Proves a refined positive mass theorem on spin manifolds.
problem Proving a refined positive mass theorem under spectral scalar curvature bounds.
method Using the Dirac operator method and spin geometry.
result Discovers the origin of a special coefficient in the spectral positivity condition.
Derives spectral density function for symplectic manifolds.
problem Calculating spectral density functions on symplectic manifolds.
method Explicit local formula derivation for spectral density function.
result Explicit formula for spectral density function.
Positive scalar curvature metrics on cobordisms yield only reducible Seiberg-Witten solutions.
problem Counting solutions to Seiberg-Witten equations on cobordisms.
method Constructing families of metrics with positive scalar curvature.
result Irreducible solutions are absent when positive scalar curvature metrics are used.
Paper generalizes spectral embedding for better graph interpretation.
problem Modeling heterophilic connectivity and negative eigenvalues in graph data.
method Generalized latent position network model (Random Dot Product Graph).
result Consistent latent position estimates with asymptotically Gaussian error.
Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.
problem Closed orientable surfaces with nonnegative curvature in spectral sense.
method Spectral condition and associated conformal metrics to prove inequalities and bounds.
result Isoperimetric inequalities, area growth theorems, and diameter bounds for surfaces.
The paper proves conditions for Kähler and Riemannian manifolds to be simply connected.
problem Conditions for Kähler and Riemannian manifolds to be simply connected.
method Spectral positivity assumptions for Kähler manifolds and a specific spectral positivity assumption for Riemannian manifolds.
result Compact Kähler manifolds and Riemannian manifolds under the specified spectral positivity assumptions are simply connected.
Improved spectral projection estimates on manifolds of non-positive curvature.
problem Estimating spectral projections on manifolds with non-positive curvature.
method New spectral projection estimates, including sharp ones for tori, using pointwise estimates and microlocal L2oLqc Kakeya-Nikodym estimates. result Stronger and more precise spectral projection estimates, including new sharp estimates for tori.
Spectral sparsification improves Gaussian graphical models under MTP2 constraints.
problem Learning accurate, sparse graphs from data under MTP2 constraints.
method Spectral graph sparsification applied to Gaussian graphical models.
result Spectral-MTP2 preserves MTP2 and approximates the original model well.
Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
problem Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
method Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
result Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
Proposes a privacy-preserving method for graph embedding.
problem Privacy leakage in adjacency spectral embedding for stochastic blockmodels.
method Differentially private adjacency spectral embedding algorithm for stochastic blockmodels.
result Estimates latent positions close to those by non-private embedding, maintaining accuracy at desired privacy levels.
New method efficiently learns positive-definite curvature for neural nets.
problem Efficiently learn positive-definite curvature for neural net training.
method Spectral-factorized positive-definite curvature learning approach.
result Efficiently applies arbitrary matrix roots and generic curvature learning.
Positive weights improve kernel quadrature's accuracy.
problem Improving kernel quadrature weights to be positive and stable.
method Using convex geometry to approximate the kernel mean embedding with positive weights.
result Positive weights lead to improved kernel quadrature bounds with Monte-Carlo-beating rates.
The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.
problem Bounding bandwidth and focal radius for manifolds with positive isotropic curvature.
method Using spectral properties of a twisted de Rham-Hodge operator.
result Upper bounds on bandwidth and focal radius are derived for hypersurfaces in PIC manifolds.
In this paper we consider a modified version of the classical optimal dividends problem of de Finetti in which the dividend payments subject to a penalty at ruin. We assume that the risk process is modeled by a general spectrally positive Levy process before dividends are deducted. Using the fluctuation theory of spect…
A new method for spectral positional encodings in directed graphs using Hermitian block Krylov subspaces.
problem Challenges in spectral positional encodings for directed graphs, including computational complexity and gauge invariance issues.
method Learnable spectral positional encodings of the form hθ(Aq)R, computed in a Hermitian block Krylov subspace from sparse matrix-vector products. result The method is gauge-invariant and converges to the exact eigendecomposition oracle as the depth grows.
The paper validates Stokes' theorem for differential subcomplexes in positively graded Lie groups.
problem Validating Stokes' theorem for differential subcomplexes in positively graded Lie groups.
method Introducing geometric conditions and spectral complexes to recover Stokes' theorem on locally smooth intrinsic graphs.
result Stokes' theorem holds for Rumin complex and new spectral complexes on Carnot groups.
Study spectral flow for Callias operators to prove obstructions to positive scalar curvature.
problem Prove obstructions to positive scalar curvature on manifolds.
method Use spectral flow and odd K-cowaist to derive obstructions.
result Infinite odd K-cowaist is an obstruction to the existence of PSC metrics.
The paper finds optimal threshold strategies for insurance companies with a positive terminal value at creeping ruin.
problem Optimizing dividend payments in an insurance company's surplus process with a positive terminal value at creeping ruin.
method Using fluctuation theory, the paper derives explicit formulas for the objective function and shows the optimality of threshold strategies.
result Threshold strategies are optimal for the dividend optimization problem under certain conditions.
New method for directed graphs using learnable spectral positional encodings.
problem Challenges in magnetic Laplacians and unitary gauge invariance for directed graphs.
method Learnable spectral PEs of the form hθ(Aq)R, computed in Hermitian block Krylov subspace.
result Gauge-invariant and computationally efficient solution for directed graphs.
New spectral mixture representation for isotropic kernels simplifies random Fourier features.
problem Applying Random Fourier Features to complex kernels.
method Decompose isotropic kernels into scale mixtures of α-stable random vectors.
result Constructive spectral sampling formula for various kernels.
New method embeds dynamic networks with stability for node behavior.
problem Embed time-evolving node representations with stability.
method Unfolded adjacency spectral embedding for dynamic networks.
result Method satisfies cross-sectional and longitudinal stability.
Study compares spectral properties of a specific tensor in geometry.
problem Comparing spectral properties of a specific tensor in geometry.
method Diameter and global weighted volume comparison with a positive lower bound on the N-Bakry-Emery Ricci tensor. result Established diameter and volume comparisons for tensors with positive lower bounds.
This paper presents non-parametric estimates of spectral risk measures applied to long and short positions in 5 prominent equity futures contracts. It also compares these to estimates of two popular alternative measures, the Value-at-Risk (VaR) and Expected Shortfall (ES). The spectral risk measures are conditioned on …
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 …
Paper studies spectral invariants and monopole Floer homology for rational homology three-spheres.
problem Tackles the existence of positive scalar curvature metrics on ribbon homology cobordisms.
method Defines an R-filtration on the equivariant complex of monopole Floer homology via Chern-Simons-Dirac functional, leading to a spectral invariant.
result Shows that the spectral invariant provides an obstruction to the existence of positive scalar curvature metrics on ribbon homology cobordisms.
Study optimizes dividend strategies for risk processes with Lévy jumps.
problem Optimizing dividend payments in risk processes with Lévy jumps.
method Analyzes spectrally positive and negative Lévy processes, using scale functions.
result Periodic barrier strategy is optimal for spectrally negative Lévy processes with completely monotone Lévy density.
Proves positive mass theorem for hyperbolic manifolds with ends.
problem Establishing positive mass theorem for complex initial data sets.
method Used spectral PSC, Jang equation, and quantitative shielding theorem.
result Proved positive mass theorem for asymptotically hyperbolic manifolds.
The paper compares spectral geometry in hyperbolic and spherical manifolds.
problem Understanding spectral geometry in spherical manifolds.
method Survey of known results and open problems.
result Analogous results hold in hyperbolic manifolds but not necessarily in spherical manifolds.
LASE improves local network structure visualization by targeting locally low-dimensional regions.
problem Global spectral embedding fails to capture local geometric features in sparse, transitive networks.
method Local Adjacency Spectral Embedding (LASE) using weighted spectral decomposition.
result LASE reveals locally low-dimensional structure, improving local reconstruction and visualization.
The paper constructs noncompact hyperbolic surfaces with uniform spectral gaps using random graph models.
problem Building noncompact hyperbolic surfaces with uniform spectral gaps.
method Introduced a random graph model Fχ,n to construct expanding families of graphs, then applied these families to create hyperbolic surfaces. result Explicitly constructed an expanding family of graphs in the critical regime, leading to a sequence of complete, noncompact hyperbolic surfaces with uniformly positive spectral gaps.
Researchers find spectral gaps in quantum flag manifolds using twisted operators.
problem Finding spectral gaps in quantum flag manifolds.
method Tensoring Laplace and Dolbeault-Dirac operators with negative Hermitian holomorphic modules.
result Twisting Dirac and Laplace operators by negative line bundles produces a spectral gap for q close to 1.
In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degener…
The paper derives geometric and spectral inequalities using Ricci curvature assumptions.
problem Estimating geometric and spectral properties of Riemannian manifolds under Ricci curvature conditions.
method Using spectral and Kato conditions on Ricci curvature, the paper derives inequalities.
result The lowest eigenvalue of the Ricci tensor leads to finite fundamental group and isoperimetric inequalities.
A coupling method and an analytic one allow us to prove new lower bounds for the spectral gap of reversible diffusions on compact manifolds. Those bounds are based on the a notion of curvature of the diffusion, like the coarse Ricci curvature or the Bakry--Emery curvature-dimension inequalities. We show that when this …
The spectral curve correspondence for finite-type solutions of the sinh-Gordon equation describes how they arise from and give rise to hyperelliptic curves with a real structure. Constant mean curvature (CMC) 2-tori in S3 result when these spectral curves satisfy periodicity conditions. We prove that the spectra…
This note summarizes results that were obtained by the author in his habilitation thesis (arXiv:1607.08792) concerning the development of a spectral theory for simply periodic, 2-dimensional, complex-valued solutions of the sinh-Gordon equation. Spectral data for such solutions are defined for periodic Cauchy data on a…
Introduce Collapsed Effective Operators for higher-order structures.
problem Existing spectral operators decompose topology into separate ranks, leaving practitioners to fuse information back to vertices.
method Introduce Collapsed Effective Operators via Schur complementation of a graded Laplacian.
result Preserves positive semi-definiteness, lowers system energy under higher-order connectivity.
Positive entropy automorphisms have virtual eigenvalues outside the unit circle.
problem Understanding the dynamics of surface automorphisms with positive entropy.
method Investigating virtual homological eigenvalues of surface automorphisms.
result Existence of virtual eigenvalues outside the unit circle for automorphisms with positive entropy.
Study isotopy and concordance for metrics on manifolds with boundary.
problem Understanding isotopy and concordance for metrics on manifolds with boundary.
method Developed tools for studying isotopy and concordance, introduced variants of minimal concordance.
result Obtained a complete picture for surfaces, introduced new variants of concordance.
Constructs equivariant spectral flow for Dirac-type operators on manifolds.
problem Calculating spectral flow for Dirac-type operators on manifolds with group actions.
method Equivariant spectral flow construction for paths of Dirac-type operators on manifolds.
result Relates delocalised η-invariants and ρ-invariants for different positive scalar curvature metrics.
This work provides a computationally efficient and statistically consistent moment-based estimator for mixtures of spherical Gaussians. Under the condition that component means are in general position, a simple spectral decomposition technique yields consistent parameter estimates from low-order observable moments, wit…
Study on complex manifolds' Gauduchon metrics and spectral sequences under deformations.
problem Understanding Gauduchon metrics and spectral sequences in complex manifold deformations.
method Two approaches: partial degeneration of Frölicher spectral sequence and h-∂∂ˉ-property. result Introduction of a positivity cone and its lower semicontinuity under deformations.
Study optimizes inventory restocking for demand processes with exponential replenishment.
problem Optimizing inventory restocking for demand processes with exponential replenishment.
method Developed periodic barrier replenishment policies for spectrally positive Lévy demand processes.
result Optimal policies and value functions are concisely written in terms of scale functions.
Phase retrieval requires at least d+o(d) measurements to recover signals with high probability.
problem Recovering signals from quadratic measurements with noisy data.
method Used Gaussian sensing vectors and spectral methods to analyze the minimum number of measurements needed.
result A sharp phase transition occurs at n = d+o(d), where a simple spectral estimator achieves positive correlation.
We use spectral embeddings to give upper bounds on the spectral function of the Laplace--Beltrami operator on homogeneous spaces in terms of the volume growth of balls. In the case of compact manifolds, our bounds extend the 1980 lower bound of Peter Li for the smallest positive eigenvalue to all eigenvalues. We also i…
Geodesic rays and chordal distances link algebraic and geometric properties of positive metrics.
problem Understanding the geometry of the space of positive metrics at infinity.
method Using Monge-Ampère equations and test configurations, algebraic descriptions of geodesic rays and chordal distances are derived.
result The Mabuchi chordal distance between geodesic rays associated with ample test configurations equals the spectral distance between their filtrations.
This paper considers magnitude, asymptotics and duration of drawdowns for some Lévy processes. First, we revisit some existing results on the magnitude of drawdowns for spectrally negative Lévy processes using an approximation approach. For any spectrally negative Lévy process whose scale functions are well-behaved at …
Study optimal periodic dividend strategies for risky businesses with transaction costs.
problem Optimal periodic dividend strategies for spectrally positive Lévy risk processes with fixed transaction costs.
method Investigates periodic (bu,bl) strategies for a Poisson arrival process of decision times. result A periodic (bu,bl) strategy is optimal with lump sum dividends net of transaction costs.