Develops trace class operators and inverse Laplacian theory for infinite dimensions.
problem Understanding trace class operators and inverse Laplacian on infinite dimensional spaces.
method Presentation of trace class operators and construction of inverse Laplacian on closed manifolds.
result Original trace computations involving the inverse Laplacian on the torus.
This work generalizes Log-Determinant divergences to infinite-dimensional settings.
problem Generalizing Log-Determinant divergences to infinite-dimensional spaces.
method Introducing a parametrized family of divergences, Alpha-Beta Log-Determinant divergences, for positive definite unitized trace class operators.
result The Alpha-Beta Log-Det divergences encompass various divergences and metrics, including the affine-invariant Riemannian distance and symmetric Stein divergence.
The paper studies the spectrum of Laplace-Beltrami operators on complex spaces.
problem Analyzing the spectrum of Laplace-Beltrami operators on compact complex spaces.
method Examined the Friedrichs extension of Laplace-Beltrami and Hodge-Kodaira Laplacians, providing estimates for eigenvalues and trace-class properties.
result Discrete spectrum and trace-class properties of Laplace-Beltrami operators on compact complex spaces.
Formula for index of Dirac-type operators on stratified spaces.
problem Calculating the index of Dirac-type operators on complex geometric structures.
method Defined a closed domain, proved self-adjoint and Fredholm properties, established index formula.
result Proved a formula for the Chern character of the index of Dirac-type operators.
New BNN architectures reduce computational cost for uncertainty quantification.
problem High computational cost in Bayesian neural networks.
method Partial trace-class Bayesian neural networks (PaTraC BNNs).
result Comparable uncertainty quantification with fewer parameters.
Bayesian inference for deep neural networks using trace-class priors and MLMC.
problem Efficient Bayesian inference for deep neural networks.
method Trace-class neural network priors and Multilevel Monte Carlo method.
result Optimal computational complexity for Bayesian inference of TNN models.
A cocycle Ω:P×G→H taking values in a Lie group H for a free right action of G on P defines a principal bundle Q with the structure group H over P/G. The Chern character of a vector bundle associated to Q defines then characteristic classes on X. This observation becomes useful in the case …
Abstract: Determinants and formulas for operators on various spaces.
problem Determinants and formulas for operators on different algebras and spaces.
method Use of Poincaré type determinants, invariant operators, and full matrix-symbols.
result Explicit formulas for determinants of elliptic operators and periodic pseudo-differential operators.
Infinite-dimensional SBDMs improve image generation across multiple resolutions.
problem Efficient image generation at high resolutions and across different levels.
method Developed SBDMs in infinite-dimensional setting, using trace class operators and operator networks.
result Improved efficiency and generalization across resolution levels.
Study of eta invariant for non-compact manifolds via Dirac-type operators.
problem Defining and studying the relative eta invariant for non-compact manifolds.
method Defined the relative eta function and studied its variation and gluing law.
result Shows the relative eta invariant coincides with a previously defined version.
The paper studies Lévy processes on compact manifolds, proving properties of their semigroups.
problem Analyzing Lévy processes on compact Riemannian manifolds.
method Proving properties of Feller semigroups and generators on Lp spaces. result The generator has a discrete spectrum of eigenvalues and the semigroup is trace-class when the process has a non-trivial Brownian part.
A new model captures forward curve dynamics with stochastic volatility.
problem Modeling continuous-time evolution of forward curves in financial markets.
method Affine stochastic volatility model with modulated dynamics.
result Model allows for maturity-specific risk and volatility clustering.
The paper introduces Causal Neural Operators to approximate operators in stochastic analysis.
problem Leveraging temporal structure in non-linear operators for deep learning models.
method Designing a deep learning model framework for infinite-dimensional linear metric spaces.
result Causal Neural Operators can uniformly approximate Hölder or smooth trace class operators.
Let V⊂CPn be an irreducible complex projective variety of complex dimension v and let g be the Kähler metric on $\reg(V)$, the regular part of V, induced by the Fubini Study metric of CPn. In this setting Li and Tian proved that $W^{1,2}_0(\reg(V),g)=W^{1,2}(\reg(V…
Study on geometric Jensen-Shannon divergence for Gaussian measures in Hilbert space.
problem Computing divergence between Gaussian measures in infinite-dimensional Hilbert space.
method Closed form expression and regularization for divergence calculation.
result Closed form expression and regularization for Geometric Jensen-Shannon divergence.
Maps with many singularities found in complex space.
problem Constructing maps with singularities in complex space.
method Created a map with infinitely many Schoen-Wolfson singularities on a disc.
result Found a Ck map with smooth trace in C2. The paper addresses statistical consistency in functional flow matching with rigorous mathematical proofs.
problem Statistical consistency in functional flow matching under scattered or adaptive refinement.
method Strong L2 convergence of finite conditional velocity targets for every strongly consistent sequence of finite-rank reconstructions. result Proves strong L2 convergence with quantitative bounds for orthogonal projections and point-sensor extensions. Let (X,h) be a compact and irreducible Hermitian complex space of complex dimension m. In this paper we are interested in the Dolbeault operator acting on the space of L2 sections of the canonical bundle of reg(X), the regular part of X. More precisely let $\overline{\mathfrak{d}}_{m,0}:L^2Ω^{m,0}(reg(X),h)\…
This paper solves nonparametric estimation of continuous DPPs using kernel methods.
problem Estimating continuous Determinantal Point Processes (DPPs) without assuming a parametric form.
method Developed a fixed point algorithm based on a representer theorem for nonnegative functions in RKHS.
result Demonstrated a finite-dimensional problem for nonparametric MLE of continuous DPPs.
Study convergence and approximations of entropic regularized Wasserstein distances for Gaussian and RKHS measures.
problem Convergence and approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
method Analysis of convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
result Strictly weaker convergence in 2-Sinkhorn divergence for Gaussian measures compared to exact 2-Wasserstein distance.
The paper advances U-statistics in dependent settings, improving spectral estimation and goodness-of-fit tests.
problem Non-asymptotic analysis of U-statistics in dependent Markov chain settings.
method Proved new concentration and exponential inequalities for U-statistics, applied to spectral estimation, online algorithms, and goodness-of-fit tests.
result Established new results for spectral estimation, online algorithms, and goodness-of-fit tests in Markov chain settings.
Logarithmic invariant for restricted quantum sl(2) constructed.
problem Constructing a logarithmic invariant for a specific quantum group.
method Combining a universal invariant and a modified trace, defined for a 3-manifold and link.
result A new logarithmic invariant for restricted quantum sl(2) at a 2p-th root of unity.
Develops hypothesis tests for conditional distributions using learning-theoretic bounds.
problem Testing differences in conditional distributions and functionals.
method Transforming learning-theoretic bounds into hypothesis tests for conditional expectations.
result Establishes comprehensive foundation for conditional testing, including theoretical guarantees and practical implementations.
New Gaussian priors for neural networks improve scalability and Bayesian inference stability.
problem Scalability and stability issues in Bayesian neural network inference.
method Introduces a new Gaussian neural network prior with decreasing variance in network width, enabling stable MCMC sampling.
result The new prior enables stable MCMC sampling for Bayesian neural network inference, improving scalability and stability.
This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slovák has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for …
New boundary operators for sixth-order GJMS operator on manifolds.
problem Developing boundary operators for sixth-order GJMS operator.
method Conformally covariant boundary operators and fractional GJMS operators.
result New realization of fractional GJMS operators and Sobolev trace inequalities.
The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.
problem Eigenvalue problems and heat trace asymptotics for different operators.
method Establishes connections and inequalities for eigenvalues and heat traces.
result Eigenvalue inequalities and three-term asymptotic formulas for heat traces of various operators.
Introduces a new elliptic operator with positive eigenvalue.
problem None explicitly stated in the abstract.
method Introduces a new elliptic operator called the two-radical Laplace operator.
result The eigenvalue of the new operator is the positive square root of the Laplace operator's eigenvalue.
Proves Kato inequalities for various conformal operators.
problem Proving inequalities for differential operators.
method Analyzes a class of first order differential operators, including Dirac and Penrose twistor operators.
result Derives Kato inequalities that interpolate between classical and refined versions.
We describe a set of conformally covariant boundary operators associated to the Paneitz operator, in the sense that they give rise to a conformally covariant energy functional for the Paneitz operator on a compact Riemannian manifold with boundary. These operators naturally give rise to a first- and third-order conform…
Study on biharmonic hypersurfaces with specific recurrent operators in Euclidean space.
problem Characterizing biharmonic hypersurfaces with recurrent operators.
method Analysis of various recurrent operators and their impact on biharmonic hypersurfaces.
result Some well-known recurrent operators play a significant role in making biharmonic hypersurfaces minimal.
Study the heat operator of a transversally elliptic operator on Lie groups.
problem Spectral properties and convergence of a heat operator on Lie groups.
method Review spectral properties, define character, estimate heat operator convergence.
result Estimate of fα(t) determines convergence of the character. Local index theorem for chiral geometric operators proved using heat kernel.
problem Proving a local index theorem for geometric first-order differential operators.
method Using Gilkey's invariance theory and heat kernel techniques.
result Supertrace of heat kernel converges to Chern-Weil form.
Study estimates eigenvalues for concave Hessian operators on convex domains.
problem Estimating eigenvalues for concave elliptic Hessian operators.
method Investigates Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators.
result Existence and properties of the first nonzero eigenvalue and eigenfunction.
The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.
problem Spaces of unbounded Fredholm operators and their properties.
method Analyzing the spaces and proving homotopy equivalences.
result Natural maps between four spaces of unbounded Fredholm operators are homotopy equivalences.
Standard Laplace operator extends Hodge and Casimir operators to broader geometric contexts.
problem Extending Laplace operator to vector bundles and Riemannian manifolds.
method Functorial approach, showing commutation with homomorphisms and differential operators.
result Standard Laplace operator commutes with a wide range of differential operators.
GJMS operators connect geometry, analysis, and physics.
problem None explicitly stated; focus on operators and their impact.
method Construction of conformally invariant differential operators.
result GJMS operators have significant impact in geometry, analysis, and physics.
Extends Calabi operator to Riemannian locally symmetric spaces.
problem Local integrability conditions on Riemannian locally symmetric spaces.
method Generalizes Calabi operator to Riemannian locally symmetric spaces.
result Generalised operator works in irreducible case and fails in products.
Method uses neural networks to fit nonlinear operators from data.
problem Finding nonlinear integro-differential operators from data.
method Parametrizes spatial operator with neural networks and Fourier transforms.
result Can recover spatial operators in fractional heat and Kuramoto-Sivashinsky equations.
Mathai, Melrose, and Singer compute the index of projective elliptic operators.
problem Computing the index of projective elliptic operators on manifolds with Azumaya bundles.
method Equivariant index of transversally elliptic operators as pullbacks of projective elliptic operators.
result Comprehensive fractional index formula for projective elliptic operators.
New spectral torsion defined for rescaled Dirac operators.
problem Defining spectral torsion for rescaled Dirac operators.
method Using three vector fields and noncommutative residue.
result Computed spectral torsion for one form rescaled Dirac operators.
Researchers create new operators from Riemannian invariants.
problem Developing new mathematical tools for Riemannian geometry.
method Introducing formally self-adjoint conformally covariant polydifferential operators.
result Found a fourth-order, conformally covariant tridifferential operator.
Constructs conformal boundary operators and fractional Laplacians.
problem Developing conformally invariant boundary operators and fractional Laplacians.
method Constructs continuously parametrised families of conformally invariant boundary operators on densities.
result Constructs odd-order conformally invariant fractional Laplacian pseudo-differential operators.
Proves formal self-adjointness of certain differential operators.
problem Verifying conjectures about differential operators.
method Proving formal self-adjointness through mathematical proof.
result Proves two conjectures about differential operators.
Study describes how operator properties depend on smoothness on surfaces.
problem Understanding operator properties on surfaces with Morse-Smale diffeomorphisms.
method Analyzes pseudodifferential operators and shift operators on closed smooth surfaces.
result Fredholm property of operators depends on Sobolev smoothness exponent.
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.
Extends index theorem to uniformly elliptic operators on manifolds.
problem Generalizing index theorem to uniformly elliptic operators.
method Local index theorem on manifolds of bounded geometry.
result Validates multigraded elliptic uniform pseudodifferential operators.
Study of Dirac-like operators on spin manifolds with large mass parameters.
problem Understanding spectra of Dirac-like operators with piecewise constant mass terms.
method Analysis of asymptotic regimes to derive effective operators.
result Extension of MIT Bag operator concept to spin geometry.