein modules for lens spaces.

Study flat metrics from right prisms, finding non-lattice surfaces with translation coverings.

The study classifies and normalizes 3D gl-regular Nijenhuis operators.

Study on GL(2) geometries on complex manifolds, focusing on Kähler-Einstein and Fano manifolds.

We construct a canonical frame for an arbitrary Gl(2)-structure thus solving the equivalence problem for Gl(2)-structures. Our treatment includes also a problem of contact equivalence of ordinary differential equations and applies to certain classes of vector distributions. Additionally we characterise Gl(2)-structures…

Class I CR manifolds have initial G-structure a certain 4-dimensional subgroup of GL_3(C). Class II CR manifolds have initial G-structure a certain 10-dimensional subgroup of GL_4(C). Class III-1 CR manifolds have initial G-structure a certain 10-dimensional subgroup of GL_5(C). Class III-2 CR manifolds have initial G-…

Defines Khovanov homology with gl(1|1) action, preserving Reidemeister moves.

We construct a right-invariant differential calculus on the quantum supergroup GL$_h(1| 1)$ and obtain the $h$-deformed superalgebra of GL$_h(1| 1)$.

Proves metric equivalence for GL^+_n matrices.

New basis for quantum gl_N invariants derived from Macdonald polynomials.

Given a projective structure on a surface $N$, we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space $M$ of a certain rank $2$ affine bundle $M \to N$. The Einstein metric has anti-self-dual conformal curvature and admits …

Simplified computation of symmetric gl_1 homology for links.

Defines new $\mathfrak{gl}_2$-foams and their algebras, showing equivalence and applications.

New homology for links in annulus discovered.

We study a natural map from representations of a free group of rank g in GL(n,C), to holomorphic vector bundles of degree 0 over a compact Riemann surface X of genus g, associated with a Schottky uniformization of X. Maximally unstable flat bundles are shown to arise in this way. We give a necessary and sufficient cond…

New diagrammatic approach connects knot Floer homology with quantum group representations.

Study GLS estimator properties in multivariate regression with heteroskedastic and autocorrelated errors.

The dualizing module of GL_n(O) varies, affecting cohomology vanishing and nonvanishing.

Extended quantum state result for gl_n weight systems.

In this paper we define an explicit basis for the $\mathfrak{gl}_n$-web algebra $H_n(\vec{k})$ (the $\mathfrak{gl}_n$ generalization of Khovanov's arc algebra) using categorified $q$-skew Howe duality. Our construction is a $\mathfrak{gl}_n$-web version of Hu--Mathas' graded cellular basis and has two major application…