TQFT signatures linked to trace fields of knots.
problem Relationship between TQFT signatures and knot trace fields.
method Analysis of Frobenius algebras and TQFTs at specific roots.
result TQFT signatures equal to trace fields of two-bridge knots.
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48 results for “Frobenius trace”
Nearly Frobenius Algebrasmath.RA
In this introductory paper we study nearly Frobenius algebras which are generalizations of the concept of a Frobenius algebra which appear naturally in topology: nearly Frobenius algebras have no traces (co-units). We survey the most basic foundational results and some of the applications they encounter in geometry, to…
Study ranks of elliptic curves via prime averages.
problem Classifying elliptic curves by rank.
method Average Frobenius trace over primes, data science experiments.
result Oscillating pattern in average trace values, correlates with rank.
Study proves certain algebraic structures are symmetric Frobenius algebras.
problem Understanding algebraic structures in bordered surfaces.
method Analyzing stated skein algebras and their fraction rings.
result Fraction ring of stated skein algebra is a symmetric Frobenius algebra.
New pivoting strategy improves trace norm contraction in low-rank approximation.
problem Finding good low-rank approximations of symmetric, positive-definite matrices.
method Choosing rows with likelihood proportional to for randomly pivoted partial Cholesky algorithm.
result Same trace norm contraction result in Frobenius norm for improved pivoting strategy.
Convolutional neural networks predict the analytic rank of elliptic curves accurately.
problem Predicting the analytic rank of elliptic curves over Q.
method Applied one-dimensional convolutional neural networks to Frobenius traces.
result High accuracy predictions for analytic rank across various conductors.
Study of Chebyshev-Frobenius homomorphism in 3-manifold skein modules.
problem Exploring the Chebyshev-Frobenius homomorphism in 3-manifold skein modules.
method Generalization of splitting homomorphism for stated skein modules of 3-manifolds.
result Existence and properties of Chebyshev-Frobenius homomorphism for 3-manifold skein modules.
New arithmetic phenomenon 'murmurations' detected using AI.
problem Detecting new arithmetic patterns in large datasets.
method Machine learning interpretability tools (PCA, saliency, convolutional filters).
result Murmurations encode Frobenius traces and connect to number theory.
Quantum invariants for surfaces in 4D 2-handlebodies.
problem Quantum invariants of ribbon surfaces in 4D 2-handlebodies.
method Unimodular ribbon categories, labeled Kirby graphs, and modified traces.
result Recovery and generalization of existing invariants.
GL-LowPopArt improves minimax-optimal estimation for trace regression.
problem Minimizing estimation error in generalized low-rank trace regression.
method Two-stage approach: nuclear norm regularization followed by matrix Catoni estimation.
result Achieves instance-wise optimal error bounds up to condition number.
Matrix completion has been well studied under the uniform sampling model and the trace-norm regularized methods perform well both theoretically and numerically in such a setting. However, the uniform sampling model is unrealistic for a range of applications and the standard trace-norm relaxation can behave very poorly …
If is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is …
The study proves nearly Frobenius algebras over certain domains are Frobenius.
problem Understanding nearly Frobenius algebras and their properties.
method Analyzing nearly Frobenius algebras over principal ideal domains with specific algebraic properties.
result Any nearly Frobenius algebra with surjective multiplication and injective comultiplication is a Frobenius algebra.
I.A.B. Strachan introduced the notion of a natural Frobenius submanifold of a Frobenius manifold and gave a sufficient but not necessary condition for a submanifold to be a natural Frobenius submanifold. This paper will give a necessary and sufficient condition and classify the natural Frobenius hypersurfaces.
The paper proves geometric and spectral alignment for deep neural networks.
problem Understanding the singular spectra of deep neural network layers.
method Proves deterministic quotient-geometric estimates for singular spectra of Frobenius-normalized layer factors.
result Exact power-law spectra form a trace-normalized Cartan orbit under Frobenius normalization.
Frobenius submanifoldsmath.DG
The notion of a Frobenius submanifold - a submanifold of a Frobenius manifold which is itself a Frobenius manifold with respect to structures induced from the original manifold - is studied. Two dimensional submanifolds are particularly simple. More generally, sufficient conditions are given for a submanifold to be a s…
Flat coordinates found for algebraic Frobenius manifolds in low dimensions.
problem Understanding algebraic Frobenius manifolds in small dimensions.
method Using reflection representations of finite Coxeter groups, finding flat coordinates of the Frobenius metric.
result Explicit relations between flat coordinates of the Frobenius metric and intersection form for most known examples up to dimension 4.
A Lie version of Turaev's -Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a \textit{-quasi-Frobenius Lie algebra} for a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius…
Quantum Frobenius map for skein modules constructed and described.
problem Constructing a quantum Frobenius map for skein modules.
method Using threading polynomials and the Frobenius map of Parshall-Wang for quantum group
result Described the quantum Frobenius map for skein modules.
A new algebra for Frobenius manifolds solves PDEs and constraints.
problem Understanding the algebraic structure of Frobenius manifolds.
method Constructing a Virasoro-like algebra and deriving PDEs and constraints.
result Solves a family of quadratic PDEs for the genus-zero free energy.
Let π: V \rightarrow M be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure (\circ_{M},e_{M}, g_{M}) and typical fiber has the structure of a Frobenius algebra (\circ_{V},e_{V},g_{V}). Using a connection D on the bundle V and a morphism α: V \rightarrow TM, we construct an almost Froben…
New braided Frobenius algebras created from specific Hopf algebras.
problem Creating new algebraic structures from Hopf algebras.
method Heap operation and Yang-Baxter operator on tensor product.
result Heap operation induces a braiding compatible with Frobenius operations.
Symplectic and Poisson structures proved for information geometry's Frobenius manifold.
problem Connecting disconnected theories in information geometry.
method Proving symplectic and Poisson structures on the Frobenius manifold.
result Established a bridge between Vinberg, Souriau, and Koszul's theories.
Curved Frobenius manifolds link to Hessian metrics in geometry.
problem Understanding curved Frobenius manifolds and their relation to Hessian metrics.
method Analyzing the relationship between curved Frobenius structures and Hessian metrics on spaces with non-vanishing curvature.
result Consistent curved Frobenius structures on constant curvature spaces are linked to Hessian metrics.
Paper proves algebraic structure of a specific Frobenius manifold.
problem Understanding the algebraic properties of a specific Frobenius manifold.
method Proves decomposition into symmetric submanifolds over ideals.
result Decomposes the fourth Frobenius manifold into symmetric submanifolds.
We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra (C, W, z, z^*) consist…
Curvature interpretation for WDVV equation in Frobenius manifolds.
problem Understanding the WDVV equation in statistical manifolds.
method Analyzing the curvature of statistical manifolds and their tangent spaces.
result WDVV equation is equivalent to zero sectional -curvature.
New integrable deformations for topological hierarchies from Frobenius manifolds.
problem Integrable deformations of topological hierarchies from Frobenius manifolds.
method Construction of integrable deformations with polynomial tau-structures.
result Conjecture of universal object for Riemann--Hopf hierarchy.
We construct a class of infinite-dimensional Frobenius manifolds on the space of pairs of certain even functions meromorphic inside or outside the unit circle. Via a bi-Hamiltonian recursion relation, the principal hierarchies associated to such Frobenius manifolds are found to be certain extensions of the dispersionle…
Introduces a new canonical connection for Riemannian manifolds and proves Frobenius theorem geometrically.
problem Geometric proof of the Frobenius theorem on Riemannian manifolds.
method Introduces a new canonical connection and applies it to prove the Frobenius theorem.
result Geometric proof of the Frobenius theorem.
Study Frobenius pencils and compatible non-homogeneous Poisson structures.
problem Compatibility of multicomponent local Poisson structures.
method Algebraic interpretation via Frobenius algebras and classification of Frobenius pencils.
result Classification of Frobenius pencils under generic conditions.
Study on Frobenius manifold structures and inversion symmetry.
problem Understanding inversion symmetry in solutions to Witten-Dijkgraaf-Verlinde-Verlinde equations.
method Using conjugacy relations on Frobenius manifold structures and flat pencils of metrics.
result Geometric interpretation of inversion symmetry.
The study classifies Kähler-Frobenius manifolds and their properties.
problem Classifying Kähler-Frobenius manifolds and understanding their structure.
method Using Topological Quantum Field Theory and Frobenius manifold theory.
result All flat compact Kähler manifolds are Frobenius manifolds and are classified.
Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.
problem Existence and uniqueness of solutions to loop equations in generalized Frobenius manifolds.
method Proves existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
result Existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
The paper develops a theory of Ehresmann structures in positive characteristic.
problem Developing a theory for Ehresmann structures in positive characteristic.
method Comparing Frobenius-Ehresmann structures with Cartan geometries and studying their equivalence.
result Formulating and proving the Ehresmann-Weil-Thurston principle for Frobenius-Ehresmann structures.
Submanifolds of Frobenius manifolds are studied. In particular, so-called natural submanifolds are defined and, for semi-simple Frobenius manifolds, classified. These carry the structure of a Frobenius algebra on each tangent space, but will, in general, be curved. The induced curvature is studied, a main result being …
New superintegrable systems derived from Frobenius structures.
problem Constructing second-order superintegrable systems.
method Using conification and direct product construction, applying to semi-simple and nilpotent algebras.
result Explicitly constructed second-order superintegrable systems in three dimensions.
Researchers find Frobenius manifold structures on orbits spaces of finite groups.
problem Understanding Frobenius manifold structures on orbits spaces of finite groups.
method Applying Dubrovin's method to various orbits spaces of linear representations of finite groups.
result Discoveries of non-trivial Frobenius manifold structures on orbits spaces.
New findings on Frobenius structures on Kodaira manifolds.
problem Understanding Frobenius structures on Kodaira manifolds.
method Extended deformation theory and Frobenius structures.
result Frobenius structure on Kodaira manifolds is trivial on degree-2 component.
New Frobenius manifold structures found on Dicyclic group orbits.
problem Finding Frobenius manifold structures on orbits spaces of Dicyclic groups.
method Applying Dubrovin's method to Dicyclic groups.
result Dicyclic group orbits spaces acquire two Frobenius manifold structures.
The paper geometrically characterizes graded manifolds and proves the Frobenius theorem.
problem Understanding and characterizing graded manifolds.
method Geometric characterization and Frobenius theorem proof.
result Frobenius theorem proven for graded distributions.
Linearizes Virasoro symmetries for semisimple Frobenius manifolds.
problem Linearizing Virasoro symmetries for semisimple Frobenius manifolds.
method Proving the existence of an infinite family of linearizable Virasoro symmetries under specific conditions.
result The Dubrovin-Zhang hierarchy associated with semisimple Frobenius manifolds has a bihamiltonian structure that can be represented by differential polynomials.
The paper studies transformations of Frobenius manifolds and their properties.
problem Analyzing transformations of Frobenius manifolds and their properties.
method Analytic theory of Legendre-type transformations for Frobenius manifolds.
result Monodromy data, Stokes matrix, and central connection matrix are shared among Legendre-type transformations.
Study bihamiltonian structures and Frobenius manifolds for specific Toda hierarchies.
problem Local bihamiltonian structures and Frobenius manifolds for asymmetric rational reductions of 2D-Toda hierarchy.
method Construct three-dimensional generalized Frobenius manifold, relate to other hierarchies via transformations.
result Explicit relation between RR2T and bi-graded Toda and constrained KP hierarchies.
Constructs integrable hierarchies for generalized Frobenius manifolds with non-flat unity.
problem Integrable hierarchies for generalized Frobenius manifolds with non-flat unity.
method Constructs a bihamiltonian integrable hierarchy of hydrodynamic type.
result Integrable hierarchy possesses Virasoro symmetries and a tau structure.
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.
Study on special coordinates for Dubrovin-Frobenius manifolds in low dimensions.
problem Characterizing and understanding Dubrovin-Frobenius manifolds in specific dimensions.
method Introduction of special local coordinates and analysis of invariant metrics.
result Special local coordinates lead to a specific form of the invariant metric.
Classifies 2-solvable Frobenius Lie algebras based on endomorphisms.
problem Classifying 2-solvable Frobenius Lie algebras.
method Semidirect sum decomposition, Jordan form, classification of MASAs.
result Complete classification of 2-solvable Frobenius Lie algebras in low dimensions.