arXiv research

Study local features of decorated representation spaces for spherical surfaces.

Introduces noncommutative coordinates for symplectic representations.

Let S be a path-connected, locally-compact CW-complex, and let M be a subcomplex with finitely-many components. A `decorated SL_2(C)-local system' is an SL_2(C)-local system on S, together with a choice of `decoration' at each component of M (a section of the stalk of an associated vector bundle). We study the (decorat…

Study geometric bases for A-polynomials in SU(3) using arcade formalism.

Study of quantum decorated character stacks and their quantizations.

We study surfaces with decorations and prove uniformization in non-Euclidean geometries.

Study strip deformations of hyperbolic polygons with decorated vertices.

Decorated TQFTs compute invariants with additional structures.

Researchers prove the arc complexes of decorated hyperbolic polygons are balls.

Constructs TQFTs for cobordisms with cohomology class decorations.

Study compares constrained and decoupled moduli spaces of manifolds with particles and discs.

The article describes how decorations on hyperbolic surfaces lead to unique tessellations and decompositions.

Corrects a 1-off error in Harer's spine dimension calculation for decorated Teichmüller spaces.

We determine the explicit transformation under duality of generic configurations of four flags in $\PGL(3,\bC)$ in cross-ratio coordinates. As an application we prove invariance under duality of an invariant in the Bloch group obtained from decorated triangulations of 3-manifolds.

This is a survey on the project `Decorated Marked Surfaces', where we introduce the decoration $Δ$ on a marked surfaces $\mathbf{S}$, to study Calabi-Yau-2 (cluster) categories, Calabi-Yau-3 (Fukaya) categories, braid groups for quivers with potential, quadratic differentials and stability conditions.

DecoR estimates causal effects in confounded time series data.

Enhanced Teichmüller space for surfaces with decorations and enhancements.

Unified framework for smooth structures on coadjoint orbits.

Study of decorated surfaces with vortices and their group structures.

Discrete conformal maps on surfaces with vertex decorations are studied.

We describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface $S$ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperpolic plane. We establish a bijection between …

New TQFT for link cobordisms without decoration.

In this paper we are interested in computing representations of the fundamental group of a 3-manifold into PSL(3;C) (in particular in PSL(2;C); PSL(3;R) and PU(2; 1)). The representations are obtained by gluing decorated tetrahedra of flags. We list complete computations (giving 0-dimensional or 1-dimensional solution …

We produce a one-parameter family of coordinates $\{Ψ_h\}_{h\in\mathbb{R}}$ of the decorated Teichmüller space of an ideally triangulated punctured surface $(S,T)$ with negative Euler characteristic, which is a deformation of Penner's simplicial coordinate \cite{P1}. If $h\geqslant0$, the decorated Teichmüller space in…

We are interested in the 3-Calabi-Yau categories $\mathcal{D}$ arising from quivers with potential associated to a triangulated marked surface $\mathbf{S}$ (without punctures). We prove that the spherical twist group ST of $\mathcal{D}$ is isomorphic to a subgroup (generated by braid twists) of the mapping class group …

The first aperiodic monotiling, introduced by Taylor, was based on a trapezoidal prototile equipped with 14 distinct decorations. A presentation of the closely related Taylor-Socolar aperiodic monotiling is based on a hexagonal prototile equipped with 7 decorations. This paper gives decoration-free algebraic descriptio…

We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi-Yau triangulated category. The motivation/application is that the…

New method calculates Chern-Simons volume for 3-manifolds with surgery diagrams.

The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements Q_λ, depending on partitions λ. We show how to find the 2-variable Homfly invariant <λ…

Given an Anosov representation $ρ\colon π_1(S) \to \PSL_{n}(\mathbb{R})$ and a maximal geodesic lamination $λ$ in a surface $S$, we construct shear deformations along the leaves of the geodesic lamination $λ$ endowed with a certain flag decoration, that is provided by the associated flag curve $\mathcal{F}_ρ\colon \Sin…

We give counterexamples to a question of Bowditch that if a non-elementary type-preserving representation $ρ:π_1(Σ_{g,n})\rightarrow PSL(2;\mathbb R)$ of a punctured surface group sends every non-peripheral simple closed curve to a hyperbolic element, then must $ρ$ be Fuchsian. The counterexamples come from relative Eu…

The paper proves a theorem for discretizing Gaussian curvature on surfaces.

Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichmüller space of puncture…

The punctured solenoid $§$ is an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmüller space of $§$ is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of $§$. Furthermore…

This paper extends the decorated Teichmüller theory developed before for punctured surfaces to the setting of ``bordered'' surfaces, i.e., surfaces with boundary, and there is non-trivial new structure discovered. The main new result identifies the arc complex of a bordered surface up to proper homotopy equivalence wit…

We define new coordinates for Fock-Goncharov's higher Teichmüller spaces for a surface with holes, which are the moduli spaces of representations of the fundamental group into a reductive Lie group $G$. Some additional data on the boundary leads to two closely related moduli spaces, the $\mathscr{X}$-space and the $\ma…

The paper introduces a new discretization of Gaussian curvature on surfaces.

We study a new bordification of the decorated Teichmüller space for a multiply punctured surface F by a space of filtered screens on the surface that arises from a natural elaboration of earlier work of McShane-Penner. We identify necessary and sufficient conditions for paths in this space of filtered screens to yield …

Study rigidity and volume optimization of hyperbolic polyhedra.

Let $M$ be a compact oriented three-manifold whose interior is hyperbolic of finite volume. We prove a variation formula for the volume on the variety of representations of $M$ in $\operatorname{SL}_n(\mathbb C)$. Our proof follows the strategy of Reznikov's rigidity when $M$ is closed, in particular we use Fuks' appro…

Study one-dimensional topological theories with linear generating functions.

The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.

We define an associative algebra AS_h(S) generated by framed arcs and links over a punctured surface S which is a quantization of the Poisson algebra C(S) of arcs and curves on S. We then construct a Poisson algebra homomorphism from C(S) to the space of smooth functions on the decorated Teichmuller space endowed with …

For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are known as homological representations. Representations of this kind have proved th…

We define a new kind of Gauss diagrams to describe knots in the solid torus with projections in the annulus. We see that it provides an efficient tool for showing that a knot diagram can be fully recovered from its decorated Gauss diagram, and we use it to establish a characterization of the decorated Gauss diagrams of…

We introduce families of decorations of a same topological space, as well as a family of sheaves over such decorated spaces. Making those families a directed system leads to the concept of emerald over a space. For the configuration space X_N of N points in the plane, connecting points of the plane with chords is a dec…

New model for Calabi-Yau-$\mathbb{X}$ categories using decorated marked surfaces.

The paper applies combinatorial Ricci flows to prove hyperbolic structures on 3-manifolds.