This is a survey on the project `Decorated Marked Surfaces', where we introduce the decoration on a marked surfaces , to study Calabi-Yau-2 (cluster) categories, Calabi-Yau-3 (Fukaya) categories, braid groups for quivers with potential, quadratic differentials and stability conditions.
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We are interested in the 3-Calabi-Yau categories arising from quivers with potential associated to a triangulated marked surface (without punctures). We prove that the spherical twist group ST of is isomorphic to a subgroup (generated by braid twists) of the mapping class group …
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 model for Calabi-Yau- categories using decorated marked surfaces.
problem Constructing models for Calabi-Yau- categories.
method Using graded decorated marked surfaces and string models.
result Isomorphism between braid twist group and spherical twist group.
Study of decorated surfaces with vortices and their group structures.
problem Understanding group structures of decorated surfaces with vortices.
method Proved isomorphism between cluster braid group, braid twist group, and fundamental group of moduli space.
result Finite presentations of isomorphic groups were given.
We study the 3-Calabi-Yau categories arising from quivers with potential associated to a decorated marked surface introduced by the first author. We prove two conjectures in the prequel, that under a bijection between certain objects in and certain arcs in $\mathb…
A new double quasi-Poisson bracket on surface groups.
problem Constructing a new mathematical structure on surface groups.
method Proposing and proving a double quasi-Poisson bracket on group algebras.
result The double quasi-Poisson bracket is a noncommutative generalization of the Goldman bracket.
New stability conditions identified from quadratic differentials on surfaces.
problem Identifying stability conditions from quadratic differentials.
method Comparison of exchange graphs from tilting hearts and flipping mixed angulations.
result Spaces of stability conditions identified with moduli spaces of quadratic differentials.
We introduce the cluster exchange groupoid associated to a non-degenerate quiver with potential, as an enhancement of the cluster exchange graph. In the case that arises from an (unpunctured) marked surface, where the exchange graph is modelled on the graph of triangulations of the marked surface, we show that the univ…
Study moduli space of quadratic differentials with new geometric insights.
problem Understanding the structure of moduli spaces of quadratic differentials.
method Using decorated marked surfaces, Abel-Jacobi map, and 3-Calabi-Yau categories.
result Fundamental group of moduli space equals kernel of Abel-Jacobi map.
Knot Floer homology is reinterpreted as immersed curves.
problem Capturing knot information using immersed curves.
method Interpreting knot Floer homology as decorated immersed curves in the marked torus.
result Knot Floer homology data can be represented by decorated immersed curves.
Infinite volume moduli spaces of hyperbolic surfaces are redefined with exponential forms.
problem Infinite volume of moduli spaces of hyperbolic surfaces with cusps.
method Introduce exponential volume form exp(-W)Vol(K,L) where W is a function of hyperbolic areas.
result Exponential volume forms make moduli spaces finite and relevant to open string theory.
Matrix formulas for super Teichmüller spaces generalize previous work and yield super λ-lengths.
problem Calculating super λ-lengths on bordered surfaces with marked points.
method Using holonomy matrices of elements in the supergroup OSp(1|2) to compute super λ-lengths in decorated super Teichmüller spaces.
result Matrix formulas for arcs on bordered surfaces yield super λ-lengths in Penner-Zeitlin's decorated super Teichmüller space.
Study of skein modules in 3-manifolds, showing non-injectivity results.
problem Understanding skein modules in 3-manifolds and their behavior under gluing.
method Extended Kauffman bracket skein modules to 3-manifolds with marking, introduced new maps and studied their properties.
result Non-injectivity of certain maps in stated skein modules, especially when quantum parameter is a root of 1.
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an addit…
Let G be a split semi-simple algebraic group over Q. Let S be a decorated surface, that is a topological oriented surface with a finite set of marked points on the boundary, considered modulo isotopy. We introduce a moduli space D(G,S) and define a collection of special rational coordinate systems on it. The moduli spa…
Wilson lines generate positive Laurent polynomials in decorated triangulations.
problem Wilson lines and their coefficients in function algebras.
method Study of Wilson lines on marked surfaces and their matrix coefficients in function algebras.
result Matrix coefficients of Wilson lines give Laurent polynomials with positive integral coefficients.
We study surfaces with decorations and prove uniformization in non-Euclidean geometries.
problem Discrete conformal equivalence in non-Euclidean geometries.
method Variational principle and continuous deformation.
result One master theory of discrete conformal equivalence across different geometries.
The article describes how decorations on hyperbolic surfaces lead to unique tessellations and decompositions.
problem Understanding the geometric structure of decorated hyperbolic surfaces.
method Developing a characterisation of canonical tessellations and dual decompositions using hyperbolic geometry.
result Decorations on hyperbolic surfaces induce unique canonical tessellations and dual decompositions.
Study local features of decorated representation spaces for spherical surfaces.
problem Local structure of moduli space of spherical surfaces with conical points.
method Analysis of decorated representation spaces of fundamental groups in SU(2).
result Smooth locus of decorated representation spaces is dense and connected.
Discrete conformal maps on surfaces with vertex decorations are studied.
problem Discrete conformal equivalence for decorated piecewise Euclidean surfaces.
method Intimate relationship between decorated PE-surfaces, canonical tessellations of hyperbolic surfaces, and convex hyperbolic polyhedra; concave variational principle.
result Proof of discrete uniformization theorem for decorated PE-surfaces.
Cluster algebras match for specific Lie algebras and surfaces.
problem Matching cluster algebras with upper cluster algebras for certain Lie algebras and surfaces.
method Proof based on moduli space function ring and Wilson lines.
result Cluster algebras match upper cluster algebras for specified Lie algebras and surfaces.
We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot…
Study of quantum decorated character stacks and their quantizations.
problem Quantization of decorated character stacks and their compatibility with cutting and gluing.
method Using stratified factorization homology, extend Fock and Goncharov's construction to include stacky points.
result Construction of categorical charts and flips on quantum decorated character stacks.
Constructs a combinatorial model for bordered Riemann surfaces with a compactification.
problem Moduli spaces of bordered Riemann surfaces.
method Symmetric metric ribbon graphs and sequences of non-contractible subgraphs.
result A combinatorial moduli space that compactifies the KSV-compactification.
Diagonal complexesmath.GT
Given an -gon, the poset of all collections of pairwise non-crossing diagonals is isomorphic to the face poset of some convex polytope called \textit{associahedron}. We replace in this setting the -gon (viewed as a disc with marked points on the boundary) with an arbitrary oriented surface with a number of la…
Corrects a 1-off error in Harer's spine dimension calculation for decorated Teichmüller spaces.
problem Incorrect dimension calculation in Harer's spine for decorated Teichmüller spaces.
method Identifies and corrects the dimension discrepancy in Harer's spine construction.
result Corrects the dimension of Harer's spine by 1 for decorated Teichmüller spaces.
Quantum cluster algebra constructed from web skein relations on surfaces.
problem Quantization of cluster structures on moduli spaces of SL3 local systems.
method Constructing a quantum cluster algebra inside the skew-field of a skein algebra of unpunctured surfaces.
result Laurent expressions of webs in clusters have positive coefficients.
Enhanced Teichmüller space for surfaces with decorations and enhancements.
problem Parameterizing and understanding Teichmüller spaces with enhancements and decorations.
method Introduced a new variation of Teichmüller space, constructed parameterization, and introduced lamination space.
result Compatibility of shear coordinates and λ-length coordinates in the new deformation space.
The paper proves a theorem for discretizing Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with nonpositive Euler number.
method Discrete conformal theory and variational principles with constraints.
result Each decorated piecewise Euclidean metric on surfaces with nonpositive Euler number is discrete conformal to a metric with a specific discrete curvature constant.
Study compares constrained and decoupled moduli spaces of manifolds with particles and discs.
problem Comparing constrained and decoupled moduli spaces of manifolds with embedded particles and discs.
method Generalized Bödigheimer--Tillmann's work to higher dimensions and different tangential structures.
result New results for surfaces with different tangential structures and higher dimensional manifolds.
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…
Study strip deformations of hyperbolic polygons with decorated vertices.
problem Understanding deformations of hyperbolic polygons with decorated vertices.
method Analyzing strip deformations of ideal hyperbolic polygons with horoballs.
result Arc complexes parameterize uniformly lengthening deformations.
The paper introduces a new discretization of Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with conic singularities.
method Discrete conformal theory and variational principles with constraints.
result Established a discrete uniformization theorem for surfaces with non-positive Euler number.
Decorated TQFTs compute invariants with additional structures.
problem Computing topological invariants with additional structures.
method Cutting and gluing to obtain decorated invariants, proposing Hilbert spaces for two-dimensional surfaces.
result Proposal for Hilbert spaces assigned to surfaces in decorated TQFTs.
A spine is constructed for a non-orientable surface's decorated Teichmüller space.
problem Constructing a spine for a non-orientable surface's decorated Teichmüller space.
method Building on Harer's work, constructing a spine and computing its dimension, showing equivariance with the pure mapping class group.
result A spine is constructed with minimal dimension for a punctured non-orientable surface.
Researchers prove the arc complexes of decorated hyperbolic polygons are balls.
problem Understanding the structure of decorated hyperbolic polygons.
method Combinatorial approach using pseudo-manifolds and shellability.
result Arc complexes of decorated hyperbolic polygons are closed piecewise linear balls.
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…
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 …
Constructs TQFTs for cobordisms with cohomology class decorations.
problem Creating TQFTs for cobordisms with cohomology class decorations.
method Starting from an abelian group and a factorizable ribbon Hopf -bialgebra , constructs a TQFT for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in .
result Our functor recovers a special case of Kerler-Lyubashenko TQFTs when restricted to trivial decorations.
New TQFT for link cobordisms without decoration.
problem Tackles the problem of defining a TQFT for link cobordisms.
method Introduces a Heegaard-Floer homology functor.
result Independently constructs a TQFT for link cobordisms.
Researchers extend parametrization of Margulis spacetimes using strip deformations.
problem Parametrize Margulis spacetimes with decorated horoballs.
method Use strip deformations to parametrize complete finite-area hyperbolic surfaces with spikes decorated with horoballs.
result Generalized parametrization of Margulis spacetimes with photons.
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…
The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.
problem Finding decorated piecewise hyperbolic metrics with prescribed combinatorial curvature.
method Introduced combinatorial α-Ricci flow with surgery to handle potential singularities and prove longtime existence and convergence.
result Existence of decorated piecewise hyperbolic metrics with prescribed combinatorial α-curvature.
We produce a one-parameter family of coordinates of the decorated Teichmüller space of an ideally triangulated punctured surface with negative Euler characteristic, which is a deformation of Penner's simplicial coordinate \cite{P1}. If , the decorated Teichmüller space in…
We present a marked analogue of Carter and Saito's movie theorem. Our definition of marking was chosen to coincide with the markings that arise in link Floer homology. In order to deal with complications arising from certain isotopies, we define three equivalence relations for marked surfaces and work over an equivalen…
A marked graph diagram is a link diagram possibly with marked -valent vertices. S. J. Lomonaco, Jr. and K. Yoshikawa introduced a method of representing surface-links by marked graph diagrams. Specially, K. Yoshikawa gave local moves on marked graph diagrams, nowadays called Yoshikawa moves. It is now known that two…
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 …