Solves Dirichlet problem for specific PSH functions on Hermitian manifolds.
problem Solving Dirichlet problem for Monge-Ampère equation for (n−1)-PSH functions. method Deriving a quantitative boundary estimate under (n−1)-PSH subsolutions assumption. result Quantitative boundary estimate confirmed for specific manifolds.
Let (X,ω) be a compact Kähler manifold. We introduce and study the largest set DMA(X,ω) of ω-plurisubharmonic (psh) functions on which the complex Monge-Ampère operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set $PSH(X,\om)$ of all…
The paper proves a function extension on Kähler manifolds.
problem Proving a function extension on Kähler manifolds.
method Analyzing strictly psh functions on compact Kähler submanifolds.
result A strictly psh function on the whole manifold can be extended from a submanifold.
Smooth solutions found for complex Monge-Ampère flows.
problem Regulating smoothness in complex Monge-Ampère flows.
method Analyzing initial ω-psh functions with zero Lelong number. result General Monge-Ampère flow solution is immediately smooth.
Establishes Yau-Tian-Donaldson conjecture for weighted metrics.
problem Constant scalar curvature Kähler metrics on polarized projective manifolds.
method Extends Chi Li's work to weighted case, uses a priori estimates and slope formulas.
result Proves Yau-Tian-Donaldson conjecture for weighted extremal Kähler metrics.
We establish plurisubharmonicity of the envelope of Poisson and Lelong functionals on almost complex manifolds. That is, we generalize the corresponding results for complex manifolds and almost complex manifolds of complex dimension two. We also provide some applications to the regularization of J-plurisubharmonic func…
Geodesics in non-Archimedean metrics are continuous.
problem Understanding geodesics in spaces of non-Archimedean metrics.
method Maximal psh segments are geodesics, and continuity of these segments is proven.
result Maximal psh segments joining continuous psh metrics are continuous.
Study on Lee classes of complex surfaces, proving connectedness and bounds.
problem Understanding Lee classes of complex surfaces with LCS structures.
method Analyzing deRham classes of Lee 1-forms and using properties of PSH functions.
result Connectedness of Lee deRham classes and explicit negative upper bound on hyperbolic Kato surfaces.
Let X be a compact complex manifold equipped with a smooth (but not necessarily positive) closed form theta of one-one type. By a well-known envelope construction this data determines a canonical theta-psh function u which is not two times differentiable, in general. We introduce a family of regularizations of u, param…
Lecture notes on using non-Archimedean geometry for complex variety degenerations.
problem Complex algebraic variety degenerations with non-Archimedean Berkovich spaces.
method Hybrid spaces and non-Archimedean pluripotential theory.
result Relation between convergence of psh metrics and Monge-Ampere measures in hybrid spaces.
Study proves long-term solutions to a specific equation on hyperKähler manifolds.
problem Proving long-term existence and uniqueness of solutions to a parabolic quaternionic Monge-Ampère equation.
method Proved long-term existence and uniqueness using parabolic quaternionic Monge-Ampère type equation.
result Solution converges smoothly to the unique solution of the Monge-Ampère equation.
The paper proves boundedness of envelopes in complex manifolds.
problem Regularity of envelopes in complex manifolds.
method Analyzes bounded functions and their envelopes in the context of cohomology classes and Laplacians.
result The α-psh envelope P(f) is locally bounded with locally bounded Laplacian on the ample locus of {α}. Study geodesic distances and convexity in contact sets.
problem Understanding geodesic distances and convexity in contact sets.
method Extending results on quasi-psh functions and big cohomology classes, studying Monge-Ampère measures on contact sets.
result Convexity of the K-energy in big and nef cohomology classes.
Study rigid motions in 3D-Heisenberg group to define and relate geometric probabilities.
problem Define geometric quantities and probabilities in the 3D-Heisenberg group.
method Define density and measure for horizontal lines, use integral geometry.
result Show relationship between geometric probability and natural geometric quantity.
Proves existence and uniqueness of solutions to a quaternionic Monge-Ampère equation.
problem Solving the quaternionic Monge-Ampère equation for (n−1)-quaternionic plurisubharmonic functions on a hyperKähler manifold. method Proves existence and uniqueness of solutions using a Cherrier-type inequality and C1 and C2 estimates. result Obtains smooth solutions to the quaternionic Monge-Ampère equation.
We study degenerate complex Monge-Ampère equations on a compact Kähler manifold (X,ω). We show that the complex Monge-Ampère operator (ω+ddc⋅)n is well-defined on the class E(X,ω) of ω-plurisubharmonic functions with finite weighted Monge-Ampère energy. The class E(X,ω) is the la…
Defines plurisubharmonic metrics on hybrid spaces and proves their canonical extensions.
problem Defining and analyzing plurisubharmonic metrics on hybrid spaces.
method Introduces a class of plurisubharmonic metrics on hybrid spaces and proves their canonical extensions.
result Canonical plurisubharmonic extensions of metrics on hybrid spaces are continuous and can be described in terms of canonical models.
Extends complex manifold structures to line bundles, revealing new projective manifolds.
problem Generalizing scalar-valued holomorphic structures to line bundles.
method Study of holomorphic p-contact and s-symplectic structures on complex manifolds with line bundles. result Holomorphic p-contact and s-symplectic manifolds can be projective. Study pluri-subharmonic envelopes and solve complex Monge-Ampère equations.
problem Solving complex Monge-Ampère equations on compact Kähler manifolds and domains.
method Approximation process and lower envelopes of super-solutions.
result Quasi-psh envelope of a viscosity super-solution is a pluripotential super-solution.
We show that the complex Monge-Ampere equation on a compact Kaehler manifold (X,ω) of dimension n admits a Holder continuous omega-psh solution if and only if its right-hand side is a positive measure with Holder continuous super-potential. This property is true in particular when the measure has locally Holder continu…
Proves properties of full mass currents in big cohomology classes.
problem Characterizing and understanding full mass currents in big cohomology classes.
method Uses a characterization of full mass currents in terms of the envelope of their singularity type and develops the theory of weak geodesics in big cohomology classes.
result Shows the inclusion of certain sets of functions and currents and characterizes big classes with additive full mass currents.
Generalizes mean-value inequality to orbifold setting.
problem Mean-value inequality for orbifold setting.
method Generalizes fundamental results in Kähler geometry to orbifolds.
result Shows mean-value inequality is insensitive to quotient singularities.
New metric measures singularity types on complex manifolds.
problem Measuring and comparing singularity types on complex manifolds.
method Introduced a pseudometric on the space of singularity types.
result The metric space of singularity types is complete in the presence of positive mass.
Paper solves complex Monge-Ampère equation on almost Hermitian manifolds.
problem Solving Dirichlet problem for complex Monge-Ampère equation.
method Properties of subsolutions for fully nonlinear elliptic equations.
result Existence of C2-smooth strictly J-plurisubharmonic subsolution. Proves C1,1 regularity for complex Monge-Ampère equations and geodesic rays.
problem Complex Monge-Ampère equations on compact Kähler manifolds with degenerate cohomology.
method Proves C1,1 estimate for solutions. result Local C1,1 regularity of geodesic rays and quasi-psh envelopes. Suppose (X,ω) is a compact Kähler manifold. Following Mabuchi, the space of smooth Kähler potentials H can be endowed with a Riemannian structure, which induces an infinite dimensional path length metric space (H,d). We prove that the metric completion of (H,d) can be identified with …
The paper studies strong topologies for complex Monge-Ampère equations on Kähler manifolds.
problem Analyzing strong topologies for complex Monge-Ampère equations on Kähler manifolds.
method Proving the Monge-Ampère operator is a homeomorphism between finite energy potentials and energy measures with their strong topologies.
result The Monge-Ampère operator produces an homeomorphism between sets of finite energy potentials and measures on Kähler manifolds.
Locally homogeneous aspherical Sasaki manifolds are quasi-regular S1-Seifert bundles.
problem Characterizing locally homogeneous aspherical Sasaki manifolds.
method Analyzing homogeneous Sasaki manifolds and their quotients by discrete subgroups.
result Compact locally homogeneous aspherical Sasaki manifolds are quasi-regular S1-Seifert bundles.
Metric study on Kähler manifolds with prescribed singularities.
problem Defining a metric space for Kähler potentials with prescribed singularities.
method Introducing a distance d and dA on the relative finite energy class and showing convergence. result The space XA is complete and converges in Gromov-Hausdorff sense. Study of non-archimedean μ-entropy and its connection to K-stability.
problem Understanding K-stability in non-archimedean settings.
method Introducing non-archimedean μ-entropy and its properties, connecting it to K-semistability.
result Established a criterion for K-semistability without vector ξ, using the non-archimedean μ-entropy.
New neural network models for complex functional data analysis.
problem Complex relations between functional predictors and responses.
method Function-on-Function regression models using neural networks with continuous hidden layers.
result Demonstrated power and flexibility in handling complex functional models.
Knot signature function defined and conditions for its existence are given.
problem Defining and characterizing the signature function of knots.
method Presentation of necessary and sufficient conditions for a function to be a knot signature function.
result Conditions for a function to be the signature function of a knot are established.
Distance function to a finite set is a topological Morse function.
problem Characterizing the topological Morse function of a finite set.
method Analyzing the distance function to a finite set in \(\mathbb{R}^n\).
result Distance function is a topological Morse function, with precise critical points and indices.
Introduces new weighted floating functions and affine surface areas.
problem Developing new mathematical concepts for convex bodies.
method Introducing weighted floating functions and weighted functional affine surface areas.
result New relations to traditional and classical affine surface areas.
Develops methods for selecting and estimating smooth functional coefficients in high-dimensional multivariate functional data.
problem Functional predictor selection and estimation of smooth functional coefficients in high-dimensional multivariate functional data.
method Functional group-sparse regression methods in a generic Hilbert space of infinite dimension.
result Consistency of estimation and selection (oracle property) under infinite-dimensional Hilbert spaces.
The paper introduces geodesic φ-convex functions and their properties.
problem Generalizing geodesic functions to φ-convex functions.
method Introducing geodesic φ-convex functions and investigating their properties.
result Characterization of geodesic φ-convex functions via their φ-epigraphs.
Neural networks can approximate functionals on RKHS with error bounds.
problem Approximating functionals on RKHS using neural networks.
method Interpolating orthogonal projections in RKHS using point evaluations.
result Explicit error bounds for various kernels (inverse multiquadric, Gaussian, Sobolev).
FFBO optimizes functions as inputs and outputs, improving on existing BO methods.
problem Optimizing functions as both inputs and outputs in complex systems.
method Function-on-function Gaussian process (FFGP) model with a separable operator-valued kernel, scalar upper confidence bound (UCB) acquisition function, and scalable functional gradient ascent algorithm (FGA).
result FFBO outperforms existing methods in synthetic and real-world data.
Analyzes properties of transnormal Finsler functions on compact manifolds.
problem Properties of transnormal Finsler functions on compact manifolds.
method Analyzes critical level sets and partition properties of transnormal functions.
result Critical level sets of an analytic transnormal function are submanifolds, and the partition of M into level sets is a Finsler partition. The study explores the Dehn functions of Kähler groups and their properties.
problem Which functions can arise as Dehn functions of Kähler groups?
method Analyzes examples of Kähler groups with various Dehn functions and proves the existence of a Kähler group with a cubic bounded Dehn function.
result There exists a Kähler group with a cubic bounded Dehn function and an exponential upper bound.
Chirped sinosoids and interferometric phase plots are functions that are not periodic, but are the composition of a smooth function and a periodic function. These functions functions factor into a pair of maps: from their domain to a circle, and from a circle to their codomain. One can easily imagine replacing the circ…
The Fridman function is bounded by the injectivity radius for certain hyperbolic manifolds.
problem Bounding the Fridman function for hyperbolic manifolds.
method Analyzing the relationship between the Fridman function and the injectivity radius function.
result The Fridman function is bounded above by the injectivity radius function for certain hyperbolic manifolds.
Optimally estimates a functional using nuisance function tuning and sample splitting.
problem Estimating optimal rates for a doubly robust functional.
method Combines nuisance function tuning and sample splitting strategies.
result Shows optimal rates of convergence for various estimators.
The paper extends mixability theory to function-valued forecasts, proving various loss functions are mixable.
problem Efficient aggregation of functional and probabilistic forecasts in online prediction games.
method Adapting mixable and exponentially concave loss functions to function-valued forecasts.
result Various loss functions used for probabilistic forecasting are mixable (exp-concave).
The paper proves isoparametric functions on Finsler space forms under specific conditions.
problem Understanding isoparametric functions in Finsler space forms.
method Proving transnormal functions as isoparametric functions and constructing global and local isoparametric functions using the distance function.
result Generalization of Theorem B to Finsler space forms.
Paper introduces a nonparametric functional graphical model for random functions.
problem Estimating probabilistic conditional independence in functional graphical models.
method Functional sufficient dimension reduction to relax Gaussian or copula Gaussian assumptions.
result Enhances estimation accuracy and retains probabilistic conditional independence.
Robustifies elicitable functionals to handle small distribution misspecifications.
problem Determining uniquely optimal forecasts under distributional misspecification.
method Integrates statistical robustness into elicitable functionals using Kullback-Leibler divergence.
result Robust elicitable functionals admit unique solutions at the boundary of uncertainty regions.
Deep neural networks with various activation functions can approximate Hölder smooth functions.
problem Expressivity of deep neural networks with general activation functions.
method Investigates approximation ability of deep neural networks with a broad class of activation functions, including Hölder smooth functions.
result Derives the required depth, width, and sparsity of deep neural networks to approximate Hölder smooth functions.