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.
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.
Non-injectivity proven for trace map on character varieties.
problem Proving non-injectivity of trace map on character varieties.
method Using Amitsur-Levitzki identity and free homotopy classes.
result Explicit construction of nonzero elements in kernel of trace map.
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.
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.
The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.
problem Existence of Poincaré embeddings for specific spaces.
method Relates total obstruction to Reidemeister trace and uses Poincaré duality.
result Diagonal maps admit Poincaré embeddings under certain conditions.
The study finds all trace field degrees for Torelli group mappings.
problem Identifying all possible trace field degrees for Torelli group mappings.
method Using Thurston-Veech construction of pseudo-Anosov maps, and providing examples of stretch factors with specific algebraic degrees.
result All integers 1≤d≤3g−3 are trace field degrees for g≥2. Study 2-loop part of Johnson cokernel using trace map.
problem Identify components of Johnson cokernel in degree 6.
method Use 2-loop trace map to capture Johnson cokernels.
result Capture all components of Johnson cokernels in degree 6.
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 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.
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.
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.
Maps with many singularities found in complex space.
problem Constructing maps with singularities in complex space.
method Created a map with infinitely many Schoen-Wolfson singularities on a disc.
result Found a Ck map with smooth trace in C2. 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.
Unified formula for higher traces of linear maps on finite-dimensional normed spaces.
problem Unified trace-average formula for higher traces of linear maps.
method Unified trace-average formula for the k-th higher trace of a linear operator A on a finite-dimensional normed space.
result Unified trace-average formula holds for all A if and only if the operator-valued average equals the identity.
New parametrization of 3-spheres using Johnson subgroups.
problem Constructing integral homology 3-spheres.
method Intrinsic description of equivalence relation on fourth Johnson subgroup.
result Intrinsic description of equivalence relation using fourth Johnson handlebody subgroups.
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. The study computes trace fields and minimal polynomials for specific knots and links.
problem Computing trace fields and minimal polynomials for specific knots and links.
method Using factorization theorems for sparse polynomials.
result Results depend on the degrees of the trace fields over Q being sufficiently large.
We give a construction to remove coincidence points of continuous maps on graphs (1-complexes) by changing the maps by homotopies. When the codomain is not homeomorphic to the circle, we show that any pair of maps can be changed by homotopies to be coincidence free. This means that there can be no nontrivial coincidenc…
We give a homotopy invariant construction of the Reidemeister trace for the coincidence of two maps between closed manifolds of not necessarily the same dimensions. It is realized as a homology class of the homotopy equalizer, which coincides with the Hurewicz image of Koschorke's stabilized bordism invariant. To defin…
In this article, we introduce an analogous problem to Yamabe type problem considered by Case, J., which generalizes the Escobar-Riemann mapping problem for smooth metric measure spaces with boundary. The last problem will be called Escobar-Riemann mapping type problem. For this purpose, we consider the generalization o…
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 …
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.
We review the current state of the art concerning the characterization of traces of the spaces W1,p(Bm−1×(0,1),N) of Sobolev mappings with values into a compact manifold N. In particular, we exhibit a new analytical obstruction to the extension, which occurs when p<m …
The flat trace of geodesic Koopman operators varies with negatively curved surfaces.
problem Understanding how the flat trace of geodesic Koopman operators changes with variations of negatively curved surfaces.
method Computing the first variation of the flat trace as a distribution and analyzing its leading singularity.
result The leading singularity coefficient is a linear functional of length variations, forcing marked lengths to be locally constant.
We give a concise introduction to the Farrell-Jones Conjecture in algebraic K-theory and to some of its applications. We survey the current status of the conjecture, and we illustrate the two main tools that are used to attack it: controlled algebra and trace methods.
This paper motivates and develops source traces for temporal difference (TD) learning in the tabular setting. Source traces are like eligibility traces, but model potential histories rather than immediate ones. This allows TD errors to be propagated to potential causal states and leads to faster generalization. Source …
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 trace systoles on surfaces, finding optimal bounds and implications.
problem Optimal systolic inequalities on hyperbolic manifolds and non-Fuchsian representations.
method Defined trace systole, used Markoff maps correspondence, computed bounds.
result Explicit optimal bounds for one-holed torus, four-holed sphere, and non-orientable surface of genus 3.
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.
Study subgroup of mapping class group related to handlebody, answering a question about Johnson homomorphism.
problem Understanding the image of the second Johnson homomorphism for a specific subgroup of mapping class groups.
method Introduced trace-like operators and used them to compute images of Johnson homomorphisms.
result Answered a question about algebraic description of the image of the second Johnson homomorphism.
The forcing relation of braids has been introduced for a 2-dimensional analogue of the Sharkovskii order on periods for maps of the interval. In this paper, by making use of the Nielsen fixed point theory and a representation of braid groups, we deduce a trace formula for the computation of the forcing order.
CausalSim corrects bias in trace-driven simulations for more accurate results.
problem Bias in trace-driven simulations due to system conditions during trace collection.
method CausalSim learns a causal model of system dynamics and latent factors from an RCT to remove bias from trace data.
result CausalSim reduces simulation errors by 53% and 61% compared to baselines, providing more accurate insights.
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.
We prove that the Farrell-Jones assembly map for connective algebraic K-theory is rationally injective, under mild homological finiteness conditions on the group and assuming that a weak version of the Leopoldt-Schneider conjecture holds for cyclotomic fields. This generalizes a result of Bökstedt, Hsiang, and Madsen, …
Let G a be subgroup of SL(2,C), the group of 2x2 matrices of determinant 1 with complex entries. Let h map onto h(G) be a homomorphism. We call h a trace preserving homomorphism if tr(h(g))=tr(g) for all g in G,where tr(g) is the trace of g. We solve the question of when a trace invariant homomorphism is a conjugation …
Deform moment map on symplectic connections using star product algebras.
problem Understanding symplectic connections and their deformations.
method Study vector bundle of Fedosov star product algebras, formal connection, curvature, and star product trace.
result Showed star product trace as a formal symplectic form and moment map.
By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface. The new skein algebra of an ideal triangle has a simple presentation. This gives …
We define a canonical map from a certain space of laminations on a punctured surface into the quantized algebra of functions on a cluster variety. We show that this map satisfies a number of special properties conjectured by Fock and Goncharov. Our construction is based on the "quantum trace" map introduced by Bonahon …
We prove that the balanced Chekhov-Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Che…
Symplectic GP regression models Hamiltonian systems for particle tracing.
problem Efficiently modeling long-term Hamiltonian flow maps for charged particles.
method Multi-output Gaussian process regression with symplectic matrix-valued covariance function.
result Symplectic methods outperform existing approaches in learning Hamiltonian functions.
The cyclotomic trace of Bökstedt-Hsiang-Madsen, the subject of Bökstedt's lecture at the congress in Kyoto, is a map of pro-abelian groups K_*(A) -> TR_*^.(A;p) from Quillen's algebraic K-theory to a topological refinement of Connes' cyclic homology. Over the last decade, our understanding of the target and its relatio…
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.
Using geometric quantization, we represent curve operators in the TQFT of Witten-Reshetikhin-Turaev with jauge group SU_2 as Toeplitz operators with symbols corresponding to trace functions. As an application, we show that eigenvectors of these operators are concentrated near the level sets of these trace functions, an…
For a hyperbolic link complement with a triangulation, there are hyperbolicity equations of the triangulation, which guarantee the hyperbolic structure of the link complement. In this paper, we explain that the number of the essential solutions of the equations is equal to or bigger than the extension degree of the inv…
PSI-KT improves KT accuracy and interpretability in learning materials.
problem Optimizing learning materials selection and timing for understanding and retention.
method Hierarchical generative approach using Bayesian inference.
result Superior multi-step predictive accuracy and scalable inference.
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.