Quantum Teichmüller theory solved by linking Bonahon-Wong trace and Gabella's solution.
problem Quantize the trace-of-monodromy function on Teichmüller space.
method Used Bonahon and Wong's mSL2 quantum trace for skein algebras and Gabella's Seiberg-Witten curves, spectral networks, and writhe of links. result Bonahon-Wong quantum trace and Gabella's solution coincide and are a twist of each other.
Quantum link invariants derived from skein algebras.
problem Defining invariants for framed links with SL2 local systems.
method Theory of representations of stated skein algebras, quantum coadjoint action, Drinfeld double, Bonahon-Wong quantum trace.
result Explicit formulas for link invariants and alternative proof of Murakami-Murakami relation.
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.
Quantum traces embed into quantum tori for surface skein algebras.
problem Embedding stated skein algebras into quantum tori.
method Two different embeddings using quantum trace maps and lambda length coordinates.
result Quantum cluster algebra of Muller equals reduced stated skein algebra.
Researchers compute quantum invariant for four-puncture sphere, verifying volume conjecture.
problem Verifying the Bonahon-Wong-Yang volume conjecture for a specific case.
method Representation theory of the Checkov-Fock algebra to compute quantum invariant.
result Verification of the volume conjecture for four-puncture sphere bundles with technical conditions.
The volume conjecture is extended for surface diffeomorphisms with quantum invariants.
problem Extending the volume conjecture for quantum invariants of surface diffeomorphisms.
method Relating asymptotics of quantum invariants to hyperbolic cone structures on mapping tori.
result The conjecture is proven for a specific case of the once-punctured torus bundle.
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. 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.
We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recovered from the Poisson structu…
The paper calculates intertwiners for a torus and proves a conjecture about their limits.
problem Relating quantum invariants to hyperbolic geometry using intertwiners.
method Explicit calculation of intertwiners for a closed torus and periodic diffeomorphisms.
result The limit superior of the trace of intertwiners is zero for certain diffeomorphisms.
Quantum Frobenius map for SL3 skein modules constructed and described.
problem Constructing a quantum Frobenius map for SL3 skein modules. method Using threading polynomials and the Frobenius map of Parshall-Wang for quantum group Oq(SL3). result Described the quantum Frobenius map for SL3 skein modules. Embeds skein algebras into quantum tori using Dehn-Thurston coordinates.
problem Studying representations of Kauffman bracket skein algebras at roots of unity.
method Using the action of the skein algebra on the skein module of the handlebody.
result Explicit reconstruction of unique representation with fixed classical shadow.
New invariants explain topological properties of pseudo-Anosov maps.
problem Understanding invariants of pseudo-Anosov maps at roots of unity.
method Numerical methods and quantum modularity conjecture.
result Descendant invariants related by Fourier transform.
Quantum trace map defined for 3-manifolds with torus boundaries.
problem Quantifying topological structures of 3-manifolds with torus boundaries.
method Defining a quantum trace map from skein module to a quantum torus module.
result Established a 3D quantum trace map for 3-manifolds with torus boundaries.
3D quantum trace map connects 3-manifold quantizations.
problem Quantization of 3-manifold character varieties.
method Study of stated skein modules and face suspensions.
result Existence of 3D quantum trace map proved.
The paper studies algebraic and geometric properties of stated skein algebras of surfaces.
problem Understanding the algebraic and geometric properties of stated skein algebras of surfaces.
method Analyzes the skein algebra of surfaces, proving isomorphisms and lifting properties, and interpreting topologically.
result The skein algebra of a surface with n boundary components is an algebra-comodule over Oq2(SL(2))⊗n. The paper shows compatibility between two quantum maps for surfaces and 3-manifolds.
problem Connecting quantum trace and UV-IR maps for surfaces and 3-manifolds.
method Analyzing compatibility under triangulation changes and using skein modules.
result Compatibility of quantum trace and UV-IR maps for surfaces and 3-manifolds.
Propose a new 3d quantum trace map that agrees with Garoufalidis and Yu's construction and extends to certain manifolds with ideal triangulated boundaries.
problem Relationship between two constructions of 3d quantum trace maps.
method Propose a new 3d quantum trace map.
result Proposed 3d quantum trace map agrees with Garoufalidis and Yu's construction and extends to certain manifolds with ideal triangulated boundaries.
Quantum trace map defines invariants for knots and links, confirming a length conjecture.
problem Defining invariants for knots and links in hyperbolic 3-manifolds.
method Introducing a quantum trace map for ideally triangulated knot complements, combining with state-integral models.
result Perturbative invariants determine an asymptotic expansion of the Jones polynomial, confirming the length conjecture.
Unified framework combines trace-induced quantum kernels for improved machine learning models.
problem Improving performance of quantum machine learning models using trace-induced kernels.
method Developed a unified framework combining various trace-induced quantum kernels, including global fidelity and local projected kernels, as Lego kernels.
result Local projected kernels can achieve comparable performance to global fidelity kernels with fewer quantum resources.
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.
Center identified in stated skein algebra for quantum traces.
problem Understanding the center of the stated skein algebra.
method Analyzing the algebra as a generalization of Kauffman bracket skein algebra, focusing on the case when the quantum parameter is a root of unity.
result Simple description and dimension calculation of the center over the center module.
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.
The abstract discusses new 3-manifold invariants and ETQFTs from Lie superalgebra representations.
problem Developing new 3-manifold invariants and ETQFTs from Lie superalgebra representations.
method Examining two m-traces in the category of representations over quantum sl(m∣n), considering quotients, and conjecturing generalizations. result Quotients of perturbative modules over quantum sl(m∣n) lead to 3-manifold invariants and ETQFTs. Direct formula found for ADO invariants from homological representations.
problem Computing ADO invariants from quantum group representations.
method Direct homological formula for ADO invariants using partial traces of homological representations.
result Direct formula for ADO invariants without further truncations.
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.
New identities link Frobenius elements to Jones-Wenzl projectors at roots of unity.
problem Understanding relationships between Frobenius elements and Jones-Wenzl projectors at roots of unity.
method Obtained skein identities relating Frobenius elements to Jones-Wenzl projectors in the Kauffman bracket skein module.
result Skein identities provide new proofs of the existence of the Chebyshev-Frobenius homomorphism.
Defines a map connecting 3d-index and skein module.
problem Connecting mathematical physics predictions with topological quantum field theory.
method Defines a map from skein module to Laurent series ring.
result The map fulfills a supersymmetry prediction and is part of a conjectural topological quantum field theory.
We study new invariants of elliptic partial differential operators acting on sections of a vector bundle over a closed Riemannian manifold that we call the relativistic heat trace and the quantum heat traces. We obtain some reduction formulas expressing these new invariants in terms of some integral transforms of the u…
We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichmüller space of a marked surface, defined by Chekhov-Fock (and Kashaev) in an abstract …
Develops trace class operators and inverse Laplacian theory for infinite dimensions.
problem Understanding trace class operators and inverse Laplacian on infinite dimensional spaces.
method Presentation of trace class operators and construction of inverse Laplacian on closed manifolds.
result Original trace computations involving the inverse Laplacian on the torus.
Quantum trace maps abelian character varieties to SL2 character varieties.
problem Classifying irreducible representations of Chekhov-Fock algebras.
method Non-commutative deformation of algebraic morphisms.
result Induces birational morphism between torus and SL2 character variety.
New methods derive a generalized Frenkel trace formula for Lie groups.
problem Deriving a generalized Frenkel trace formula for Lie groups.
method Applying supersymmetric localization to quantum mechanical and gauged sigma models.
result Presented two complementary approaches for the derivation of the trace formula.
We consider two different quantizations of the character variety consisting of all representations of surface groups in SL_2. One is the skein algebra considered by Przytycki-Sikora and Turaev. The other is the quantum Teichmuller space introduced by Chekhov-Fock and Kashaev. We construct a homomorphism from the skein …
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.
Introduces a new model for mapping matrices to matrices, subsuming linear regression.
problem Learning matrix-to-matrix mappings from data.
method Partial trace regression model, leveraging quantum information theory.
result Relevance demonstrated in matrix-to-matrix regression and positive semidefinite matrix completion.
Witt algebra acts on categorified quantum groups in type A.
problem Action of Witt algebra on categorified quantum groups.
method Construction of action on categorified quantum group and foams.
result Action of Witt algebra on foams recovers previous results.
Proves super-version of index theorem from algebraic cobordism invariants.
problem Cobordism invariants in supersymmetric quantum mechanics.
method Trace methods for deformation quantization.
result Recovery of cobordism invariant using trace methods.
Spectral sequence connects knot homologies via algebraic geometry.
problem Connecting algebraic and geometric knot homologies.
method Bigraded spectral sequence from gl(0)-homology to knot Floer homology.
result Constructs a Bockstein-type spectral sequence.
Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a~pair: bicategory C and endobifunctor Σ:C→C. For a graded linear bicategory and a fixed invertible parameter q, we quantize this theory by using the endofunctor Σq such th…
Quantum ML predicts data with improved speed and accuracy.
problem Predicting data using maximum likelihood in a quantum setting.
method Quantum states embedding and minimization of quantum relative entropy.
result Unified framework for classical and quantum LLMs with performance guarantees.
Researchers compute the rank and trace of Kauffman bracket skein algebra over its center.
problem Computing the rank and trace of Kauffman bracket skein algebra.
method Using finite type surfaces and complex roots of unity, they compute the rank and trace of Kζ(F) over its center. result They extend a theorem about the skein algebra having a splitting coming from two pants decompositions of F. The paper studies properties of stated SL(n)-skein algebras and their centers.
problem Properties of stated SL(n)-skein algebras and their centers.
method Quantum trace maps and embeddings into quantum tori.
result Finitely generation and PI-degrees of centers of stated SL(n)-skein algebras.
We show that the reduced quantum hyperbolic invariants of pseudo-Anosov diffeomorphisms of punctured surfaces are intertwiners of local representations of the quantum Teichmüller spaces. We characterize them as the only intertwiners that satisfy certain natural cut-and-paste operations of topological quantum field theo…
Abstract: Topological quantum field theory connects graph evaluations to polynomial identities.
problem Graph evaluations in topological quantum field theory.
method Relates SO(3) topological quantum field theory trace evaluations to topological Tutte polynomial evaluations.
result Generalizes the Tutte golden identity for graphs on the torus.
Study geometric quantization of Hamiltonian flows using Berezin-Toeplitz operators.
problem Quantum dynamics of Hamiltonian flows over symplectic manifolds.
method Geometric quantization, Berezin-Toeplitz operators, parallel transport.
result Established a Gutzwiller trace formula for Kostant-Souriau operator.
In earlier work, Helen Wong and the author discovered certain "miraculous cancellations" for the quantum trace map connecting the Kauffman bracket skein algebra of a surface to its quantum Teichmueller space, occurring when the quantum parameter q is a root of unity. The current paper is devoted to giving a more repr…
Quantum stochastic flow computes heat kernel traces for Ricci flat manifolds.
problem Computing heat kernel traces for Ricci flat manifolds.
method Quantum stochastic differential equation (qsde) on Fock space over L2 differential 1-forms, adapted flow construction. result Trace of the connection Laplacian heat kernel can be computed over any compact Ricci-flat Riemannian manifold.