New model for rational tropical points using sp4-webs and measures.
problem Understanding rational tropical points of Fock-Goncharov moduli space.
method Introducing rational bounded sp4-laminations and defining tropical coordinate systems. result Established a bijection between rational tropical points and sp4-webs. Spaces of positive and tropical points described as Teichmüller and lamination spaces with pinnings.
problem Describing spaces of positive and tropical points in moduli space with pinnings.
method Topological description of Teichmüller and lamination spaces with pinnings.
result Formulae relating functions on Teichmüller space with pinnings and topological description of tropicalized amalgamation map.
Interprets SL3-web intersections on surfaces.
problem Interpreting SL3-web intersections on surfaces.
method Intersection pairing between reduced SL3-webs and tropical sets.
result Provides a new proof of flip equivariance.
Geometric interpretation of Fock-Goncharov positivity and disk stabilization in symmetric space.
problem Understanding Fock-Goncharov positivity and its geometric implications.
method Geometric interpretation and bending deformations of Fuchsian representations.
result Stabilization of a uniform Finsler quasi-convex disk in the symmetric space.
Dominant representations found via Fock-Goncharov coordinates.
problem Finding dominant representations of surface groups.
method Linear-algebraic approach using Fock-Goncharov coordinates.
result Explicit description of dominating representations.
New proof of SL(n) skein algebra for twice punctured sphere, showing it's a polynomial algebra.
problem Proving the structure of SL(n) skein algebra for a specific surface.
method Constructing a linear basis of explicit SL(n) webs, proving spanning and linear independence.
result SL(n) skein algebra of twice punctured sphere is a commutative polynomial algebra in n-1 generators.
The area of a convex projective surface of genus g≥2 is at least (g−1)π2/2+∥τ∥2/8 where τ=(logti) is the vector of triangle invariants of Bonahon-Dreyer and ti are the Fock-Goncharov triangle coordinates.
New coordinates for SL3-web graphs on surfaces defined by Fock-Goncharov.
problem Characterizing functions on SL3-character varieties of surfaces.
method Introduced tropical Fock-Goncharov coordinates based on Knutson-Tao rhombus inequalities and congruence conditions.
result Tropical coordinates naturally index the commutative algebra of functions on SL3-character varieties.
Geometric model of unbounded sl3 laminations with tropical coordinates.
problem Modeling unbounded laminations in cluster varieties.
method Introducing tropical cluster coordinates and geometric gluing procedures.
result Established a geometric gluing procedure for unbounded sl3 laminations.
Let P(S) be the space of convex projective structures on a surface S with negative Euler characteristic. Goldman and Bonahon-Dreyer constructed two different sets of global coordinates for P(S), both associated to a pair of pants decomposition of the surface S. The article explicitly describes the coordinate change bet…
Quantum trace map connects Teichmüller theory and quantum groups.
problem Connecting quantum groups to Teichmüller theory for knots.
method Quantum snakes technology to relate Fock-Goncharov monodromy matrices to quantum SL_n.
result Quantized Fock-Goncharov matrices satisfy quantum SL_n relations.
Analyzes how quadratic differential trajectories change with variation, proving a wall-crossing formula.
problem Analyzing how the number of trajectories of quadratic differentials changes with variation.
method Proves an analytic wall-crossing formula using Fock-Goncharov coordinates and characterizes birational automorphisms.
result Characterizes certain birational automorphisms and computes Stokes automorphisms.
The paper constructs bases for cluster varieties using mSL3-webs and laminations.
problem Cluster varieties associated to mSL3-local systems on surfaces. method Introducing mSL3-laminations, developing quantum and classical trace maps, and constructing bases. result Bases of regular functions on mPGL3 cluster varieties constructed from mSL3-laminations. Quantum traces map skein algebras to Fock-Goncharov spaces.
problem Establishing quantum traces between skein algebras and Fock-Goncharov spaces.
method Defining and proving properties of quantum traces for SLn-skein algebras. result Existence and properties of quantum traces for SLn-skein algebras. Study geometrically characterizes piecewise circular curves with decreasing curvature.
problem Characterizing piecewise circular curves with decreasing curvature.
method Introducing moduli spaces and relating them to Legendrian polygons.
result Proves the moduli space contains a connected component homeomorphic to the Fock-Goncharov space of positive flags.
Let G be a simply connected, simple, complex Lie group of rank 2. We give explicit Fock-Goncharov coordinates for configurations of triples and quadruples of affine flags in G. We show that the action on triples by orientation preserving permutations corresponds to explicit quiver mutations, and that the same holds for…
New representations defined for groups and graphs, with applications to stable representations.
problem Defining and constructing new types of representations for groups and graphs.
method Introducing (R,Λ)-directed Anosov representations and using Fock-Goncharov positivity to construct them. result Constructs large families of primitive stable representations from F2 to PGL(V), including non-discrete and non-faithful examples. This paper deals with non-Archimedean representations of punctured surface groups in PGL(3), associated actions on Euclidean buildings (of type A2), and degenerations of real convex projective structures on surfaces. The main result is that, under good conditions on Fock-Goncharov generalized shear parameters, non-Arch…
We explain that spectral networks are a unifying framework that incorporates both shear (Fock-Goncharov) and length-twist (Fenchel-Nielsen) coordinate systems on moduli spaces of flat SL(2,C) connections, in the following sense. Given a spectral network W on a punctured Riemann surface C, we explain the process of "abe…
A cluster variety of Fock and Goncharov is a scheme constructed by gluing split algebraic tori, called seed tori, via birational gluing maps called mutations. In quantum theory, the ring of functions on seed tori are deformed to non-commutative rings, represented as operators on Hilbert spaces. Mutations are quantized …
New solutions to 3D integrability equations using quantum cluster algebras.
problem Constructing solutions to the tetrahedron and 3D reflection equations.
method Extending quantum cluster algebra approach to Fock-Goncharov quivers and investigating cluster transformations.
result Explicit formulas for matrix elements of solutions derived for typical representations.
We generalise in this article the Mc Shane-Mirzakhani identities in hyperbolic geometry to arbitrary cross ratios. We give an expression of them in the case of Hitchin representations of surface groups in PSL(n, R) in a suitable choice of Fock-Goncharov coordinates.
Extends quantum trace map to SL3(C) for 3D surfaces.
problem Generalizing quantum trace map to higher dimensions.
method Definition of SL3(C) quantum trace invariant.
result Construction of SL3(C) quantum trace map.
Unified view of spectral networks linking geometry and gauge theory.
problem Understanding BPS states in gauge theories.
method Unified geometric and physical approaches, focusing on spectral networks.
result Spectral networks provide a framework for determining BPS spectra.
The paper generalizes Thurston's earthquake map to cluster algebras of finite type.
problem Tackling Thurston's earthquake map in the context of cluster algebras of finite type.
method Introducing a cluster algebraic generalization of Thurston's earthquake map, defined by gluing exponential maps.
result Proves an analogue of the earthquake theorem for cluster algebras of finite type, showing the cluster earthquake map is a homeomorphism.
The {\em rank n swapping algebra} is a Poisson algebra defined on the set of ordered pairs of points of the circle using linking numbers, whose geometric model is given by a certain subspace of (Kn×Kn∗)r/GL(n,K). For any ideal triangulation of Dk---a disk wit…
A cluster variety of Fock and Goncharov is a scheme constructed from the data related to the cluster algebras of Fomin and Zelevinsky. A seed is a combinatorial data which can be encoded as an n×n matrix with integer entries, or as a quiver in special cases, together with n formal variables. A mutation is a c…
Quantum trace maps for surfaces are shown to be compatible under triangulations.
problem Constructing and understanding quantum trace maps for surfaces.
method Developed quantum mutation maps between subalgebras of quantum torus algebras for different triangulations.
result Quantum trace maps are natural and independent of triangulation choices.
Natural coordinates for SL3-webs on surfaces are shown to be consistent under triangulation changes.
problem Natural coordinates for SL3-webs on surfaces.
method Showed natural coordinates are consistent under triangulation changes using cluster transformations.
result Natural coordinates for SL3-webs on surfaces are consistent under triangulation changes.
Bracelets and theta bases match in various cluster algebras.
problem Matching bracelet and theta bases in cluster algebras.
method Comparing skein and cluster algebras, defining quantum bracelets, and analyzing cluster scattering diagrams.
result Quantum bracelets coincide with theta functions in various cluster algebras.
We show that the Borel sums of the Voros symbols considered in the theory of exact WKB analysis arise naturally as Fock-Goncharov coordinates of framed PGL2(C)-local systems on a marked bordered surface. Using this result, we show that these Borel sums can be meromorphically continued to any point of $\math…
Quantizes geodesic lengths in Teichmüller spaces using algebraic methods.
problem Constructing quantized geodesic lengths for Teichmüller spaces.
method Developed quantum trace maps and investigated algebraic structures.
result Showed a recursion relation and commutation properties for quantized trace-of-monodromy.
Study of Hamiltonian flows on character varieties for self-intersecting curves.
problem Analyzing periodic orbits of Hamiltonian flows on character varieties.
method Explicit computations in Fock-Goncharov coordinates.
result Hamiltonian flows of trace functions associated to self-intersecting curves on a pair of pants have periodic orbits.
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 …
Study examines Hilbert area of inscribed polygons in projective geometry.
problem Understanding Hilbert area of inscribed polygons in projective geometry.
method Examined correspondence between Fock-Goncharov and Cartesian coordinates, analyzed degeneration and Hilbert area of inscribed quadrilaterals, developed microlocal condition.
result Sequence of strictly convex domains with bounded Hilbert area and divergent Goldman parameters.
It was shown by Fock, Goncharov and Fomin, Shapiro, Thurston that some cluster algebras arise from triangulated orientable suraces. Subsequently Dupont and Palesi generalised this construction to include unpunctured non-orientable surfaces, giving birth to quasi-cluster algebras. Previously we linked this framework to …
Using the work of Bonahon-Dreyer and Fock-Goncharov, one can construct a real-analytic parameterization for the PSL(n,R) Hitchin component of a surface S, that is explicitly analogous to the Fenchel-Nielsen coordinates on the Teichmuller space of S. Given a Hitchin representation, we give a lower bound on the "length" …
Proves Poincaré duality for Hopf algebroids with bijective antipode.
problem Proving Poincaré duality for Hopf algebroids.
method Using twisted Poincaré duality and bijective antipode properties.
result Recovering and extending known Poincaré dualities for Hopf algebroids.
New proof of chain duality for simplicial complexes.
problem Proving the existence of chain duality for chain complexes over simplicial complexes.
method Geometric and conceptual treatment of chain duality.
result Fundamental for Ranicki's surgery exact sequence.
Introduces Kähler duality between domains in complex space.
problem None explicitly stated; focuses on concept introduction.
method None explicitly stated; focuses on concept introduction.
result Introduces Kähler duality between domains in complex space.
Research on dualities in geometric stereotypes.
problem Understanding dualities in geometric stereotypes.
method Continuation of previous research on stereotype spaces and algebras.
result New insights into geometric stereotype dualities.
Duality restored in gauge theory, gravity, and string theory models.
problem Restoring duality invariance in theories coupled to matter.
method Extending phase space to allow for violations of the algebraic Bianchi identity and considering the axion as the duality current.
result Duality current in NS-NS gravity is the divergence of the axion.
Unified proof of four Bavard dualities and new results on quasimorphisms.
problem Comparing stable commutator length and quasimorphisms on groups.
method Expository account and new strengthening of Bavard duality, providing complete proofs.
result Generalized mixed Bavard duality, recovering all previous dualities.
Verma Howe duality connects tensor products of Verma modules to LKB representations.
problem Understanding the relationship between tensor products of Verma modules and LKB representations.
method Established a quantized version of Verma Howe duality and used it to prove the simplicity of LKB representations.
result LKB representations arise from the quantized Verma Howe duality and are shown to be simple modules.
Cohomological and homological spectral sequences are shown to be isomorphic.
problem Cohomological and homological Atiyah-Hirzebruch spectral sequences are not always isomorphic.
method Spanier-Whitehead duality is used to establish an isomorphism between the two spectral sequences.
result Cohomological and homological Atiyah-Hirzebruch spectral sequences are isomorphic for finite spectra.
The paper proves T-duality and Hori formulae for winding loop spaces.
problem Realizing T-duality and Hori formulae for loop spaces.
method Proving T-duality and Hori formulae for winding q-loop spaces.
result T-duality and Hori formulae for winding q-loop spaces are proven.
Equivariant T-duality connects bundles with twists.
problem Establishing a relationship between bundles with twists.
method Formulating T-duality in equivariant K-theory for compact Lie group actions.
result T-duality is an isomorphism in equivariant K-theory for compact Lie group actions.
We give the definition of a duality that is applicable to arbitrary k-forms. The operator that defines the duality depends on a fixed form Ω. Our definition extends in a very natural way the Hodge duality of n-forms in 2n dimensional spaces and the generalized duality of two-forms. We discuss the properties of …