Quantum heat traces study new invariants from elliptic operators.
problem New invariants of elliptic operators on Riemannian manifolds.
method Relativistic and quantum heat traces, integral transforms, asymptotic expansion.
result Coefficients of asymptotic expansion determined by local and global invariants.
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.
Researchers found a Weyl law for Liouville quantum gravity eigenvalues.
problem Understanding the spectral geometry of Liouville quantum gravity.
method Obtained a Weyl law for eigenvalues of Liouville Brownian motion.
result The n-th eigenvalue grows linearly with n, with a constant determined by the Liouville area and a specific cγ. Study heat trace expansion on manifolds with conic points.
problem Analyzing heat diffusion on manifolds with sharp corners.
method Using the Singular Asymptotics Lemma to derive an expansion.
result Detailed asymptotic expansion reveals geometric insights.
Heat trace analysis reveals singularity properties on wedge-shaped manifolds.
problem Analyzing heat trace asymptotics on manifolds with wedge-like singularities.
method Heat trace expansion and asymptotic analysis on manifolds with wedge-like singularities.
result Two terms in the heat trace expansion contain information about the singularity.
Study heat traces for drifting Laplacian and Schrödinger operators on manifolds.
problem Analyzing heat traces for drifting Laplacian and Schrödinger operators on manifolds.
method Proved asymptotic expansions and remainder estimates for heat traces under different regularity conditions.
result The asymptotic behavior of the remainder is determined by higher regularity of the potential or weight function.
Study on heat trace on sub-Riemannian manifolds using probabilistic methods.
problem Analyzing heat trace on sub-Riemannian manifolds with smooth measures.
method Probabilistic approach using S. Watanabe's distributional Malliavin calculus.
result Proved a short time asymptotic expansion of the heat trace up to any order.
The paper analyzes heat trace asymptotics for de Rham and Dolbeault complexes in both real and complex settings.
problem Examining heat trace asymptotics for de Rham and Dolbeault complexes in different geometric settings.
method Analyzing the derived heat trace asymptotics for generalized Witten perturbations in both real and complex settings.
result The integral of the local density for the derived heat trace asymptotics is related to the Euler characteristic and characteristic numbers of the tangent and twisting vector bundles.
Study on heat trace expansion for thermoelastic Dirichlet-to-Neumann map.
problem Asymptotic expansion of heat trace for thermoelastic Dirichlet-to-Neumann map.
method Provided a method to obtain all coefficients of the asymptotic expansion.
result Explicitly gave the first two coefficients involving volume and total mean curvature of the boundary.
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.
New heat trace coefficients reveal curvature effects in polygonal domains.
problem Understanding heat trace behavior in polygonal domains with curved corners.
method Local heat trace expansion through order t1/2, analyzing both Dirichlet and Neumann boundary conditions. result Sharp sign law for the Dirichlet angular factor of the first corner-curvature heat invariant.
Study spectral invariants for polygons and orbisurfaces.
problem Compute spectral invariants for polygons and orbisurfaces.
method Study the heat kernel and asymptotic expansion of the heat trace.
result Explicit formulas for heat invariants of polygons and orbisurfaces.
The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.
problem Eigenvalue problems and heat trace asymptotics for different operators.
method Establishes connections and inequalities for eigenvalues and heat traces.
result Eigenvalue inequalities and three-term asymptotic formulas for heat traces of various operators.
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.
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.
We show that the linear trace Harnack quadratic on a steady gradient Ricci soliton satisfies the heat equation. Similar result holds for shrinkers. We also present an interpolation between Perelman's and Cao--Hamilton's Harnacks on a steady soliton.
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.
Predicts coherence from quantum heat engine noise using machine learning.
problem Predicting coherence in quantum heat engines from nonequilibrium fluctuations.
method Developed a machine learning protocol using K-Nearest Neighbor (KNN) model.
result Machine learning successfully predicts coherence from quantum heat engine noise.
The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold $H^3/\Ga$ are evaluated in the case in which the discrete group $\Ga$ contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace…
The paper calculates spectral invariants of magnetic Steklov eigenvalues on Riemannian manifolds.
problem Investigating spectral invariants of magnetic Steklov eigenvalues on Riemannian manifolds.
method Established an effective procedure to calculate all coefficients of the heat trace asymptotic expansion.
result Explicitly provided expressions for the first four coefficients of the heat trace asymptotic expansion.
We consider the Hodge Laplacian on manifolds with incomplete edge singularities, with infinite dimensional von Neumann spaces and intricate elliptic boundary value theory. We single out a class of its algebraic self-adjoint extensions. Our microlocal heat kernel construction for algebraic boundary conditions is guided …
We develop a new method for the calculation of the heat trace asymptotics of the Laplacian on symmetric spaces that is based on a representation of the heat semigroup in form of an average over the Lie group of isometries and obtain a generating function for the whole sequence of all heat invariants.
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.
Study of spectral geometry on noncommutative tori using functional metrics.
problem Understanding the spectral properties of noncommutative tori.
method Introduction of functional metrics and analysis of their Laplace type operators and spectral invariants.
result Explicit computation of scalar curvature and total scalar curvature for certain functional metrics.
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.
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.
The paper proves new Harnack inequalities for various nonlinear heat equations on manifolds.
problem Analyzing and proving new Harnack inequalities for nonlinear heat equations.
method Proving constrained trace, matrix, and interpolated Harnack inequalities for specific nonlinear heat equations.
result Derives new differential Harnack inequalities with time-exponential correction terms.
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.
Let G be a compact connected Lie group equipped with a bi-invariant metric. We calculate the asymptotic expansion of the heat kernel of the laplacian on G and the heat trace using Lie algebra methods. The Duflo isomorphism plays a key role.
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. 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.
Researchers derived heat kernel expansions for non-compact spaces using Witten deformation.
problem Heat kernel expansions on non-compact spaces, especially for Witten Laplacians.
method Introduced parabolic distance and used it to derive asymptotic expansions.
result Derived an asymptotic expansion of trace of heat kernel for small-time t. 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.
Study quantum diffusion on spectral triples and spinor bundles.
problem Characterize quantum diffusion on almost commutative spectral triples.
method Spin geometry, C *-Dirichlet forms, quantum stochastic flows.
result Existence of covariant quantum stochastic flows on spinor bundles.
The paper proves recurrence relations for heat kernels on hyperbolic and spherical spaces.
problem Understanding recurrence relations of heat kernels on different space forms.
method Direct proof and computation of recurrence relations for heat kernels on hyperbolic and spherical spaces.
result Computed diagonal of heat kernels for odd dimensional hyperbolic spaces and heat trace asymptotic expansions for odd dimensional spheres.
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.
Study calculates heat coefficients from geodesic polygon corners.
problem Heat coefficients from geodesic polygon corners.
method Formulas for cone points contributions to heat trace.
result Determines Laplacian of Gauss curvature at corners.
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. 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 …
The paper calculates heat kernel and closed geodesic asymptotics for nilpotent coverings.
problem Heat kernel and closed geodesic asymptotics for nilpotent coverings.
method Finite-dimensional rational Floquet-Bloch theory, Pytlik functional, and spectral sums.
result Genuinely local, pointwise higher-order heat-kernel expansions.
The paper calculates the full asymptotics of analytic torsions for compact orbifolds.
problem Analytic torsions of compact locally symmetric orbifolds.
method Using Selberg's trace formula and geometric localization, the paper evaluates the heat trace and orbital integrals.
result Explicit formula for the asymptotic Ray-Singer analytic torsion of compact orbifolds.
Heat kernels are used in this paper to express the analytic index of projectively invariant Dirac type operators on G-covering spaces of compact manifolds, as elements in the K-theory of certain unconditional completions of the twisted group algebra of G. This is combined with V. Lafforgue's results in the untwisted ca…
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.
The paper studies heat kernel behavior on symmetric spaces.
problem Large-time behavior of heat operator traces on symmetric spaces.
method Uses representation theory and Carmona's proof of Vogan's lambda map.
result Provides an asymptotic formula for heat kernel behavior.
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.