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Niky Kamran

Publications and source records attributed to Niky Kamran.

At least 19 recordsLinked to original sources
arXiv2026

The $\mathcal{L}$-Calculus for Causal Variational Principles: An Exterior Differential Calculus on Non-Smooth Spaces

A differential calculus for causal variational principles is developed, which generalizes the exterior calculus of differential forms and some of the associated differential topological structures to non-smooth spaces. Our calculus includes the exterior derivative, de Rham cohomology, glueing constructions (restrictions and extensions of differential forms, Mayer-Vietoris sequence), a K\"unneth formula and Poincar{\'e}'s lemma. Moreover, we prove versions of Stokes' theorem and the Gau{\ss} divergence theorem. The constructions and results are illustrated by several examples.

math-ph↗
arXiv2026

A Local-to-Global Propagation Principle for Dirichlet-to-Neumann Maps

We establish four local-to-global propagation results for Dirichlet--to--Neumann maps. Our first two results are proved in the general setting of smooth compact Riemannian manifolds with boundary. The first shows that if two smooth Riemannian metrics coincide in a collar neighborhood of a connected boundary component \(\Gamma\), then equality of the corresponding local Dirichlet--to--Neumann maps on a nonempty open subset of \(\Gamma\) propagates to equality of the associated global Dirichlet--to--Neumann maps on all of \(\Gamma\). The proof combines unique continuation and self-adjointness arguments. The second replaces the geometric collar assumption by an exponential spectral assumption on the difference of the corresponding global Dirichlet--to--Neumann maps. The proof relies on the spectral unique continuation theory of Jerison--Lebeau, through the formulation of Le~Rousseau--Lebeau. Our third and fourth results establish local-to-global propagation principles under Ingham-type quasi--analytic spectral assumptions. Assuming that the boundary manifold is respectively a compact Riemannian symmetric space or a compact quasi--analytic Riemannian manifold, they rely on the propagation theorems of Ganguly--Thangavelu and of Bhowmik--Pradhan. As an application, we consider a class of conformally warped product metrics. In this setting, the local Borg--Marchenko theorem and Weyl--Titchmarsh theory relate the required Ingham-type spectral decay to a suitable quasi--analytic boundary closeness of the conformal factors, yielding new local-to-global uniqueness results for Dirichlet--to--Neumann maps.

math.AP↗
arXiv2026

A Sharp Regularity Threshold for Uniqueness in Riemannian Calder\'on-type Problems

We prove a sharp regularity threshold for uniqueness in two anisotropic Calder\'on-type inverse problems in dimension $n\ge 3$. The main setting is the Riemannian Schr\"odinger problem with fixed scalar potential: for a prescribed nonconstant analytic function $V$, we study whether the Dirichlet-to-Neumann map of $-\Delta_g+V$ on a domain $\Omega\subset\mathbb{R}^n$ determines the unknown metric $g$. The natural gauge is the group of boundary-fixing diffeomorphisms preserving $V$. We show that, while analytic metrics are uniquely determined modulo this gauge by a minor adaptation of the Lassas--Uhlmann reconstruction theorem, uniqueness fails densely in every non-analytic Gevrey class $G^\sigma$, $\sigma>1$. In fact, our counterexamples are not isometric in the sense that they are not connected by the pushforward of any diffeomorphism of $\overline\Omega$. We also prove the analogous sharp threshold for the anisotropic Calder\'on problem at fixed nonzero frequency, thereby upgrading the previously known finite-regularity counterexamples to Gevrey and $C^\infty$ regularity. The two constructions use different scalar mechanisms: for fixed potentials, the nonconstant potential itself provides a local coordinate, while at nonzero frequency one uses a compactly supported prescribed-Jacobian lemma in Gevrey spaces. Thus analyticity is the exact threshold for uniqueness in both problems.

math.AP↗
arXiv2026

Stability in the anisotropic Calder\'{o}n problem for Painlev\'e-Liouville Riemannian manifolds

We study the question of stability of the global and partial anisotropic Calder\'on inverse problems for the class of Painlev\'e-Liouville Riemannian manifolds, that is compact $n$-dimensional manifolds with boundary $(M,g)$, where $M=[0,1]\times K\,$, $K$ is any smooth closed connected orientable manifold of dimension $n-1$ endowed with a Riemannian metric $g_K$, and $g=\alpha^4 g_{0}$ is any conformal deformation of the product metric $g_{0}=dx^2+g_{K}$ on $M$ which is compatible with the Painlev\'e block-separability of the Laplace-Beltrami operator $\Delta_{g_0}$. Given a pair of Painlev\'e-Liouville Riemannian manifolds $(M,g)$ and $(M,\tilde{g})$ satisfying some technical hypothesis, denoting the corresponding Dirichlet-to-Neumann maps by $\Lambda_{g}$ and $\Lambda_{\tilde{g}}$, and assuming that $\lVert \Lambda_{g}-\Lambda_{\tilde{g}}\rVert_{\mathcal{B}(H^{1/2}(\partial M), H^{-1/2}(\partial M))}\ = \epsilon $, we show a logarithmic stability result for the global anisotropic Calder\'on problem which says that there exists constants $C$ and $0<\theta<1$ such that $\| \alpha - \tilde{\alpha} \|_{C^{0,r}(M)} \leq C \left( \ln \frac{1}{\epsilon} \right)^{-\theta}$ for some $0<r<1$. Similar results are obtained for the partial anisotropic Calder\'on problem, corresponding to the case where the data are measured on only one connected component of the boundary.

math.AP↗
arXiv2025

Inverse problems for the Bakry-\'Emery Laplacian on manifolds with boundary -- uniqueness and non-uniqueness

We study the questions of uniqueness and non-uniqueness for a pair of closely related inverse problems for the Bakry-\'Emery Laplacian $-\Delta_{\mathcal E}$ on a smooth compact and oriented Riemannian manifold with boundary $(\overline{M},g)$, endowed with a volume form $\mathfrak{m}=e^{-V}\omega_g$. These consist in recovering the Taylor coefficients of metric $g$ and weight $V$ along the boundary of $\overline{M}$ from the knowledge of a pair of operators that can be viewed as geometrically natural Dirichlet-to-Neumann maps associated to $-\Delta_{\mathcal E}$.

math.AP↗
arXiv2024

Holographic Mixing and Fock Space Dynamics of Causal Fermion Systems

A limiting case is considered in which the causal action principle for causal fermion systems describing Minkowski space gives rise to the linear Fock space dynamics of quantum electrodynamics. The quantum nature of the bosonic field is a consequence of the stochastic description of a multitude of fluctuating fields coupled to non-commuting operators, taking into account dephasing effects. The scaling of all error terms is specified. Our analysis leads to the concept of holographic mixing, which is introduced and explained in detail.

math-ph↗
arXiv2024

Global counterexamples to uniqueness for a Calder\'on problem with $C^k$ conductivities

Let $\Omega \subset R^n$, $n \geq 3$, be a fixed smooth bounded domain, and let $\gamma$ be a smooth conductivity in $\overline{\Omega}$. Consider a non-zero frequency $\lambda_0$ which does not belong to the Dirichlet spectrum of $L_\gamma = -{\rm div} (\gamma \nabla \cdot)$. Then, for all $k \geq 1$, there exists an infinite number of pairs of non-isometric $C^k$ conductivities $(\gamma_1, \gamma_2)$ on $\overline{\Omega}$, which are close to $\gamma$ such that the associated DN maps at frequency $\lambda_0$ satisfy \begin{equation*} \Lambda_{\gamma_1,\lambda_0} = \Lambda_{\gamma_2,\lambda_0}. \end{equation*}

math.AP↗
arXiv2023

A Positive Quasilocal Mass for Causal Variational Principles

A new inequality for a nonlinear surface layer integral is proved for minimizers of causal variational principles. This inequality is applied to obtain a new proof of the positive mass theorem with volume constraint. Next, a positive mass theorem without volume constraint is stated and proved by introducing and using the concept of asymptotic alignment. Moreover, a positive quasilocal mass and a synthetic definition of scalar curvature are introduced in the setting of causal variational principles. Our notions and results are illustrated by the explicit examples of causal fermion systems constructed in ultrastatic spacetimes and the Schwarzschild spacetime. In these examples, the correspondence to the ADM mass and similarities to the Brown-York mass are worked out.

math-ph↗
arXiv2023

Boundary rigidity, and non-rigidity, of projective structures

We investigate the property of boundary rigidity for the projective structures associated to torsion-free affine connections on connected analytic manifolds with boundary. We show that these structures are generically boundary rigid, meaning that any automorphism of a generic projective structure that restricts to the identity on the boundary must itself be the identity. However, and in contrast with what happens for example for conformal structures, we show that there exist projective structures which are not boundary rigid. We characterise these non-rigid structures by the vanishing of a certain local projective invariant of the boundary.

math.DG↗
arXiv2022

Local H\"older Stability in the Inverse Steklov and Calder\'on Problems for Radial Schr\"odinger operators and Quantified Resonances

We obtain H\"older stability estimates for the inverse Steklov and Calder\'on problems for Schr\"odinger operators corresponding to a special class of $L^2$ radial potentials on the unit ball. These results provide an improvement on earlier logarithmic stability estimates obtained in \cite{DKN5} in the case of the the Schr\"odinger operators related to deformations of the closed Euclidean unit ball. The main tools involve: i) A formula relating the difference of the Steklov spectra of the Schr\"odinger operators associated to the original and perturbed potential to the Laplace transform of the difference of the corresponding amplitude functions introduced by Simon \cite{Si1} in his representation formula for the Weyl-Titchmarsh function, and ii) A key moment stability estimate due to Still \cite{St}. It is noteworthy that with respect to the original Schr\"odinger operator, the type of perturbation being considered for the amplitude function amounts to the introduction of a finite number of negative eigenvalues and of a countable set of negative resonances which are quantified explicitly in terms of the eigenvalues of the Laplace-Beltrami operator on the boundary sphere.

math.AP↗
arXiv2022

Entangled Quantum States of Causal Fermion Systems and Unitary Group Integrals

This paper is dedicated to a detailed analysis and computation of quantum states of causal fermion systems. The mathematical core is to compute integrals over the unitary group asymptotically for a large dimension of the group, for various integrands with a specific scaling behavior in this dimension. It is shown that, in a well-defined limiting case, the localized refined pre-state is positive and allows for the description of general entangled states.

math-ph↗
arXiv2022

A Green's function for the source-free Maxwell equations on $AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$

We compute a Green's function giving rise to the solution of the Cauchy problem for the source-free Maxwell's equations on a causal domain $\mathcal{D}$ contained in a geodesically normal domain of the Lorentzian manifold $AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$, where $AdS^5$ denotes the simply connected $5$-dimensional anti-de-Sitter space-time. Our approach is to formulate the original Cauchy problem as an equivalent Cauchy problem for the Hodge Laplacian on $\mathcal{D}$ and to seek a solution in the form of a Fourier expansion in terms of the eigenforms of the Hodge Laplacian on $\mathbb{S}^3$. This gives rise to a sequence of inhomogeneous Cauchy problems governing the form-valued Fourier coefficients corresponding to the Fourier modes and involving operators related to the Hodge Laplacian on $AdS^5 \times \mathbb{S}^2$, which we solve explicitly by using Riesz distributions and the method of spherical means for differential forms. Finally we put together into the Fourier expansion on $\mathbb{S}^3$ the modes obtained by this procedure, producing a $2$-form on $\mathcal{D}\subset AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$ which we show to be a solution of the original Cauchy problem for Maxwell's equations.

math.AP↗
arXiv2021

Fermionic Fock Spaces and Quantum States for Causal Fermion Systems

It is shown for causal fermion systems describing Minkowski-type spacetimes that an interacting causal fermion system at time $t$ gives rise to a distinguished state on the algebra generated by fermionic and bosonic field operators. The proof of positivity of the state is given and representations are constructed.

math-ph↗
arXiv2021

The Linear Dynamics of Wave Functions in Causal Fermion Systems

The dynamics of spinorial wave functions in a causal fermion system is studied. A so-called dynamical wave equation is derived. Its solutions form a Hilbert space, whose scalar product is represented by a conserved surface layer integral. We prove under general assumptions that the initial value problem for the dynamical wave equation admits a unique global solution. Causal Green's operators are constructed and analyzed. Our findings are illustrated in the example of the regularized Minkowski vacuum.

math-ph↗
arXiv2021

Complex Structures on Jet Spaces and Bosonic Fock Space Dynamics for Causal Variational Principles

Based on conservation laws for surface layer integrals for critical points of causal variational principles, it is shown how jet spaces can be endowed with an almost-complex structure. We analyze under which conditions the almost-complex structure can be integrated to a canonical complex structure. Combined with the scalar product expressed by a surface layer integral, we obtain a complex Hilbert space $\mathfrak{h}$. The Euler-Lagrange equations of the causal variational principle describe a nonlinear time evolution on $\mathfrak{h}$. Rewriting multilinear operators on $\mathfrak{h}$ as linear operators on corresponding tensor products and using a conservation law for a nonlinear surface layer integral, we obtain a linear norm-preserving time evolution on bosonic Fock spaces. The so-called holomorphic approximation is introduced, in which the dynamics is described by a unitary time evolution on the bosonic Fock space. The error of this approximation is quantified. Our constructions explain why and under which assumptions critical points of causal variational principles give rise to a second-quantized, unitary dynamics on Fock spaces.

math-ph↗
arXiv2020

Ramified local isometric embeddings of singular Riemannian metrics

In this paper, we are concerned with the existence of local isometric embeddings into Euclidean space for analytic Riemannian metrics $g$, defined on a domain $U\subset \mathbf{R}^n$, which are singular in the sense that the determinant of the metric tensor is allowed to vanish at an isolated point (say the origin). Specifically, we show that, under suitable technical assumptions, there exists a local analytic isometric embedding $u$ from $(U',Π^*g)$ into Euclidean space $\mathbf{E}^{(n^2+3n-4)/2}$, where $Π:U' \to U\backslash\{0\}$ is a finite Riemannian branched cover of a deleted neighborhood of the origin. Our result can thus be thought of as a generalization of the classical Cartan-Janet Theorem to the singular setting in which the metric tensor is degenerate at an isolated point. Our proof uses Leray's ramified Cauchy-Kovalevskaya Theorem for analytic differential systems, in the form obtained by Choquet-Bruhat for non-linear systems.

math.DG↗
arXiv2019

Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations

Painlevé metrics are a class of Riemannian metrics which generalize the well-known separable metrics of Stäckel to the case in which the additive separation of variables for the Hamilton-Jacobi equation is achieved in terms of groups of independent variables rather than the complete orthogonal separation into ordinary differential equations which characterizes the Stäckel case. Our goal in this paper is to carry out for Painlevé metrics the generalization of the analysis, which has been extensively performed in the Stäckel case, of the relation between separation of variables for the Hamilton-Jacobi and Helmholtz equations, and of the connections between quadratic first integrals of the geodesic flow and symmetry operators for the Laplace-Beltrami operator. We thus obtain the generalization for Painlevé metrics of the Robertson separability conditions for the Helmholtz equation which are familiar from the Stäckel case, and a formulation thereof in terms of the vanishing of the off-block diagonal components of the Ricci tensor, which generalizes the one obtained by Eisenhart for Stäckel metrics. We also show that when the generalized Robertson conditions are satisfied, there exist $r<n$ linearly independent second-order differential operators which commute with the Laplace-Beltrami operator and which are mutually commuting. These operators admit the block-separable solutions of the Helmholtz equation as formal eigenfunctions, with the separation constants as eigenvalues. Finally, we study conformal deformations which are compatible with the separation into blocks of variables of the Helmholtz equation for Painlevé metrics, leading to solutions which are $R$-separable in blocks. The paper concludes with a set of open questions and perspectives.

math-ph↗