Study examines metrics and functionals for Hodge-Dirac operator on manifolds.
problem Examining metrics and functionals for Hodge-Dirac operator on manifolds.
method Analyzing the metric and Einstein bilinear functionals of differential forms for Hodge-Dirac operator d+δ. result The functionals reproduce those for the canonical Dirac operator on a spin manifold up to a numerical factor.
This paper extends Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.
problem Generalizing theorems to manifolds with boundary.
method Extending results of Dabrowski etc. to 4D oriented Riemannian manifolds with boundary.
result Proof of Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary.
Defines metrics and Einstein tensors on Riemannian manifolds, proving vanishing for non-commutative two-torus.
problem Defining metrics and Einstein tensors on Riemannian manifolds.
method Defines bilinear functionals of vector fields and differential forms, generalizing to non-commutative geometry.
result Proves the vanishing of the Einstein functional for the conformally rescaled geometry of the noncommutative two-torus.
Defines spectral Einstein functional for manifolds with boundary.
problem Calculating the spectral Einstein functional for manifolds with boundaries.
method Defined spectral Einstein functional associated with the Dirac operator and proved a theorem for 4D manifolds.
result Proof of Kastler-Kalau-Walze type theorem for spectral Einstein functional.
Proves Kastler-Kalau-Walze theorem for spectral Einstein functional on low-dimensional manifolds.
problem Proving Kastler-Kalau-Walze type theorems for spectral Einstein functional.
method Defining spectral Einstein functional associated with Dirac operator and proving theorem for low-dimensional manifolds.
result Proves Kastler-Kalau-Walze type theorem for spectral Einstein functional on low-dimensional manifolds with boundary.
Defines spectral Einstein functionals for sub-Dirac operators on manifolds with boundary.
problem Calculating Einstein-like functionals for sub-Dirac operators.
method Introduced spectral Einstein functional for sub-Dirac operators on manifolds with boundary.
result Proved a theorem for spectral Einstein functions on four-dimensional manifolds.
Paper defines spectral triple and computes functional for nonminimal de Rham-Hodge operator.
problem Computing spectral functions for nonminimal de Rham-Hodge operators.
method Definitions and computations of spectral triple and functional.
result Computed spectral Einstein functional for even-dimensional compact manifolds.
Computes spectral Einstein functional for Witten deformation on even-dimensional spin manifolds.
problem Calculating the spectral Einstein functional for a specific deformation.
method Computes the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
result Computed the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
The paper introduces spectral Einstein functionals for Dirac operators on manifolds with boundary.
problem Analyzing perturbations of Dirac operators on manifolds with boundary.
method Introduction of spectral Einstein functionals and proof of Dabrowski-Sitarz-Zalecki type theorems.
result Proof of theorems associated with spectral Einstein functionals for perturbations of Dirac operators.
New spectral functionals for Dirac operators with inner fluctuations computed.
problem Spectral functionals and Dirac operators with inner fluctuations.
method Extension of spectral functionals for Dirac operators with inner fluctuations.
result Computed spectral Einstein functional for Dirac operator with inner fluctuations on even-dimensional spin manifolds.
Extends spectral Einstein functionals computation to 4D spin manifolds with boundary.
problem Computing spectral Einstein functionals for 4D spin manifolds with boundary.
method Generalizes Dabrowski's results to 4D spin manifolds with boundary using noncommutative residue.
result Generalized spectral Einstein functionals computation for 4D spin manifolds with boundary.
The paper calculates a functional for a specific Dirac operator.
problem Computing a spectral Einstein functional for a Dirac operator with torsion.
method Computing the spectral Einstein functional for even-dimensional spin manifolds without boundary.
result The spectral Einstein functional for the Dirac operator with torsion is computed.
The paper computes metrics and Einstein tensors on even-dimensional manifolds.
problem Computing metrics and Einstein tensors on even-dimensional Riemannian manifolds.
method Development of spectral Einstein functionals and equivariant Bismut Laplacian.
result Explicit computation of the equivariant noncommutative residue density.
The paper defines a new functional and proves related theorems for manifolds with boundary.
problem Defining and proving theorems for manifolds with boundary.
method Defining the spectral Einstein functional and relating it to the noncommutative residue.
result Proof of Dabrowski-Sitarz-Zalecki type theorems for spectral Einstein functional on 4D manifolds with boundary.
New method deforms function algebras on manifolds using spectral decomposition.
problem Deforming function algebras on compact Riemannian manifolds.
method Introducing a bilinear product on the finite spectral core of smooth functions using unimodular phases.
result The product extends to a Sobolev algebra and admits iteration under certain conditions.
Proves new theorems for manifolds with boundary.
problem No boundary conditions for manifolds.
method Generalizes Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.
result Proof of new theorems for even and odd dimensional manifolds with boundary.
New spectral functionals related to noncommutative residue and torsion Dirac operators.
problem Spectral functionals and Dirac operators with torsion.
method Noncommutative residue and Dirac operators with torsion.
result Extension of spectral functionals to noncommutative realm with torsion.
Study tackles distribution shift in combinatorial settings using matrix completion techniques.
problem Tackling distribution shift in combinatorial settings with rigorous statistical guarantees.
method Develops novel algorithms and theoretical results for extrapolating to test distributions not covered in training.
result Achieves bilinear combinatorial extrapolation under gradual spectral decay in high-dimensional data.
New mathematical proposal for TQFTs using TMF-modules.
problem Constructing new types of TQFTs at the intersection of topology, algebra, physics, and homotopy theory.
method Defines TMF-modules associated with symmetric bilinear forms and assigns them to closed 3-manifolds and maps of TMF-modules to 4-dimensional cobordisms.
result Invariants of 4-manifolds arising from 6-dimensional superconformal field theories, conjecturally generalizing the theta function of a lattice.
Defines and computes a generalized spectral action for Lorentz warped products.
problem Computing spectral actions for Lorentz warped products.
method Defines and computes the bimetric spectral Einstein-Hilbert action for Lorentz warped products.
result Derives a Kastler-Kalau-Walze type theorem for Lorentz warped products.
The theory of harmonic symmetric bilinear forms on a Riemannian manifold is an analogue of the theory of harmonic exterior differential forms on this manifold. To show this, we must consider every symmetric bilinear form on a Riemannian manifold as a one-form with values in the cotangent bundle of this manifold. In thi…
Extends Einstein-Hilbert action to higher-order spectral triples.
problem No specific problem stated; focuses on extending action.
method Introduced two second-order spectral triples and computed their Einstein-Hilbert actions.
result Demonstrated applicability of the theoretical framework.
Promotes spectral functionals to noncommutative fields and proves a theorem.
problem Spectral functionals on noncommutative fields and manifolds with boundary.
method Carries out promotion to spectral functionals, associates with noncommutative residue, and proves theorem.
result Proves Dabrowski-Sitarz-Zalecki type theorem for statistical de Rham Hodge operators on manifolds with boundary.
Study spectral functionals on manifolds with torsion.
problem Extending classical results to geometries with torsion.
method Using Wodzicki residue, investigate spectral functionals of Dirac and Laplace-type operators.
result Local densities recover fundamental geometric tensors.
Proves another theorem for odd dimensional manifolds with boundary.
problem Spectral Einstein functional on odd dimensional manifolds with boundary.
method Proof of a theorem using the Dirac operator.
result Another general Dabrowski-Sitarz-Zalecki type theorem proved.
The paper introduces a trilinear functional to recover torsion in spectral triples.
problem Recovering torsion in noncommutative spectral triples.
method Introduces a trilinear functional for spectral triples and demonstrates its application to recover torsion.
result The trilinear functional recovers the torsion of the linear connection in canonical spectral triples.
In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension n+1>3. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold (X,g) is less than $\ndemi -1$ if and only …
In this paper we study the spectral asymmetry of (possibly nonselfadjoint) elliptic PsiDO's in terms of the difference of zeta functions coming from different cuttings. Refining previous formulas of Wodzicki in the case of odd class elliptic PsiDO's, our main results have several consequence concerning the local indepe…
We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restriction…
We consider the equations, arising as the conformal invariance conditions of the perturbed curved beta-gamma system. These equations have the physical meaning of Einstein equations with a B-field and a dilaton on a hermitian manifold, where the B-field 2-form is imaginary and proportional to the canonical form associat…
Let (M,g) be a compact Einstein manifold with smooth boundary. We consider the spectrum of the p form valued Laplacian with respect to a suitable boundary condition. We show that certain geometric properties of the boundary may be spectrally characterized in terms of this data where we fix the Einstein constant.
Harmonic gauge simplifies geometric analysis of Riemannian metrics.
problem Analyzing the Hilbert-Einstein functional and its stability.
method Developed a harmonic gauge to eliminate divergence terms and induce elliptic structure.
result Positivity of curvature operator implies spectral stability of the functional.
We extend a result of Patodi for closed Riemannian manifolds to the context of closed contact manifolds by showing the condition that a manifold is an η-Einstein Sasakian manifold is spectrally determined. We also prove that the condition that a Sasakian space form has constant φ-sectional curvature c is spectral…
The paper studies Gauss sums and their applications in algebra and topology.
problem Computing signatures of bilinear forms and Poincaré spaces.
method Investigates properties of Gauss sums and applies them to algebraic and topological problems.
result Reproves and generalizes results on signatures mod 8 of bilinear and Poincaré spaces.
The study examines spectral rigidity in Ricci solitons and Einstein-type manifolds.
problem Determining sectional curvature from eigenvalues of the p-Laplacian.
method Analyzes spectral rigidity under gradient shrinking Ricci soliton and cohomologically Einstein conditions.
result With some exceptions, sectional curvature can be determined by eigenvalues of the p-Laplacian.
The paper studies invariant Einstein metrics on Lie supergroups.
problem Investigating invariant Einstein metrics on Lie supergroups.
method Parameterized left invariant metrics, derived Levi-Civita connection and Ricci tensor, reduced Einstein condition to algebraic system.
result Most real basic classical Lie superalgebras admit at least two distinct Einstein metrics.
We first show that hypergeometric functions appear naturally as spectral functions when applying pseudo-differential calculus to decipher heat kernel asymptotic in the situation where the symbol algebra is noncommutative. Such observation leads to a unified (works for arbitrary dimension) method of computing the modula…
New gauge preserves Einstein metrics' interactions, proving rigidity on negatively curved manifolds.
problem Stability and deformation theory of Einstein metrics.
method Introduces Chen-Nagano gauge condition, linking Lichnerowicz Laplacian to shifted scalar operator.
result Chen-Nagano gauge collapses to classical transverse-traceless gauge under spectral pinching assumptions.
We show how the well-known classical field equations as Einstein and Yang-Mills ones, which arise as the conformal invariance conditions of certain two-dimensional theories, expanded up to the second order in the formal parameter, can be reformulated as Generalized/formal Maurer-Cartan equations (GMC), where the differ…
We are concerned in this article with a classical question in spectral geometry dating back to McKean-Singer, Patodi and Tanno: whether or not the constancy of holomorphic sectional curvature of a complex n-dimensional compact Kähler manifold can be completely determined by the eigenvalues of its p-Laplacian for a …
New argument suggests torsion cannot be part of gravity models.
problem The presence of torsion in gravity models is debated.
method Used spectral geometry and pseudo-differential calculus.
result No well-defined functional for torsion in spectral formulation.
We introduce RSE to measure robustness in estimation problems.
problem Estimating statistical models from observed data.
method Developed theory for spectral functions of measures to compute RSE.
result RSE reveals a reciprocal relationship with problem complexity.
Study heat flow on collapsing K3 surfaces, handling conic singularities.
problem Analyzing heat flow on K3 surfaces as they collapse.
method Using semi-flat product approximations and conic-renormalized bilinear functionals.
result Heat operators converge to base Laplacian on regular locus.
Maps dBKP solutions to MS system solutions, defining Einstein-Weyl structures.
problem Constructing solutions and structures for dBKP and MS systems.
method Map construction and spectral characterisation of reductions.
result Defines Einstein-Weyl structures for dBKP and BMS systems.
We are interested in approximation of a multivariate function f(x1,…,xd) by linear combinations of products u1(x1)⋯ud(xd) of univariate functions ui(xi), i=1,…,d. In the case d=2 it is a classical problem of bilinear approximation. In the case of approximation in the L2 space the bili…
Bilinear MLPs offer a new way to interpret deep learning models without complex nonlinearities.
problem Lack of mechanistic understanding in how MLPs compute.
method Introduced bilinear MLPs without element-wise nonlinearities, analyzed their weights using tensor and eigendecomposition.
result Bilinear MLPs provide interpretable weight structures and enable adversarial attacks and overfitting analysis.
Investigates nearly Kähler and parallel G2 manifolds using Hitchin functionals.
problem Stability analysis of nearly Kähler and parallel G2 manifolds.
method Gradient flow of Hitchin functionals, spectral decomposition of Hessians, Hitchin index.
result Hitchin index provides a lower bound for the Einstein co-index.
New method identifies latent treatment effects from proxy models.
problem Identifying heterogeneous treatment effects under unobserved confounding.
method Compressed observable operator and spectral analysis of treatment effects.
result Eigenvalues of the operator represent latent treatment effects.