New framework generalizes neural network parameters to C∗-algebra for more efficient feature learning.
problem Efficient feature learning and adaptability of neural network models.
method Generalizes neural network parameters to C∗-algebra-valued parameters and combines models continuously. result Shows improved feature learning with limited data using the new framework.
Unified geometric framework for quantum states using dual number algebras.
problem Representing quantum states in a geometrically unified way.
method Smooth embeddings into higher-order dual number algebras and algebraic flows.
result Established nilpotent dual algebras as a geometric landscape for quantum kinematics.
We define new differential graded algebras A(n,k,S) in the framework of Lipshitz-Ozsváth-Thurston's and Zarev's strands algebras from bordered Floer homology. The algebras A(n,k,S) are meant to be strands models for Ozsváth-Szabó's algebras B(n,k,S); indeed, we exhibit a quasi-isomorphism from B(n,k,S) to A(n,k,S). We …
This dissertation explores Clifford bundles and spinor fields in geometric and algebraic contexts.
problem Understanding spinor fields and their classification in geometric frameworks.
method Combines algebraic and geometric approaches to study Clifford structures on bundles and spinor fields.
result Identifies new spinor field classes in warped flux compactifications.
A new algebraic framework models LOBs with physics and stochastic processes.
problem Capturing the dynamics of limit order books (LOBs).
method Algebraic framework using Dirac notation and generating functions.
result Exact simulations of market scenarios using the Gillespie algorithm.
Similarity algebra extends algebraic structures with quantitative bounds.
problem Exact algebraic structures with strict axioms.
method Framework for approximate algebraic and Lie structures with ε-estimates. result Similarity structures converge to classical algebraic objects as εightarrow0. 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.
Study uses geometric algebra to analyze credit cycles, revealing dangerous feedback loops.
problem Understanding and predicting dangerous feedback loops in credit cycles.
method Represent economic states as multi-vectors in Clifford algebra, focusing on bivector elements for rotational coupling.
result Geometric relationship between unemployment and credit contraction shifts from simple correlation to dangerous rotational dynamics during crises.
In this thesis, we study deformations of compact holomorphic Poisson manifolds and algebraic Poisson schemes in the framework of Kodaira-Spencer's analytic deformation theory and Grothendieck's algebraic deformation theory.
Study Poisson algebras for Hamiltonian systems linearization.
problem Linearize dynamics along Poisson submanifolds.
method Use contravariant derivative to characterize Poisson algebras.
result Infinitesimal Poisson algebras provide a framework for Hamiltonization.
This paper is a presentation, where we compute the HOMFLYPT Skein module of singular links in the 3-sphere. This calculation is based on some results previously proved by Rabenda and the author on Markov traces on singular Hecke algebras, as well as on classical techniques that allow to pass from the framework of Marko…
New framework shows C∗-simplicity for groups without certain subalgebras.
problem Characterizing C∗-simplicity of groups. method Introducing confined subalgebras and Uniformly Recurrent States.
result A countable discrete group is C∗-simple if it has no non-trivial amenable confined subalgebras. It was shown by Fomin, Shapiro and Thurston that some cluster algebras arise from orientable surfaces. Subsequently, Dupont and Palesi extended this construction to non-orientable surfaces. We link this framework to Lam and Pylyavskyy's Laurent phenomenon algebras, showing that both orientable and non-orientable unpunc…
Geometric AD framework simplifies derivative computation in JAX.
problem Efficient and accurate automatic differentiation.
method Jet functors and Weil algebras for geometric analysis.
result Unified view of derivative propagation with algebraic exactness.
Introduces Lie-Yamaguti algebra bundles and their cohomology.
problem Defining and studying Lie-Yamaguti algebra bundles.
method Defined Lie-Yamaguti algebra bundles and their cohomology groups.
result Lie-Yamaguti algebra bundles naturally arise from geometric considerations.
Study bubbling Kahler metrics using algebraic geometry.
problem Analyzing the degeneration of Kahler metrics with Euclidean volume growth.
method Algebraic construction of birational modifications to simplify degenerations, comparing with analytic constructions.
result Provide a framework to compare algebraic and analytic approaches to bubbling phenomena.
New algebraic structures called cyclic Lie-Rinehart algebras are defined.
problem Defining new algebraic structures.
method Construct Lie-Rinehart algebras with a specific cyclic submodule.
result Found a connection to differential operators in physics.
Defines formal vertex laws related to Lie conformal algebras.
problem No specific problem stated; focuses on definitions and proofs.
method Definitions and proofs of vertex/conformal versions of classical Lie theory results.
result Proves vertex/conformal versions of important Lie theory results.
We solve higher-order morphisms for twisted Courant algebras.
problem Construct canonical L∞-morphisms for higher Courant algebroids. method Develop a general framework for arbitrary r. result Affirmative answer to Zambon's question for higher degrees.
Develops method to construct Lie algebra weight system kernel using Vogel algebra.
problem Detecting correlators and distinguishing knots in 3D Chern-Simons theory.
method Uses Vogel's Λ algebra and Jacobi diagrams.
result Explicitly provides Jacobi diagrams in the kernel of sl_N weight system.
In this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle. It has been proven previously that this noncommutative algebra generalizes in a natural way the ordinary geometry of connections. We use explicitely some geometric constructions us…
New algebra defined for Legendrian submanifolds, preserving key invariants.
problem Defining a new algebra to preserve invariants of Legendrian submanifolds.
method Combining string topology techniques with combinatorial methods to count holomorphic disks.
result The new algebra PDA is a filtered, differential graded algebra that captures invariants of Legendrian submanifolds. Defines and extends flat pseudo-Riemannian F-Lie algebras.
problem Generating weakly flat Lorentzian non-abelian bi-nilpotent F-Lie algebras.
method Constructs double extensions of flat pseudo-Riemannian F-Lie algebras.
result Provides a framework for generating all weakly flat Lorentzian non-abelian bi-nilpotent F-Lie algebras.
It was shown by Fock, Goncharov and Fomin, Shapiro, Thurston that some cluster algebras arise from triangulated orientable suraces. Subsequently Dupont and Palesi generalised this construction to include unpunctured non-orientable surfaces, giving birth to quasi-cluster algebras. Previously we linked this framework to …
The Temperley-Lieb algebra is a fundamental component of SU(2) topological quantum field theories. We construct chain complexes corresponding to minimal idempotents in the Temperley-Lieb algebra. Our results apply to the framework which determines Khovanov homology. Consequences of our work include semi-orthogonal deco…
We compute the equivariant cohomology Chern character of the index of elliptic operators along the leaves of the foliation of a flat bundle. The proof is based on the study of certain algebras of pseudodifferential operators and uses techniques for analizing noncommutative algebras similar to those developed in Algebra…
A novel geometric algebra-based KG embedding framework improves link prediction.
problem KG embedding to model entities and relations in a low-dimensional space.
method Utilizes multivector representations and geometric product in geometric algebra.
result Outperforms state-of-the-art models in link prediction experiments.
This paper defines the concept of an oriented quantum algebra and develops its application to the construction of quantum link invariants. We show that all known quantum link invariants can be put into this framework.
Develops a SageMath framework for computing characteristic classes.
problem Computing characteristic classes in computer algebra systems.
method Chern-Weil approach, symbolic calculus, vector bundles, connections, mixed differential forms.
result Implementation and computation of characteristic classes in SageMath.
New calculus framework for vector bundles with metrics.
problem Developing calculus for vector bundles with fiber metrics.
method Adapting differential calculus to graded commutative algebras and focusing on diole and triole algebras.
result Triole algebra provides a suitable environment for vector bundle calculus with fiber metrics.
Abstract: Geometrically reformulates estimation theory for finite-dimensional C*-algebras.
problem Estimation theory for finite-dimensional C*-algebras.
method Geometrical formulation of estimation theory.
result Derivation of Cramer-Rao and Helstrom bounds.
The ordinary (or classical) Birman-Wenzl-Murakami algebras were initially conceived as an algebraic framework for the Kauffman link invariant. They also appear as centralizer algebras for representations of quantum universal enveloping algebras of orthogonal or symplectic types. It was shown by Morton and Wassermann th…
The paper quantizes Hessian structures on R^2 using KV-algebras.
problem Quantizing Hessian structures on a 2D space.
method Deformation quantization within Koszul-Vinberg algebras.
result Established links between deformation theory and Hessian geometry.
We propose a tensor neural network (t-NN) framework that offers an exciting new paradigm for designing neural networks with multidimensional (tensor) data. Our network architecture is based on the t-product (Kilmer and Martin, 2011), an algebraic formulation to multiply tensors via circulant convolution. In this $t…
New C∗-algebra approach unifies machine learning strategies.
problem Lack of diverse and information-rich data models in machine learning.
method Integrates C∗-algebra into machine learning frameworks. result Unified learning strategies and new data models.
The paper explores geometric and algebraic structures on Lie groups.
problem Investigating F-manifolds and Fextman-algebras on Lie groups. method Constructing a canonical connection and analyzing curvature and holonomy.
result Established the integrability of a Poisson-algebra distribution.
Revives Vogel's diagrammatic technique for universal Lie algebra computations.
problem The universality of Lie algebra quantities remains open, despite many being described.
method Diagrammatic algebra based on Vogel's Λ-algebra.
result Diagrammatic technique enables truly universal computations in Lie theory.
This research introduces Lie brackets on spaces of biderivations in Lie algebras.
problem Understanding higher-order infinitesimal symmetries in Lie algebras.
method Study of right biderivations and Lie brackets on their spaces.
result New Lie algebra framework for biderivations with applications in deformation theory.
Groupoids provide a more appropriate framework for differential geometry than principal bundles. Synthetic differential geometry is the avant-garde branch of differential geometry, in which nilpotent infinitesimals are available in abundance. The principal objective in this paper is to show within our favorite framewor…
Extends differential calculus to triole algebras.
problem No specific problem stated; focuses on extending differential calculus.
method Generalizes diolic differential calculus to triole algebras with fiber metrics.
result Established a conceptual framework for calculus on bundles with vector-valued fiber metrics.
Study embeds PC matrices into Grassmannian manifold for geometric interpretation.
problem Understanding algebraic consistency of pairwise comparisons matrices.
method Leverages Plücker coordinates and geometric interpretation of Grassmannian manifold.
result Algebraic consistency condition is equivalent to geometric consistency in G(2,n). Develops a new method to study algebraic tangent cones of sheaves using valuations.
problem Analyzing tangent cones of torsion-free sheaves on algebraic varieties.
method Introduces a slope stability theory and uses it to define a canonical tangent cone for quasi-regular valuations.
result Shows the existence of a canonical tangent cone for torsion-free sheaves, up to equivalence.
The paper extends Chern-Weil-Lecomte map to L∞-algebras.
problem Defining characteristic classes for L∞-algebra extensions. method Using the Chern-Weil-Lecomte map to define characteristic classes in an L∞-algebra setting. result Unified definition of several known cohomology classes.
Research decouples Lie algebroids using bicocycle double cross product theory.
problem Understanding decoupling and coupling phenomena in Lie algebroids.
method Bicocycle double cross product realization method.
result Unified product, double cross product, semi-direct product, and cocycle extension frameworks are instances of the general method.
Researchers prove formulas for flag area measures, extending previous work.
problem Proving additive kinematic formulas for flag area measures.
method Introducing an algebraic framework to compute these formulas explicitly.
result Existence and explicit computation of additive kinematic formulas for flag area measures.
We develop an algebraic framework for the description and analysis of financial behaviours, that is, behaviours that consist of transferring certain amounts of money at planned times. To a large extent, analysis of financial products amounts to analysis of such behaviours. We formalize the cumulative interest compliant…
Unified framework for complex, split-complex, and dual numbers.
problem Analytic and geometric scope of real-analytic functions.
method Generalized Cauchy-Riemann structure and unified real algebra family.
result Milnor-Le type fibration theorem for nondegenerate algebras.
New framework for data-driven hyperparameter tuning with structured loss.
problem Statistical foundations for multi-dimensional hyperparameter tuning remain limited.
method General framework using real algebraic geometry for semi-algebraic function classes.
result First general guarantees for multi-dimensional hyperparameter tuning.