Study perturbs Dirac operator on 4D manifolds, proving Kastler-Kalau-Walze theorems.
problem Analyzing perturbations of Dirac operator on compact manifolds.
method Defining pseudo-differential perturbations and proving Kastler-Kalau-Walze theorems.
result Proved Kastler-Kalau-Walze theorems for 4D compact manifolds with boundary.
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.
problem Computing properties of spectral triples and their associated Hodge operators.
method Introduces a new multilinear functional for spectral triples and computes its properties using noncommutative residue and perturbed de-Rham Hodge operators.
result Recover two forms, torsion of the linear connection, and four forms by the noncommutative residue and perturbed de-Rham Hodge Dirac triple.
Differentiable clustering method using perturbed spanning forests.
problem Efficient clustering in trainable pipelines with noisy data.
method Stochastic perturbations of minimum-weight spanning forests.
result Method performs well even in challenging settings.
Study complex structures with perturbed differential operators to compute curvature-like operators and obtain vanishing results.
problem Analyzing complex structures with perturbed differential operators.
method Perturbing the standard differential operator to a first-order operator Dη and computing Bochner-Kodaira-Nakano-type formulae. result Obtained vanishing results for certain harmonic spaces and Dolbeault cohomology.
The paper calculates spectral torsion for rescaled Dirac operators on manifolds.
problem Computing spectral torsion for rescaled Dirac operators.
method Using trilinear Clifford multiplication and functional of differential one-forms.
result Computed spectral torsion for four types of rescaled Dirac operators.
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.
problem Maintaining Hodge kernel isomorphism for smooth bundles with non-smooth geometric data.
method Analyzing nilpotent differential operators and Hodge-Dirac-type operators under perturbations of geometric data.
result Kernels of Hodge-Dirac operators remain isomorphic under uniform perturbations of geometric data.
Differentiable perturbed optimizers enable end-to-end learning of discrete decisions.
problem Discrete decisions in machine learning pipelines break back-propagation.
method Transform optimizers into differentiable operations using stochastically perturbed optimizers.
result Smoothness of derivatives can be tuned via noise amplitude.
The paper defines and analyzes abla-Sobolev spaces and operators on manifolds.
problem Defining and analyzing Sobolev spaces and differential operators on manifolds.
method Coordinate-free approach using connections, proving properties of abla-Sobolev spaces and operators. result Equivalent definitions of abla-Sobolev spaces and operators under certain conditions. Optimizes material distribution on surfaces using topological derivatives.
problem Optimal distribution of two materials on smooth submanifolds in Rd. method Topological derivative approach for shape optimization constrained by PDEs.
result Numerical solution of topology optimization problem on surfaces.
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.
problem Decoupling of curvature scalars in Kerr spacetime.
method Hidden Kahler structure in Kerr spacetime, showing decoupling as a consequence of Kahler geometry.
result Decoupling of Teukolsky equations on Kahler background.
Proposes differentiable and sparse top-k operators for neural networks.
problem Discontinuity of top-k operator makes it unsuitable for end-to-end training with backpropagation.
method Formulates top-k as a linear program over permutahedron, introduces p-norm regularization, and uses isotonic optimization.
result Successfully applied to neural network pruning, fine-tuning, and routing.
The paper explores how polynomial roots and operator eigenvalues change with parameters.
problem How do roots of polynomials and eigenvalues of operators vary with parameter changes?
method Analyzes parameter dependence of polynomials and linear operators, covering real analytic to differentiable of finite order.
result Definitive optimal results for perturbation theory of polynomials and linear operators, including hyperbolic polynomials.
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.
problem Analyzing conformal perturbations of Dirac and signature operators.
method Developed Kastler-Kalau-Walze type theorems for specific operator types.
result Established new theorems for six-dimensional manifolds with boundary.
Fast algorithm for rescaling vectors with clipping, improving training efficiency.
problem Efficiently rescale vectors to a desired length while maintaining them within a domain after clipping.
method Analytical solution for optimal rescaling using fast and differentiable algorithm.
result Optimal rescaling can be found analytically, improving training efficiency for neural networks.
Study delocalized eta invariants for signature operators on proper manifolds.
problem Define and analyze delocalized eta invariants for signature operators on proper manifolds.
method Develop detailed heat-kernel analysis and apply to proper manifolds with boundary.
result Prove index formulas relating delocalized eta invariants to Atiyah-Patodi-Singer indices.
The paper studies perturbations of de Rham Hodge operators on manifolds with or without boundaries.
problem Analyzing perturbations of de Rham Hodge operators on manifolds with boundaries.
method Lichnerowicz type formulas and Kastler-Kalau-Walze type theorems for 4D and 6D compact manifolds with or without boundaries.
result Proves Kastler-Kalau-Walze type theorems for perturbations of de Rham Hodge operators on 4D and 6D manifolds with or without boundaries.
Proves K-K-W type theorems for specific types of operators.
problem Analyzes conformal perturbations of twisted Dirac operators.
method Uses Kastler-Kalau-Walze type theorems for four-dimensional manifolds.
result Establishes theorems for both with and without boundary conditions.
Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.
problem Understanding perturbations of Dirac operators in various dimensions.
method Analyzes canonical perturbations of Dirac operators on Hermitian Clifford modules.
result Characterizes the low-energy spectrum of these operators on complete surfaces.
Study perturbs APS boundary conditions for Lorentzian Dirac operators.
problem Maintaining Fredholmness of Dirac operators under perturbations of APS boundary conditions.
method Develop criteria for perturbing compact pairs of projections to remain Fredholm.
result Criteria for perturbing APS boundary conditions without losing Fredholmness.
The paper proves two theorems for modified Novikov operators under conformal perturbations.
problem Proving theorems for modified Novikov operators under conformal perturbations.
method Two Kastler-Kalau-Walze type theorems for conformal perturbations of modified Novikov Operators on 4D and 6D compact manifolds.
result Obtained two Kastler-Kalau-Walze type theorems for conformal perturbations of modified Novikov Operators.
Method finds domain of Laplace-Beltrami operator on 2D almost-Riemannian manifolds.
problem Determining the domain of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds with tangency points.
method Using tools from Lie groupoids, natural domains of perturbations are found.
result Method allows treatment of geometries with tangency points.
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.
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.
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.
problem Analyzing perturbations of order ≤ 1 in noncommutative geometry.
method Develops a Feynman-Kac formula for differential operators of order ≤ 1 on complex metric vector bundles over Riemannian manifolds.
result Explicit Feynman-Kac type formula for holomorphic semigroups generated by Q. Gradient noise improves privacy-protected optimization performance.
problem Improving privacy in convex optimization while maintaining utility.
method We analyze the effect of gradient perturbation on differentially private convex optimization, focusing on expected curvature.
result Gradient perturbation can achieve a significantly improved utility guarantee for differentially private convex optimization.
The paper proves new theorems for Dirac operators on even-dimensional manifolds with boundary.
problem Proving theorems for Dirac operators on manifolds with boundary.
method Establishing general Kastler-Kalau-Walze type theorems for conformal perturbations of Dirac operators.
result Proof of new theorems for Dirac operators on even-dimensional manifolds with boundary.
We generalize stochastic smoothing for gradient estimation of non-differentiable functions.
problem Gradient estimation for non-differentiable functions.
method Developed a general framework for relaxation and gradient estimation of non-differentiable black-box functions using stochastic smoothing with reduced assumptions.
result Empirically validated the effectiveness of variance reduction strategies for various non-differentiable tasks.
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…
Introduces a new geometric framework for non-perturbative BV-theory.
problem Non-perturbative generalization of BV-theory in infinite-dimensional spaces.
method Derived differential geometry and homotopical algebraic geometry.
result Concrete model of derived smooth stacks for encoding non-perturbative BV-theory.
Recently, the infinitesimal moduli space of heterotic G2 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.
problem Computing indices of mixed order Dirac-type operators and tensor fields.
method Using Hilbert complexes and differential operators of mixed order, computing indices with cohomology groups of tensor fields.
result Computation of indices for elasticity and biharmonic complexes.
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.
problem Concentration in complex mathematical structures.
method Construct conjugate-linear perturbations of twisted spinc Dirac operators using the conjugate-linear Hodge star operator.
result These perturbations satisfy the concentration principle.
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, Δ4+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.
problem Protecting both training data and model parameters while maintaining privacy.
method Add noise to training data and train with perturbed data for differential privacy.
result Achieves (ε,δ)-differential privacy on the final model with privacy on original data.
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.
problem Finding conditions for conformal isometry and existence of metrics.
method Introduces perturbations of Fefferman's conformal circle bundle and investigates existence of metrics.
result Provides conditions for existence of metrics satisfying Einstein equations.