This is a sequel to the paper "The signature package on Witt spaces, I. Index classes" by the same authors. In the first part we investigated, via a parametrix construction, the regularity properties of the signature operator on a stratified Witt pseudomanifold, proving, in particular, that one can define a K-homology …
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48 results for “signature classes”
Study proves equality of signature classes in codimension two.
problem Proving equality of signature classes in codimension two.
method C*-algebraic K-theory and homotopy equivalences.
result Developed a variation of signature classes for noncompact manifolds.
New method calculates signatures of biquotients.
problem Signature calculation for homogeneous spaces.
method Generalization of Hirzebruch's computation to biquotients.
result Signature of biquotients computed for equal rank cases.
Constructs symplectic surface bundles with positive signatures.
problem Symplectic surface bundles over surfaces with positive signatures.
method Constructs symplectic surface bundles with specific genera.
result Determines commutator lengths of new mapping classes.
New signatures for knotted graphs linked to classical knot signatures.
problem Defining invariants for knotted trivalent graphs.
method Using branched covers to define and relate new signatures to classical knot signatures.
result Computable invariants for Kinoshita's knotted theta graph.
This note proves that, as K-theory elements, the symbol classes of the de Rham operator and the signature operator on a closed manifold of even dimension are congruent mod 2. An equivariant generalization is given pertaining to the equivariant Euler characteristic and the multi-signature.
We give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction is inductive. It is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that…
Formula for signature of handlebody bundles, interpreting cohomology.
problem Signature of handlebody bundles and their monodromy.
method Explicit formula using homological monodromy.
result Cobounding function of Meyer's signature cocycle on handlebody group.
Researchers develop Malliavin calculus for signatures, simplifying option Greeks computation.
problem Lack of tractability and explicit representations in Malliavin calculus.
method Focus on finite linear combinations of time-extended Brownian motion signatures, derive explicit formulas for Malliavin derivative, and compute Greeks for path-dependent options.
result Closed-form expressions for classical operators of Malliavin calculus, providing algebraic formulations.
The paper studies signature cocycles on mapping class groups and symplectic groups.
problem Computing the signature of surface bundles over surfaces.
method Geometric and algebraic constructions of signature cocycles, group cohomology analysis.
result Provides a complete answer for N = 2, 4, and 8, showing this is the best possible using this method.
Paper uses algebraic signatures to identify probabilistic structures in empirical data.
problem Identifying probabilistic structure from observed binomials in empirical probability tensors.
method Treating vanishing binomials as algebraic signatures, matching signatures to identify models without parameter estimation.
result The method successfully identified rank-one structures in real language data, revealing interpretable sets of words.
We define generalized Atiyah-Patodi-Singer boundary conditions of product type for Dirac operators associated to C*-vector bundles on the product of a compact manifold with boundary and a closed manifold. We prove a product formula for the K-theoretic index classes, which we use to generalize the product formula for th…
New exotic 4-manifolds with zero signature found.
problem Finding exotic 4-manifolds with specific properties.
method Special small Lefschetz fibrations built via positive factorizations in the mapping class group.
result Smallest known exotic closed simply connected 4-manifolds with signature zero.
New method defines Gysin maps for stratified spaces, preserving signatures.
problem Defining and preserving signatures in stratified spaces.
method Smooth atlas stratified spaces, fiber bundles, Gysin maps, KK-theory.
result Gysin maps preserve analytic signature classes of Witt spaces.
The signature of a surface bundle over a surface is known to be divisible by 4. It is also known that the signature vanishes if the fiber genus is less than or equal to 2 or the base genus is less than or equal to 1. In this article, we construct new smooth 4-manifolds with signature 4 which are surface bundles over su…
In a strengthening of the G-Signature Theorem of Atiyah and Singer, we compute, at least in principle (modulo certain torsion of exponent dividing a power of the order of G), the class in equivariant K-homology of the signature operator on a G-manifold, localized at a prime idea of R(G), in terms of the classes in non-…
In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction, which is inductive over the `depth' of the singularity…
A hyperelliptic broken Lefschetz fibration is a generalization of a hyperelliptic Lefschetz fibration. We construct and compute a local signature of hyperelliptic directed broken Lefschetz fibrations by generalizing Endo's local signature of hyperelliptic Lefschetz fibrations. It is described by his local signature and…
Study explores embedding signature-changing manifolds into higher-dimensional spaces.
problem Smooth metric signature changes in spacetimes.
method Global isometric embeddings into higher-dimensional pseudo-Euclidean spaces.
result Explicit constructions of global embeddings into Minkowski and Misner spaces.
New geometric proof and generalization of Chen signature theorem.
problem Chen signature theorem and its generalizations.
method Topology on loops, Fréchet-Lie group, principal bundle with connection.
result Alternative geometric proof and generalization of Chen signature theorem.
Signature portfolios approximate optimal wealth in non-Markovian markets.
problem Approximating optimal wealth in non-Markovian markets.
method Linear path-functional portfolios based on signatures of market weights.
result Signature portfolios can uniformly approximate any continuous portfolio function.
Surface bundles' signatures tied to their Euler characteristics.
problem Relating signatures and Euler characteristics of surface bundles.
method Using bounded cohomology and Morita's results.
result The inequality for surface bundles.
The Signature of a Manifoldmath.DG
Let us consider a compact oriented riemannian manifold M without boundary and of dimension n=4k. The signature of M is defined as the signature of a given quadratic form Q. Two different products could be used to define Q and they render equivalent definitions: the exterior product of 2k-forms and the cup product of co…
We define families of invariants for elements of the mapping class group of S, a compact orientable surface. Fix any characteristic subgroup H of pi_1(S) and restrict to J(H), any subgroup of mapping classes that induce the identity modulo H. To any unitary representation, r of pi_1(S)/H we associate a higher-order rho…
A new deep learning approach enhances signature transforms for data analysis.
problem Improving data analysis through more flexible signature transforms.
method Combining deep learning with signature transforms to select terms data-dependently.
result Signature transform can be used as a pooling operation within neural networks.
We construct a family of split signature Einstein metrics in four dimensions, corresponding to particular classes of third order ODEs considered modulo fiber preserving transformations of variables.
A new graph signature invariant to graph automorphisms.
problem Graph symmetry and feature generation.
method Power spectrum signature derived from squared graph Fourier transform.
result Power spectrum signature is stable under graph perturbations.
Proves geometric invariance of signature and cohomology for Riemannian foliations.
problem Defining and proving invariance of geometric invariants for Riemannian foliations.
method Analyzes basic signature and Lichnerowicz cohomology under homotopy equivalence.
result Foliated homotopy invariance of basic signature and cohomology.
The study constructs geometrically decomposable aspherical 4-manifolds with non-zero signature and explores their properties.
problem Characterizing geometrically decomposable aspherical 4-manifolds with non-zero signature.
method Constructing examples and proving inequalities for geometrically decomposable aspherical 4-manifolds.
result All geometrically decomposable aspherical 4-manifolds with non-zero signature satisfy the inequality \( \chi \geq 3|σ| \).
We derive a linear estimate of the signature of positive knots, in terms of their genus. As an application, we show that every knot concordance class contains at most finitely many positive knots.
Explicit formulas for Hattori-Stong integrability conditions and manifolds' signature properties.
problem Understanding and calculating the signature of stably almost-complex manifolds.
method Combinatorial techniques to derive explicit formulas for Chern class coefficients and signature conditions.
result An evenness condition for the signature of stably almost-complex manifolds in terms of Chern numbers.
Kernel for Lévy rough paths derived from PDE system.
problem Computing similarity measures for Lévy rough paths.
method Developed a PDE system for the expected signature of inhomogeneous Lévy processes.
result Gaussian martingales' expected signature kernel satisfies a Goursat PDE.
Generalizes global hyperbolicity to higher signatures and proves compactness.
problem Global hyperbolicity in higher pseudo-Riemannian signatures.
method Generalization and proof of compactness of causal diamonds.
result Existence of solutions to a Plateau problem and characterization of holonomies.
Universal approximation for rough paths and Lévy processes.
problem Approximating continuous functionals of càdlàg paths.
method Linear functionals of time-extended signatures.
result Universal approximation theorem for continuous functionals of càdlàg paths.
Novikov conjecture reduced to Lipschitz cohomology of groups.
problem Novikov higher signature conjecture for groups.
method Introducing Lipschitz cohomology classes and reducing the conjecture.
result Reduced Novikov conjecture to Lipschitz cohomology.
The paper extends higher signature invariants to stratified spaces and establishes a new surgery exact sequence.
problem Extending higher signature invariants to stratified spaces.
method Revisiting and extending the surgery exact sequence for stratified spaces.
result Established a natural transformation between the surgery exact sequence and a long exact sequence of K-theory groups for stratified spaces.
Estimates exotic option prices without a model using market data.
problem Pricing exotic derivatives without a model.
method Uses rough path signatures and implied expected signatures from market prices.
result Prices exotic derivatives using market data and implied expected signatures.
The paper develops methods for novelty detection on path space using signature-based statistics.
problem Novelty detection on path space as a hypothesis testing problem.
method Signature-based test statistics, transportation-cost inequalities, CVaR, one-class SVM algorithms.
result Established lower bounds on type- error and general power bounds.
Let p be a finite regular covering on a 2-sphere with at least three branch points. In this paper, we construct a local signature for the class of fibrations whose general fibers are isomorphic to the covering p.
New criteria ensure uniqueness of curve signatures, robust to metric variations.
problem Ensuring uniqueness of curve signatures in differential geometry.
method Introducing new methods through differential equations and higher order derivatives.
result New criteria for curve signature uniqueness in general settings.
Study on averaging geometric structures in Finsler spaces with Lorentzian signature.
problem Averaging geometric structures in Finsler spaces with Lorentzian signature.
method Definition of an average connection without using the timelike vector field.
result No direct relation between the two averaged objects.
The study identifies volatility models from path geometry using signature-based methods.
problem Identifying different stochastic volatility models from observed data.
method Mapping volatility trajectories into a feature space via truncated path signatures and applying a gradient boosting classifier.
result The method achieves high classification accuracy across various volatility dynamics and parameter settings.
We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation of a compact Riemannian manifold with coefficients in a leafwise U(p,q)-flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences …
It is shown that the signature of a manifold with a symplectic circle action having only isolated fixed points, equals the alternating sum of the Novikov numbers corresponding to the cohomology class of the generalized moment map. The same is true for more general fixed point sets.
The paper studies surface bundles and Dehn twists, providing new bounds and factorizations.
problem Understanding stable commutator lengths of Dehn twists and their gaps in mapping class groups.
method Examples of surface bundles, factorizations of Dehn twists, and asymptotic bounds.
result Improved upper bounds for stable commutator lengths and a gap in mapping class groups.
Paper develops a new classifier for time series using topological signatures and Sinkhorn divergences.
problem Classifying time series from chaotic systems with unknown models and noise.
method Topological signatures as weighted KDEs over persistent homology diagrams, predicting labels with Sinkhorn divergences.
result The method accurately discriminates between chaotic system states close in parameter space, robust to noise.
Signature-Based Optimal Execution for Statistical Arbitrage with Path-Dependent Trading Signalsq-fin.TR
Develops a new trading strategy for statistical arbitrage with path-dependent signals.
problem Optimal execution in statistical arbitrage strategies with dynamic predictive signals.
method Signature-based framework modeling alpha and trading speed as linear functionals of truncated signature of market path.
result Fitted policy achieves higher return on turnover compared to a z-score benchmark.
Researchers extend geodesic orbit properties to pseudo-Riemannian nilmanifolds of specific signature.
problem Extending geodesic orbit properties to pseudo-Riemannian manifolds of signature (n-2,2).
method Analyzing trans-Lorentz nilmanifolds with reductive decomposition and nilpotent subgroup.
result Characterization of nilpotent subgroups and structure of nilmanifolds.