Eigenvalues of Steklov eigenproblems change predictably with boundary tweaks.
problem Understanding how Steklov eigenvalues respond to boundary changes.
method Analyzing smooth boundary perturbations of Steklov eigenvalues.
result Steklov eigenvalues are generically simple under such perturbations.
The paper examines stability of Yamabe boundary problem under perturbations.
problem Stability of Yamabe boundary problem under perturbations of mean curvature and scalar curvature.
method Analyzes stability of the Yamabe boundary problem with respect to perturbations of mean curvature and scalar curvature.
result The stability of the Yamabe boundary problem is proven under perturbations from below, but not from above.
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 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.
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.
Groups' boundary actions are stable under small perturbations.
problem Stability of group actions on their boundaries.
method Proving semi-conjugacy of perturbed actions to the standard action.
result Perturbed actions are globally semi-conjugate to the standard action.
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.
The study examines how perturbations of lattice actions on group boundaries behave.
problem Understanding how perturbations of lattice actions on group boundaries affect semi-conjugacy.
method Analyzes continuous factorization of perturbed actions onto original actions by semi-conjugacy.
result Perturbations of lattice actions on group boundaries can be C0 semi-conjugate or not. This paper introduces a general perturbative quantization scheme for gauge theories on manifolds with boundary, compatible with cutting and gluing, in the cohomological symplectic (BV-BFV) formalism. Explicit examples, like abelian BF theory and its perturbations, including nontopological ones, are presented.
We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the Weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n>10.
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…
Deep networks have recently been shown to be vulnerable to universal perturbations: there exist very small image-agnostic perturbations that cause most natural images to be misclassified by such classifiers. In this paper, we propose the first quantitative analysis of the robustness of classifiers to universal perturba…
We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is n>6 and the trace-free part of the second fundamental form is non-zero everywhere on the boundary.
Compact solutions persist even with linear perturbations of the mean curvature term.
problem Compactness of solutions to the Yamabe problem on manifolds with boundary.
method Linear perturbation of the mean curvature term, proving compactness of solutions.
result Set of solutions remains compact even with negative perturbations.
In this paper, we define lower dimensional volumes of compact Riemannian manifolds with boundary. For five dimensional spin manifolds with boundary, we prove a Kastler-Kalau-Walze type theorem associated with one-form perturbations of Dirac operators in this case.
Analyzes perturbed contact instantons with Legendrian boundary conditions using geometric analysis.
problem Analyzing nonlinear elliptic systems associated with contact Hamiltonian trajectories.
method Identifies correct action and energy functionals, develops elliptic regularity theory, and proves asymptotic convergence.
result Established C∞ convergence of perturbed contact instantons under finite energy hypothesis. Study on compactness and blow-up of solutions for Yamabe problems on manifolds with non-umbilic boundaries.
problem Compactness and blow-up behavior of solutions to the Yamabe boundary problem on manifolds with non-umbilic boundaries.
method Analysis of stability and blow-up sequences for solutions under perturbations of mean curvature and scalar curvature.
result Existence of a blowing-up sequence of solutions when perturbing the mean curvature from above or below with a function having a large positive maximum.
On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator DB in L2 depends Riesz continuously on L∞ perturbations of local boundary conditions B. The Lipschitz bound for the map ${…
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.
The goal of this paper is to analyze the geometric properties of deep neural network classifiers in the input space. We specifically study the topology of classification regions created by deep networks, as well as their associated decision boundary. Through a systematic empirical investigation, we show that state-of-t…
Unique minimal surfaces near quadratic cones are identified.
problem Identifying minimal surfaces near quadratic cones.
method Analyzing minimal hypersurfaces inside the unit ball with perturbed boundary conditions.
result Minimal surfaces are uniquely determined by their boundary conditions.
Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.
problem Reduces the dimension of the singular set of area-minimizing hypersurfaces.
method Perturbs a smooth hypersurface to minimize the Minkowski dimension of the singular set.
result The singular set of the perturbed minimizing current has Minkowski dimension less than n-9.
This work proves Kerr black holes are dynamically stable under certain perturbations.
problem Dynamical stability of Kerr black holes under axially symmetric perturbations.
method Dimensional reduction to 2+1 Einstein-wave map system, construction of positive-definite energy functional, proving boundary terms vanish.
result Strictly conserved positive energy for axially symmetric linear perturbations of Kerr black holes.
Proves existence of solutions to Einstein constraints with specific boundary conditions and verifies Penrose inequality.
problem Existence of asymptotically hyperbolic solutions to Einstein constraints with marginally outer trapped boundaries.
method Constant mean curvature conformal method.
result Verification of Penrose inequality for certain Schwarzschild-AdS black hole perturbations.
Formula calculates invariant for 3-manifolds with torus boundaries.
problem Calculating an invariant for 3-manifolds with torus boundaries.
method Generalized Chern-Simons invariant and provided a gluing formula.
result A gluing formula for the invariant d(M,ρ). The study reveals how adversarial perturbations can include class features for generalization.
problem Understanding why adversarial examples deceive neural networks and transfer between networks.
method A one-hidden-layer network trained on mutually orthogonal samples.
result Adversarial perturbations, even of a few pixels, contain sufficient class features for generalization.
Constructs perturbations of a minimal surface with triple junctions.
problem Minimal surfaces with triple junctions in curved spaces.
method Constructs stationary perturbations with given boundary conditions.
result Constructs minimal surfaces with triple junctions in R2imesS1. The paper proves the existence and uniqueness of certain spacelike hypersurfaces with specific curvature and boundary conditions.
problem Existence and uniqueness of convex, entire, spacelike hypersurfaces with constant σk curvature. method Investigation of hypersurfaces with prescribed set of lightlike directions and perturbation on the ideal boundary at infinity.
result Existence and uniqueness of complete entire spacelike constant σk curvature hypersurfaces with prescribed lightlike directions and perturbation. 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.
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.
This is a survey of recent results on zeta- and eta-function poles and values for realizations of Laplace- and Dirac-type operators defined by pseudodifferential projection boundary conditions (including the Atiyah-Patodi-Singer operator and its square). Section 1 recalls some useful results for ps.d.o.s on closed mani…
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.
The paper solves a curvature problem on a ball's surface near constant values.
problem Prescribing almost constant curvatures on a manifold with boundary.
method Perturbative approach and ansatz by Han and Li.
result New existence results for conformal metrics when curvatures are near constants.
Study how nodal domains change on surfaces under perturbations.
problem How eigenfunction nodal domains change on surfaces under smooth perturbations.
method Sector/graph count near nodal critical points, upper semicontinuity proof, branch-free on spectral clusters, wavelength-scale analysis.
result Upper semicontinuity of nodal domain count, no new domains created at wavelength scale, stable count in noncritical cases.
The paper identifies potential adversarial samples near decision boundaries of neural networks.
problem Vulnerability of deep neural networks to small perturbations of inputs.
method Developed a method to explore near decision boundaries of trained classifiers to identify potential adversarial samples.
result Potential adversarial samples represent only 61% of the test data but cover more than 82% of adversarial samples produced by iFGSM and 92% of those by DeepFool on CIFAR10.
We show that a small perturbation of the boundary distance function of a simple Finsler metric on the n-disc is also the boundary distance function of some Finsler metric. (Simple metric form an open class containing all flat metrics.) The lens map is map that sends the exit vector to the entry vector as a geodesic c…
The paper constructs new bimetric conformal invariants using metric perturbations.
problem Developing new conformal invariants in Riemannian geometry.
method Using linear metric perturbations and conformal invariants.
result New bimetric conformal invariants on 4D manifolds are derived.
Solves boundary Yamabe problem with minimal boundary scenario.
problem Existence of solutions to a conformal metric equation with constant scalar curvature.
method Iteration schemes and perturbation methods to construct sub-solutions and super-solutions.
result Complete solution for three cases based on the sign of the first eigenvalue of the conformal Laplacian.
Improved neural network robustness with instance-specific perturbation margins.
problem Adversarial training fails to generalize well to unperturbed test set.
method Instance adaptive adversarial training with sample-specific perturbation margins.
result Test accuracy improves with a marginal drop in robustness.
3D spherical caps are rigid under certain perturbations.
problem Rigidity of 3D spherical caps under specific perturbations.
method Gromov's μ-bubble technique
result 3D spherical caps are rigid under perturbations that maintain metric, scalar curvature, and mean curvature.
Solve supercritical Yamabe problem on manifolds with non-umbilic boundary.
problem Solving supercritical Yamabe problem on manifolds with non-umbilic boundary.
method Building blowing-up solutions for a supercritical perturbation of the Yamabe problem.
result Constructed solutions for a supercritical perturbation of the Yamabe problem on manifolds with non-umbilic boundary.
Study shows how tangle moduli spaces relate to boundary surfaces.
problem Relating tangle moduli spaces to boundary surfaces.
method Use of composition in the Weinstein category and SU(2) holonomy perturbations.
result Holonomy perturbed SU(2) traceless flat moduli space is a Lagrangian immersion.
Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple descripti…
Solves a complex mathematical problem on curved surfaces.
problem Boundary Yamabe problem with constant scalar and mean curvature.
method Iterative schemes and perturbation methods to solve nonlinear elliptic PDEs.
result Existence of a real, positive, smooth solution in compact manifolds.
A fast method computes class-specific adversarial perturbations for deep networks.
problem Computing robust adversarial perturbations for deep networks.
method Linear function of weights, no training data, no hyper-parameters.
result Obtains 34% to 51% fooling rate on ImageNet, transfers across models.
Solutions grow for a special type of math problem on curved spaces.
problem Yamabe problem on manifolds with umbilic boundary
method Building blowing-up solutions for a supercritical perturbation
result Existence of solutions for n>7 and non-vanishing Weyl tensor
Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the fam…
Given a state-of-the-art deep neural network classifier, we show the existence of a universal (image-agnostic) and very small perturbation vector that causes natural images to be misclassified with high probability. We propose a systematic algorithm for computing universal perturbations, and show that state-of-the-art …