Study Frobenius determinants and their connections to Bessel functions.
problem Understanding Frobenius determinants and their geometric and differential equations.
method Investigation of Frobenius determinants and their relation to twistor theory and Bessel functions.
result Extensions and new insights into Bessel functions.
Study geometry and PDEs from group-determinants and representation theory.
problem Geometry and PDEs from group-determinants and representation theory.
method Analysis of group-determinants and representation theory.
result Spectral theory of operators linked to finite Fourier transform theory.
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.
The paper determines shape operators for a special class of hypersurfaces in complex projective space.
problem Identifying shape operators for Stark hypersurfaces.
method Completely determined shape operators for Stark hypersurfaces in CPn. result Stark hypersurfaces in CP2 constructed as integrals of a Frobenius exterior differential system. For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skei…
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.
The study explores autonomous systems and their connections to contact geometry and Frobenius manifolds.
problem Understanding the connections between autonomous systems and geometric structures.
method Investigation of the Darboux-Halphen-Ramanujan system, contact geometry, and Frobenius manifolds.
result Highlighting the role of contact geometry in autonomous systems.
We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair (p,q), 2<p<q, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of compact Riemann surfaces which can be realized as a semi-regular pq-fold covering…
The paper calculates the Saito determinant for Coxeter discriminant strata.
problem Calculating the Saito determinant for specific geometric strata.
method Using the Saito flat metric and Lie derivatives, the paper finds the determinant of the metric restricted to Coxeter discriminant strata.
result The determinant of the Saito metric on Coxeter discriminant strata is proportional to a product of linear factors in flat coordinates.
Motivated by the Moore-Segal axioms for an open-closed topological field theory, we consider planar open string topological field theories. We rigorously define a category 2Thick whose objects and morphisms can be thought of as open strings and diffeomorphism classes of planar open string worldsheets. Just as the categ…
New potential functions reveal Frobenius-like structure in weighted hyperplane arrangements.
problem Understanding the Frobenius algebra of functions on critical sets of weighted hyperplane arrangements.
method Constructing two potential functions and proving their matrix coefficients are derived from derivatives of these functions.
result The potential functions completely determine the Frobenius algebra and are local in the sense of contributions from elementary subarrangements.
A regular F-manifold is an F-manifold (with Euler field) (M, \circ, e, E), such that the endomorphism {\mathcal U}(X) := E \circ X of TM is regular at any p\in M. We prove that the germ ((M,p), \circ, e, E) is uniquely determined (up to isomorphism) by the conjugacy class of {\mathcal U}_{p} : T_{p}M \rightarrow T_{p}M…
New method to determine if a spray can be Finslerian.
problem Determining if a spray can be Finslerian in 2D.
method Associate a Berwald frame and distribution, then check Frobenius integrability.
result Characterizes Finsler metrizability of 2D sprays.
We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat…
We explicitly compute the intrinsic volume of the set of real (and real symmetric) matrices of Frobenius norm one and given corank (the case of matrices with zero determinant as a special case). We give asymptotic formulas for our computations and we discuss several examples and applications.
This article announces a series of articles aiming at introducing the concept of symplectic spinors into symplectic topology resp. the concept of Frobenius structures. We will give lower bounds for the number of fixed points of a Hamiltonian diffeomorphism on the cotangent bundle over a compact manifold M by defining…
We apply representation theory to study the homology of equivariant Dehn-fillings of a given finite, regular cover of a compact 3-manifold with boundary a torus. This yields a polynomial which gives the rank of the part of the homology carried by the solid tori used for Dehn-filling. The polynomial is a symmetrized for…
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.
Paper defines nearly Frobenius algebras and explores their properties.
problem Defining and studying algebras without traces.
method Survey of foundational results and applications.
result Characterization of nearly Frobenius algebras and their properties.
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 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…
Lie algebras with G-action are studied.
problem Lie algebra structures on symplectic Lie groups.
method Definition of g-quasi-Frobenius Lie algebras. result Induced D(g)-action on g-quasi-Frobenius Lie algebras. Study on Riemannian properties of SU_n using bi-invariant metric.
problem Properties of SU_n with a specific bi-invariant metric.
method Analyzes distance, diameter, and geodesics in SU_n.
result Parametrizes minimizing geodesic segments using a complex Grassmannian.
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.
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. Khovanov homology from diagonalisable Frobenius algebras is shown to be degenerate.
problem The degeneracy of Khovanov homology constructed from certain Frobenius algebras.
method Provided a short elementary proof.
result Khovanov-type link homology constructed from diagonalisable Frobenius algebras is degenerate.
Study infinite-dimensional Toda manifold at irregular singularity, revealing non-uniqueness of formal solutions.
problem Non-uniqueness of formal solutions to the Dubrovin equation at irregular singularity.
method Revisited canonical coordinates, formal solutions analysis, Borel resummation, Stokes matrices computation.
result Infinite-dimensional Stokes matrices computed from resummed formal solutions.
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.
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.
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 K-curvature. Lower bound found for Perron-Frobenius degrees of certain complex numbers.
problem Finding lower bounds for the Perron-Frobenius degree of complex numbers.
method Using Doug Lind's idea, proving results for both cubic and biPerron numbers.
result Arbitrary large Perron-Frobenius degrees for certain complex numbers.
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.
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.
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.
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.
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.
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 …
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.
Frobenius theorem proven for a generalized supergeometry.
problem Integrable distributions on Z2n-supermanifolds. method Proof of the Frobenius theorem for Z2n-supermanifolds. result Local and global versions of the theorem are established.
New groups and manifolds from Weyl groups, with Frobenius structures.
problem Understanding new extended affine Weyl groups and their properties.
method Developed new Weyl groups and constructed Frobenius manifold structures.
result Existence of Frobenius manifold structures on orbit spaces of new Weyl groups.