Study spherical T-duality and Massey products in iterated sphere bundles.
problem Understanding spherical T-duality and Massey products in iterated sphere bundles.
method Analyzing Gysin sequences and Massey products to find T-dual iterated sphere bundles.
result For certain iterated sphere bundles, spherical T-duality can be represented by Massey products.
New method uses product embeddings to predict bundle success.
problem Designing effective product bundles in large retail settings.
method Leverage historical purchases and clickstream data to generate product embeddings, then use heuristics for complementarity and substitutability.
result Embeddings-based heuristics predict bundle success, robust across categories and retailers.
We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. We find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. We prove the characterization theorem…
Spherical T-duality for iterated sphere bundles
problem T-duality for iterated sphere bundles
method Repackaging cohomological data into Massey products
result Found T-dual iterated sphere bundles associated to Massey products
Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical bundles. We study categorical principal bundles that are product bundles in the categor…
Spinor bundle constructed on loop space for string manifolds.
problem Constructing spinor bundles for string manifolds.
method Using ideas from Stolz and Teichner, a spinor bundle and fusion product are defined for a string manifold.
result Existence of spinor bundle with fusion product on a manifold X if and only if X admits a string structure.
The paper constructs a Poisson algebra bundle for multilocal observables.
problem Representing multilocal observables in classical field theory.
method Working with unordered configuration spaces and using symmetric algebras with respect to two tensor products.
result Obtained a Poisson 2-algebra bundle mimicking Peierls bracket.
We present two formulas for Chern classes of the tensor product of two vector bundles. In the first formula we consider a matrix containing Chern classes of the first bundle and we take a polynomial of this matrix with Chern classes of the second bundle as coefficients. The determinant of this expression equals the Che…
Study of semi-principal bundles using group actions and wreath products.
problem Understanding bundles with fibers as free G-spaces. method Defining semi-principal bundles, bases, and frame bundles; using wreath products and functors.
result Semi-principal bundles can be retracted to principal bundles, preserving parallel transport.
This work merges 3-anchored bundles into 3-Lie algebroids.
problem Combining two 3-anchored bundles into a unified structure.
method Develops algebraic framework for merging bundles with mutual actions and cocycle terms.
result Unified setting for 3-Lie algebroids and special cases.
Constructs a fusion product on spinor bundle over loop space.
problem Describes a fusion product on the spinor bundle over loop space.
method Uses Connes fusion of von Neumann bimodules and string structures.
result Establishes a novel relation between string structures, loop fusion, and Connes fusion of Fock spaces.
We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.
Introduces tensor product for quiver representations and applies to stable bundles and character varieties.
problem Stability and classification of quiver bundles and their subvarieties.
method Definition of tensor product for quiver representations and application to stability and character varieties.
result Tensor products of polystable quiver bundles are polystable and provide insights into character varieties.
We prove explicit formulas for Chern classes of tensor products of vector bundles, with coefficients given by certain universal polynomials in the ranks of the two bundles.
Localized signal representation on graph bundles using Fourier analysis.
problem Representing signals on graph bundles with twists.
method Partition of unity and product factorization over the base graph.
result Lifted bases for signal spaces of graph bundle components.
The geometric product, defined by Graf on the space of differential forms, endows the sections of the exterior bundle by a structure that is necessary to construct a Clifford algebra. The Graf product is introduced and revisited with a suitable underlying framework that naturally encompasses a coframe in the cotangent …
Study of equivariant Poisson 2-algebra bundles over configuration spaces.
problem Understanding Poisson structures on equivariant vector bundles over configuration spaces.
method Construction of induced-equivariance functor, Hadamard and Cauchy tensor products, symmetric 2-monoidal structure, free commutative 2-algebra, compatible Poisson bracket.
result Construction of free commutative 2-algebra and Poisson bracket on equivariant Poisson 2-algebra bundles.
Paper develops techniques for singular metrics on vector bundles.
problem Developing techniques for singular metrics on vector bundles.
method Introducing non-pluripolar products and defining I-good singularities. result Derives a Chern--Weil type formula for Hermitian vector bundles with I-good singularities. New product manifolds can have non-negative curvature.
problem Finding metrics with non-negative curvature on product manifolds.
method Analyzing stable classes of real vector bundles and using properties of homogeneous manifolds.
result Product manifolds can have non-negative sectional curvature under certain conditions.
New rigidity results for warped product domains.
problem Scalar curvature rigidity of domains in warped products.
method Developed a new connection on a twisted spinor bundle and associated Dirac operator.
result Obtained Llarull type scalar curvature rigidity for a general class of domains in a warped product.
Defines a bundle map for currents on manifolds using higher covariant derivatives.
problem Defining a bundle map for currents on manifolds.
method Using higher covariant derivatives on a manifold equipped with a torsion-free connection.
result The bundle of generalized Weyl algebras and its properties.
Classifies smooth manifolds homotopy equivalent to sphere products
problem Classifying smooth manifolds homotopy equivalent to sphere products
method Using normal-invariant map and explicit families of manifolds
result Classifies smooth manifolds up to almost diffeomorphism
Proves elliptic operator images are closed on Hilbert bundles.
problem Closedness of images of elliptic operators on Hilbert bundles.
method Analyzes tensor product of elliptic operators and compares images.
result Establishes closedness of images with respect to natural topology.
In this paper we give some examples of almost para-hyperhermitian structures on the tangent bundle of an almost product manifold, on the product manifold M×R, where M is a manifold endowed with a mixed 3-structure and on the circle bundle over a manifold with a mixed 3-structure.
Currents on Lie groups form a Hopf algebra structure.
problem Understanding algebraic structure of currents on Lie groups.
method Defined Hopf algebra structure on currents using convolution and wedge product.
result Explicit formulas for Hopf algebra operations on currents are derived.
The interior Kasparov product formula is extended for foliated ρ-classes on Riemannian bundles.
problem Extending the Kasparov product formula for foliated ρ-classes.
method Construction of asymptotic morphisms and adiabatic deformation groupoids.
result Interior Kasparov product formula for foliated ρ-classes on Riemannian bundles.
Study examines null vector fields on Lorentzian manifolds.
problem Understanding the structure of null vector fields on Lorentzian manifolds.
method Investigates the bundle structure and ternary product of nowhere vanishing null vector fields.
result Null tangent bundle is a non-polynomial graded bundle with a para-associative ternary product.
Diffeology explores k-forms and bundles with more information than traditional differential forms.
problem Understanding k-forms and bundles in diffeological spaces. method Developed theory of diffeological vector pseudo-bundles, including limits and colimits, and various operations.
result Sections of bundles of k-forms contain more information than differential forms. Paper proposes a method to estimate consumer valuations from bundle sales data.
problem Estimating consumer valuations from bundle sales data using classical methods is challenging.
method Proposes an approach using EM algorithm and Monte Carlo simulation to estimate consumer valuations from bundle sales data.
result The approach can recover the distribution of consumers' valuations and is robust to unobserved no-purchases and clustered market segments.
The paper studies regular contact manifolds and their products.
problem Characterizing regular contact manifolds and their products.
method Elementary proof for compact manifolds, topological tools for general manifolds.
result Regular contact manifolds are principal bundles with S1 or R structure group. Introduces Lie semiheaps and their relation to Lie groups and bundles.
problem Defining and understanding Lie semiheaps and their properties.
method Introducing Lie semiheaps and proving their properties in relation to Lie groups and bundles.
result Established the existence of left-invariant vector fields on Lie semiheaps.
New method constructs degenerate Sasakian manifolds from hyperkähler bundles.
problem Constructing degenerate 3-(α,δ)-Sasakian manifolds. method Using fiber products of Boothby-Wang bundles over hyperkähler manifolds.
result No non-trivial compact examples exist, and one family of nilpotent Lie groups with this geometry is identified.
We determine which three-manifolds are dominated by products. The result is that a closed, oriented, connected three-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of the product of the two-sphere and the circle. This characterization can als…
Study Higgs sections and flat sections for nonlinear harmonic bundles.
problem Equivalence of Higgs and flat sections for nonlinear harmonic bundles.
method Analyze harmonic vector bundles, generalize to sub-fibrations and morphisms.
result Vanishing of a degree obstruction for general nonlinear harmonic bundles.
Study Ricci flow on CP1-bundles over Kähler-Einstein manifolds.
problem Preserving an initial metric on CP1-bundles.
method Ricci flow on CP1-bundles over a product of Kähler-Einstein manifolds.
result The ansatz is preserved along the Ricci flow.
Study on Kuranishi spaces of complex structures and vector bundles, showing isomorphisms and counterexamples.
problem Understanding Kuranishi spaces of complex structures and vector bundles.
method Analyzing Kuranishi spaces of pairs (M,E) of compact Kähler manifolds and vector bundles. result Isomorphisms and counterexamples of Kuranishi spaces of pairs (M,E) of nilmanifolds and trivial vector bundles. Generalized contact bundles are odd dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local splitting theorem similar to those appearing in Poisson geometry. In particular…
The Bergman kernels of holomorphic vector bundles are studied to extend the Fubini-Study map.
problem Extending the Fubini-Study map to a closed range for general inner products.
method Associate Bergman kernels with general inner products on the dual space.
result FS is an injective immersion but not necessarily closed in the space of positive definite inner products.
In this paper, we introduce horizontal and vertical warped product Finsler manifold. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemmanian curvatures of doubly warped product Finsler manifold and its components, and consi…
Study shows hyperbolic subgroups can be free products of surface and free groups.
problem Characterizing hyperbolic subgroups within larger groups.
method Analyzing fiber bundles and using properties of hyperbolic groups.
result Non-elementary hyperbolic commensurated subgroups are virtually free products of surface and free groups.
This masters thesis reviews bundle gerbe theory and the well-known basic bundle gerbe over SU(n). We introduce the cup product bundle gerbe, and show it is stably isomorphic to the pullback of the basic bundle gerbe by the Weyl map. This result enriches our understanding of the basic bundle gerbe, which has numerous ap…
We show that the set of the equivalence classes of multifoliate structures is in one-to-one correspondence with the set of equivalence classes of finite complete projective systems of vector space epimorphisms. After that we give the complete description of all product preserving bundle functors on the categories of mu…
Estimates Lelong numbers of Monge-Ampère products for Kähler manifolds.
problem Estimating Lelong numbers of Monge-Ampère products on Kähler manifolds.
method Analyzes generalized Monge-Ampère products and applies estimates to Chern and Segre currents of pseudoeffective vector bundles.
result Generalizes a recent result about pseudoeffective vector bundles and their nefness.
In this paper we provide an explicit description of normal almost contact structures obtained from Cartan-Ehresmann connections (gauge fields) on principal S1-bundles over complex flag manifolds. The main feature of our approach is to employ elements of representation theory of complex simple Lie algebras in order…
Study of discrete analogues of Atiyah sequence in principal bundles.
problem Discrete analogues of vector bundles and connections in principal bundles.
method Analysis in two categories: fiber bundles with sections and local Lie groupoids, defining discrete curvature and splittings.
result Correspondence between splittings of discrete Atiyah sequence and discrete connections with trivial curvature.
Study CR submanifolds of nearly Kähler S³ × S³ using properties of its almost product structure.
problem Characterize CR submanifolds of nearly Kähler S³ × S³.
method Analyze CR submanifolds in relation to the canonical almost product structure on vector bundles.
result Identify key properties of CR submanifolds related to the almost product structure.
The study finds quasi-Einstein metrics on sphere bundles.
problem Finding quasi-Einstein metrics on specific types of manifolds.
method Adapting Hall's work, the study explores quasi-Einstein metrics on sphere bundles over Fano Kaehler-Einstein manifolds and their blow-downs.
result The discovery of quasi-Einstein metrics on sphere bundles.
We introduce the notion of Hermitian Higgs bundle as a natural generalization of the notion of Hermitian vector bundle and we study some vanishing theorems concerning Hermitian Higgs bundles when the base manifold is a compact complex manifold. We show that a first vanishing result, proved for these objects when the ba…