A bicategory approach to differential cohomology is presented. Based on the axioms of Bunke-Schick, a symmetric monoidal groupoid is associated to differential refinements of cohomology theories. It is proven that such differential refinements are unique up to equivalence of the corresponding symmetric monoidal groupoi…
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48 results for “differential refinements”
Develops differential KO-theory with constructions and applications.
problem Constructing differential refinements in KO-theory.
method Explicit constructions and refinements of spectral sequences.
result Identification of differentials in differential KO-theory.
Proposes a new Hodge conjecture in Bott-Chern cohomology.
problem Hodge conjecture in Bott-Chern cohomology.
method Characterization of real holomorphic chains, atomic section theory, refined Bott-Chern classes.
result Proof of a new Hodge conjecture in Bott-Chern cohomology.
Refined algebraic domains with finite sets in the boundaries respecting differential geometrymath.AG
Research on refined algebraic domains respecting differential geometry.
problem Understanding shapes and regions of real algebraic curves.
method Investigates points in two curves, singular points, inflection points, and points of double tangent lines, considering differential geometry.
result Proves fundamental properties and investigates examples of refined algebraic domains.
Study refines Siegel-Veech constants for abelian differentials.
problem Computing Siegel-Veech constants for abelian differentials.
method Intersection theory and quasimodular forms.
result New identity for Siegel-Veech constants of cylinders.
We prove a refined Kato inequality for closed and coclosed differential forms on a Kahler manifold.
Improves spline quality and accuracy in computational microscopy.
problem Detecting slender, overlapping structures in microscopy images.
method Differentiable rendering approach for spline refinement.
result Achieves high reliability and sub-pixel accuracy.
Odd -theory has the interesting property that it admits an infinite number of inequivalent differential refinements. In this paper we provide a bundle theoretic model for odd differential -theory using the caloron correspondence and prove that this refinement is unique up to a unique natural isomorphism. We chara…
Cech cocycles for differential characteristic classes -- An infinity-Lie theoretic constructionmath.AT
What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and …
We establish refinements of the classical Kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first-order linear differential operator. Our main result is a general expression which gives the value of the constants appearing in the refined inequalities. These consta…
New complexes refine multicomplexes for subRiemannian geometry.
problem Analyzing subRiemannian geometry on Carnot groups.
method Spectral complexes from truncated multicomplexes.
result Retains cohomology of multicomplexes and refines Rumin complex.
Refines neural network predictions using background knowledge for improved accuracy.
problem Compensate for lack of labeled data in neural networks.
method Introduces differentiable refinement functions and Iterative Local Refinement (ILR) algorithm to refine predictions efficiently and accurately.
result ILR finds competitive results in MNIST addition task and refines predictions on complex SAT formulas.
The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de Rham cocycles -- known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: The definition of equivariant characteristic fo…
Refines deep generative models to improve data density precision.
problem Achieving precise representation of data probability density in deep models.
method Iterated generative modeling to refine latent space, addressing topological obstructions.
result Latent Space Refinement (LaSeR) protocol improves generative model precision.
Refines Haupt's theorem for surface characters.
problem Characterization of holomorphic 1-forms on surfaces.
method Refines Haupt's theorem with divisor data.
result Necessary and sufficient conditions for surface characters.
Smooth models refine topological K-theory for differential extensions.
problem Constructing a smooth version of differential K-theory.
method Using norm completions of stable Grassmannian and stable unitary group models.
result Isomorphic to unique differential extension of K-theory.
Let h be a rationally even cohomology theory and h^ the natural differential refinement, as defined by Hopkins and Singer. We consider the possible definitions of the relative differential cohomology groups, generalizing the analogous picture for the Deligne cohomology, and we show the corresponding long exact sequence…
DP-SEP privatizes EP by refining a single factor per data point.
problem Private inference of complex models with limited memory.
method Stochastic Expectation Propagation with differential privacy.
result DP-SEP provides better posterior estimates with guaranteed privacy.
Recalls and refines the concept of algebraically rectifiable curves.
problem Classical notion of algebraically rectifiable plane curves.
method Provides new criteria, relates to quadratic differentials, and generalizes to higher order differentials.
result Generalization and new criteria for algebraic rectifiability.
Constructs a model for differential KO-theory using Clifford modules.
problem Refining Atiyah and Singer's families index with differential structure.
method Builds a model using families of Clifford modules with superconnection.
result Affords a differential refinement of Atiyah and Singer's families index.
We characterize primary operations in differential cohomology via stacks, and illustrate by differentially refining Steenrod squares and Steenrod powers explicitly. This requires a delicate interplay between integral, rational, and mod p cohomology, as well as cohomology with U(1) coefficients and differential forms. A…
Develops differential K-theory for noncommutative algebras.
problem Creating a differential extension of algebraic K-theory for noncommutative algebras.
method Introduces secondary transgression forms and a differential refinement of the smooth Serre--Swan correspondence.
result Subsumes differential K-theory for smooth manifolds and fits into a noncommutative differential cohomology hexagon diagram.
In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structure…
A GPU-based workflow for building physics emulators of hypersonic flows
problem Resolving complex physical phenomena in hypersonic flows
method Fully GPU-based workflow integrating accelerated data generation and neural emulators
result Physics emulators remain reliable beyond their training distribution
Let be a flat complex vector bundle over a closed oriented odd dimensional manifold endowed with a flat connection . The refined analytic torsion for was defined and studied by Braverman and Kappeler. Recently Mathai and Wu defined and studied the analytic torsion for the twisted de Rham complex…
We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we call "cohesive". Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive struct…
Differential K-theory gets a -ring structure.
problem Establishing a -ring structure in differential K-theory.
method Splitting principle for differential K-theory, Adams operations construction.
result Differential -ring admits a -ring structure.
New method recovers differential cohomology from diffeological spaces.
problem Recovering differential cohomology from diffeological spaces.
method Introducing skeletal diffeologies and thin homotopies to recover differential cohomology.
result Ordinary differential cohomology can be recovered in terms of the homotopy theory of skeletal diffeological spaces.
In this paper we introduce the Cheeger-Simons cohomology of a global quotient orbifold. We prove that the Cheeger-Simons cohomology of the orbifold is isomorphic to its Beilinson-Deligne cohomology. Furthermore we construct a string connection (à la Segal) from a global gerbe with connection over the loop orbifold, ref…
Refined scattering diagrams and theta functions from asymptotic analysis of Maurer-Cartan equationsmath.AG
New geometric proofs and interpretations of scattering diagrams and theta functions.
problem Analyzing the asymptotic behavior of Maurer-Cartan elements for differential graded Lie algebras.
method Asymptotic analytic approach and differential geometric proofs.
result Alternative proofs of consistent completion of scattering diagrams and geometric interpretations of theta functions.
A new framework speeds up Bayesian inference in probabilistic programs.
problem Efficient Bayesian inference in complex probabilistic programs.
method Embeds a sampler within a refined variational posterior approximation, using automatic differentiation for parameter tuning.
result Significantly speeds up mixing time and improves efficiency in various probabilistic program tasks.
A new subdivision scheme for Heisenberg group values with central smoothness loss.
problem Regularity of limit curves in Heisenberg group-valued subdivision schemes.
method Interpolatory subdivision scheme with central correction based on group law.
result Central part of limit curve converges to a continuous limit with logarithmic modulus of continuity.
We consider spectral sequences in smooth generalized cohomology theories, including differential generalized cohomology theories. The main differential spectral sequences will be of the Atiyah-Hirzebruch (AHSS) type, where we provide a filtration by the Cech resolution of smooth manifolds. This allows for systematic st…
We update and refine the work of T. Hamada concerning *-Einstein hypersurfaces in complex space forms CP^n and CH^n. We also address existence questions using the methods of moving frames and exterior differential systems.
Probabilistic modeling is iterative. A scientist posits a simple model, fits it to her data, refines it according to her analysis, and repeats. However, fitting complex models to large data is a bottleneck in this process. Deriving algorithms for new models can be both mathematically and computationally challenging, wh…
Proves Kato inequalities for various conformal operators.
problem Proving inequalities for differential operators.
method Analyzes a class of first order differential operators, including Dirac and Penrose twistor operators.
result Derives Kato inequalities that interpolate between classical and refined versions.
Loop Differential K-theorymath.AT
In this paper we introduce an equivariant extension of the Chern-Simons form, associated to a path of connections on a bundle over a manifold M, to the free loop space LM, and show it determines an equivalence relation on the set of connections on a bundle. We use this to define a ring, loop differential K-theory of M,…
Curvature defined for Hilbert modules and Kasparov modules.
problem Defining and studying curvature in Hilbert modules and Kasparov modules.
method Introduced curvature for densely defined universal connections on Hilbert -modules relative to spectral triples.
result Curvature only depends on the represented form of the universal connection modulo junk forms.
Differentially private fair binary classification algorithm developed.
problem Balancing privacy and fairness in binary classification.
method Decoupling technique for fairness, refinement for differential privacy.
result Algorithm maintains fairness, privacy, and utility guarantees.
Locally-verifiable conditions ensure exactness of spline discrete de Rham complex.
problem Ensuring cohomological equivalence of spline discrete complex to continuous de Rham complex.
method Theoretical analysis and locally-verifiable sufficient conditions for exactness.
result Locally-verifiable conditions guarantee exactness of hierarchical B-spline discrete de Rham complex.
We study the local differential geometry of varieties with degenerate secant and tangential varieties. We show that the second fundamental form of a smooth variety with degenerate tangential variety is subject to certain rank restrictions. The rank restrictions imply a slightly refined v…
GENESIS-V2 infers unordered object representations without iterative refinement.
problem Unsupervised learning of unordered object representations for complex images.
method Stochastic stick-breaking process for clustering pixel embeddings.
result GENESIS-V2 outperforms recent baselines in unsupervised image segmentation and scene generation.
The paper introduces a multilevel initialization method for deep neural networks.
problem Training very deep neural networks with layer-parallel methods.
method Continuous interpretation of training as optimal control, using time-dependent ODEs for neural network discretization, and a refinement strategy across the time domain.
result The method creates deep networks with good initializations from coarser networks, reducing training time and providing regularization.
DeepXDE uses neural networks to solve complex differential equations.
problem Solving differential equations using machine learning.
method Physics-informed neural networks (PINNs) with adaptive refinement.
result PINNs can solve various types of PDEs and inverse problems.
We apply a suitable modification of the functional delta method to statistical functionals that arise from law-invariant coherent risk measures. To this end we establish differentiability of the statistical functional in a relaxed Hadamard sense, namely with respect to a suitably chosen norm and in the directions of a …
Upper bounds on nullhomotopy volumes in nilpotent spaces are refined.
problem Bounding volumes of nullhomotopies in nilpotent spaces.
method Extension of the Shadowing Principle to nilpotent spaces.
result Improved bounds on nullhomotopy volumes, nearly meeting simply connected settings.
Twisted differential generalized cohomology theories and their Atiyah-Hirzebruch spectral sequencemath.AT
We construct the Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential generalized cohomology theories. This generalizes to the twisted setting the authors' corresponding earlier construction for differential cohomology theories, as well as to the differential setting the AHSS for twisted generalized coho…
Let h^{*} be a multiplicative cohomology theory, h_{*} its dual homology theory and \hat{h}^{*} a differential refinement. We first construct the natural pairing between h_{*} and the flat part of \hat{h}^{*}, generalizing the holonomy of a flat Deligne cohomology class. Then, in order to generalize the holonomy of any…