arXiv research

Study perturbs Dirac operator on 4D manifolds, proving Kastler-Kalau-Walze theorems.

In this thesis, we study singular pseudo-differential operators defined by groupoids satisfying the Lauter-Nistor condition, by a method parallel to that of manifolds with boundary and edge differential operators. The example of the Bruhat sphere is studied in detail. In particular, we construct an extension to the cal…

Paper introduces a new multilinear functional for spectral triples and computes its properties.

Differentiable clustering method using perturbed spanning forests.

Study complex structures with perturbed differential operators to compute curvature-like operators and obtain vanishing results.

The paper calculates spectral torsion for rescaled Dirac operators on manifolds.

We provide criteria for self-adjointness and τ-Fredhomness of first and second order differential operators acting on sections of infinite dimensional bundles, whose fibers are modules of finite type over a von Neumann algebra A endowed with a trace τ. We extend the Callias-type index to operators acting on sections of…

Over a closed manifold, we consider the sectorial projection of an elliptic pseudo-differential operator A of positive order with two rays of minimal growth. We show that it depends continuously on A when the space of pseudo-differential operators is equipped with a certain topology which we explicitly describe. Our ma…

Smooth bundles with rough data maintain Hodge kernel isomorphism.

Differentiable perturbed optimizers enable end-to-end learning of discrete decisions.

The paper defines and analyzes $abla$-Sobolev spaces and operators on manifolds.

Optimizes material distribution on surfaces using topological derivatives.

We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$…

Kahler geometry explains decoupling of Kerr perturbations.

Proposes differentiable and sparse top-k operators for neural networks.

The paper explores how polynomial roots and operator eigenvalues change with parameters.

The goal of the paper is to calculate the limit sectrum of the Hodge-Laplace operator under the perturbation of collapse of one part of a connected sum. This gives some new results concerning the 'conformal spectrum' on differential forms.

The paper proves new theorems about specific types of operator perturbations.

Fast algorithm for rescaling vectors with clipping, improving training efficiency.

Study delocalized eta invariants for signature operators on proper manifolds.

The paper studies perturbations of de Rham Hodge operators on manifolds with or without boundaries.

Proves K-K-W type theorems for specific types of operators.

Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.

Study perturbs APS boundary conditions for Lorentzian Dirac operators.

The paper proves two theorems for modified Novikov operators under conformal perturbations.

Method finds domain of Laplace-Beltrami operator on 2D almost-Riemannian manifolds.

We propose a novel framework for the differentially private ERM, input perturbation. Existing differentially private ERM implicitly assumed that the data contributors submit their private data to a database expecting that the database invokes a differentially private mechanism for publication of the learned model. In i…

The paper introduces spectral Einstein functionals for Dirac operators on manifolds with boundary.

We explore the nonperturbative aspects of the chiral algebras of N = (0,2) sigma models, which perturbatively are intimately related to the theory of chiral differential operators (CDOs). The grading by charge and scaling dimension is anomalous if the first Chern class of the target space is nonzero. This has some nont…

The paper develops a Feynman-Kac formula for perturbations of order ≤ 1 in noncommutative geometry.

Gradient perturbation, widely used for differentially private optimization, injects noise at every iterative update to guarantee differential privacy. Previous work first determines the noise level that can satisfy the privacy requirement and then analyzes the utility of noisy gradient updates as in the non-private cas…

The paper proves new theorems for Dirac operators on even-dimensional manifolds with boundary.

In this paper, we prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-fo…

We generalize stochastic smoothing for gradient estimation of non-differentiable functions.

Introduces a new geometric framework for non-perturbative BV-theory.

Recently, the infinitesimal moduli space of heterotic $G_2$ compactifications was described in supergravity and related to the cohomology of a target space differential. In this paper we identify the marginal deformations of the corresponding heterotic nonlinear sigma model with cohomology classes of a worldsheet BRST …

Given a two-dimensional quantum field theory with (0,2) supersymmetry, one can construct a chiral (or vertex) algebra. The chiral algebra of a (0,2) supersymmetric sigma model is, perturbatively, the cohomology of a sheaf of chiral differential operators on a string Kähler manifold. However, it vanishes in some cases w…

Computes indices of mixed order Dirac-type operators and related tensor fields.

Differential privacy is concerned about the prediction quality while measuring the privacy impact on individuals whose information is contained in the data. We consider differentially private risk minimization problems with regularizers that induce structured sparsity. These regularizers are known to be convex but they…

New operators help focus on specific areas in complex math problems.

Forward Automatic Differentiation (AD) is a technique for augmenting programs to compute derivatives. The essence of Forward AD is to attach perturbations to each number, and propagate these through the computation. When derivatives are nested, the distinct derivative calculations, and their associated perturbations, m…

We consider perturbed quadharmonic operators, $Δ^4 + V$, acting on sections of a Hermitian vector bundle over a complete Riemannian manifold, with the potential $V$ satisfying a bound from below by a non-positive function depending on the distance from a point. Under a bounded geometry assumption on the Hermitian vecto…

In this paper, for a compact Lie group action,we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equiva…

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigen…

We show that the residue density of the logarithm of a generalised Laplacian on a closed manifold defines an invariant polynomial valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulae provide a …

Paper introduces input perturbation for privacy in machine learning models.

We present a version of the equivariant gradient degree defined for equivariant gradient perturbations of an equivariant unbounded self-adjoint operator with purely discrete spectrum in Hilbert space. Two possible applications are discussed.

Generalizes Fefferman's structure to CR three-manifolds with additional data.