Study shows various weak solutions to complex flows match, proving viscosity equals pluripotential.
problem Comparing weak solutions to complex Monge-Ampère flows.
method Examined various notions of weak subsolutions and showed they coincide.
result Viscosity solution equals pluripotential solution.
Continuity of complex Monge-Ampère potentials on Kähler manifolds.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler manifolds.
method Extending DiNezza-Lu's approach to big cohomology classes, proving continuity on Zariski open sets.
result Singular Kähler-Einstein metrics have continuous potentials on the ample locus outside of the non-klt part.
Study solves complex equation on specific types of manifolds.
problem Solving complex Monge-Ampère equation on Kähler manifolds.
method Flow-based arguments to establish existence of smooth solutions.
result Existence of smooth solutions under decreasing right-hand side.
The paper constructs invariant Calabi-Yau structures on complexified symmetric spaces.
problem Constructing invariant Calabi-Yau structures on complexified symmetric spaces.
method Solutions of a Monge-Ampère type equation.
result Existence of solutions to the Monge-Ampère type equation.
Stability proven for complex equations on Kähler manifolds.
problem Stability of solutions to complex Monge-Ampère equations.
method Elliptic and parabolic complex Monge-Ampère equations on compact Kähler manifolds.
result Stability result applies to Kähler-Ricci flow.
Develops theory for Kähler-Ricci flow on singular varieties.
problem Analyzing Kähler-Ricci flow on varieties with log terminal singularities.
method Parabolic pluripotential theory and complex Monge-Ampère equations.
result Establishes a parabolic theory analogous to Bedford-Taylor's.
We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a G×G-equivariant Fano compactification of a complex connected reductive group G in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the …
Develops parabolic pluripotential theory for complex flows.
problem Complex Monge-Ampère equations in degenerate settings.
method Study of semi-concave envelopes and unique solutions.
result Shows semi-concave envelopes as unique solutions.
Study complex Monge-Ampère flows on Kähler manifolds using Perron method.
problem Complex Monge-Ampère flows in big cohomology classes.
method Perron method for pluripotential subsolutions.
result Upper envelope of subsolutions is a unique pluripotential solution with regularity.
N. V. Efimov \cite{Ef1} proved that there is no complete, smooth surface in R3 with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if M3 has sectional curvature between two constants K2 and K3, then there exists K1<min(K2,0) such that $M…
New methods solve complex equations, proving some have solutions.
problem Solving complex equations in infinite dimensions.
method Infinite-dimensional prequantum line bundles and moment maps.
result Proves some perturbed equations have solutions for small parameters.
Solves complex Monge-Ampère equations on Kähler manifolds.
problem Behavior of singularities in solutions to degenerate equations.
method Analyzes singularities of solutions to degenerate complex Monge-Ampère equations.
result Resolves unresolved problem from Yau's work.
Method solves high-dimensional nonlinear PDEs using neural networks.
problem Solving high-dimensional fully nonlinear PDEs.
method Backward induction with multi-layer neural networks to estimate solution and its gradient, with Hessian approximated by automatic differentiation.
result Method extends previous work on semi-linear PDEs to fully nonlinear cases, demonstrating accuracy on various examples.
Develops Kähler geometry on new varieties for canonical metrics.
problem No specific problem stated; focuses on new varieties.
method Introduces new varieties, develops Kähler geometry, associates convex functions with metrics.
result Provides expression for Mabuchi functional and combinatorial sufficient condition of properness.
The paper studies mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.
problem Mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.
method Analyzes the parabolic equation and Monge-Ampère type equation, proving smooth solutions and convergence to self-expanding solutions.
result Smooth solutions u(x,t) for specific nonlinear equations and convergence to self-expanding solutions. The grid integration of intermittent Renewable Energy Sources (RES) causes costs for grid operators due to forecast uncertainty and the resulting production schedule mismatches. These so-called profile service costs are marginal cost components and can be understood as an insurance fee against RES production schedule u…
In this paper, we investigate two hyperbolic flows obtained by adding forcing terms in direction of the position vector to the hyperbolic mean curvature flows in \cite{klw,hdl}. For the first hyperbolic flow, as in \cite{klw}, by using support function, we reduce it to a hyperbolic Monge-Ampeˋre equation …
The paper re-evaluates eigenvalue estimates and rigidity of Poincare-Einstein metrics.
problem Eigenvalue estimates and rigidity of Poincare-Einstein metrics on spin manifolds.
method Revisits eigenvalue estimates of the Dirac operator and proves rigidity results under weaker conditions.
result Poincaré-Einstein metrics are rigid under specific eigenvalue conditions.
We study conformal Spin-subgeometry of submanifolds in a semi-Riemannian Spin-manifold, focusing on conformal Spin-manifolds (M,[h]) and their Poincaré-Einstein metrics (X,g+). Our approach is based on the spectral theory of Dirac operator in the ambient Spin-manifold, and associated spinor valued meromorp…
This paper re-visits the spectral method for learning latent variable models defined in terms of observable operators. We give a new perspective on the method, showing that operators can be recovered by minimizing a loss defined on a finite subset of the domain. A non-convex optimization similar to the spectral method …
It is natural to ask: what kinds of matrices satisfy the Restricted Eigenvalue (RE) condition? In this paper, we associate the RE condition (Bickel-Ritov-Tsybakov 09) with the complexity of a subset of the sphere in Rp, where p is the dimensionality of the data, and show that a class of random matrices with indep…
In this paper we provide a review of asymptotic results of Toeplitz operators and their applications in TQFT. To do this we review the differential geometric construction of the Hitchin connection on a prequantizable compact symplectic manifold. We use asymptotic results relating the Hitchin connec- tion and Toeplitz o…
Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re(f) is Morse, we prove the existence of a stratified gradient-like vector field for Re(f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimensi…
Study of free particle's geometry and its perturbations using complex projective structures.
problem Understanding the geometry of a free particle and its perturbations.
method Use of complex projective structures and quasiconformal geometry to study perturbations.
result Main results loosely modeled on algebraic transformation theory, foundational for geometric understanding of the exact WKB method.
Improved DeepONets for PDE solution operators with adaptive re-weighting and new architecture.
problem Training DeepONets for PDE solution operators without paired data.
method Adaptive re-weighting of training examples and novel network architecture.
result Consistently improved predictive accuracy by a factor of 10-50x.
Let U2(H) be the Banach-Lie group of unitary operators in the Hilbert space H which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit {upu∗:u∈U2(H)}, of an infinite projection p in H. This orbit coincides with t…
A new framework for recycling Gaussian process approximations.
problem Efficiently combining multiple Gaussian process approximations.
method Construct variational ensembles using a dictionary of fitted Gaussian processes.
result Framework allows for various tasks and scalability.
Adaptive ML learns complex time-varying systems without new data.
problem Applying ML to time-varying systems with shifting distributions.
method Mapping high-dimensional inputs to low-dimensional latent space, actively tuning latent space based on feedback.
result Learning correlations and tracking system evolution in real-time without new data.
Paper shows re-solving heuristics have constant regret for price-based revenue management.
problem Optimal pricing policies for revenue management with time constraints.
method Proves re-solving heuristics have O(1) regret compared to optimal policies. result Improved regret bound to O(1) from O(lnT), complemented by Ω(lnT) gap with fluid model. Model for dynamic pricing across multiple RE groups to maximize revenue.
problem Maximizing revenue from multiple RE pricing groups.
method Mathematical model incorporating multiple pricing groups, revenue goals, and time value of money.
result Algorithm for constructing a pricing policy for multiple RE groups.
Physics-informed neural networks and neural operators speed up solving parametric PDEs by orders of magnitude.
problem Solving PDEs for varying parameters is computationally expensive.
method Physics-informed neural networks and neural operators learn solution mappings across parameter spaces.
result Neural operators achieve computational speedups of 10^3 to 10^5 times faster than traditional methods.
An on-going debate in the energy economics and power market community has raised the question if energy-only power markets are increasingly failing due to growing feed-in shares from subsidized renewable energy sources (RES). The short answer to this is: No, they are not failing. Energy-based power markets are, however…
One-step Bellman alignment improves online RL by reducing task mismatch.
problem Online RL struggles with task similarity defined by rewards or transitions.
method One-step Bellman alignment and re-weighted targeting (RWT) to correct task mismatch.
result Regret bounds show task shift complexity, not target MDP, affects performance.
A monitoring procedure improves machine learning forecasts for digital platforms.
problem Maintaining accurate and stable forecasts for data streams at digital platforms.
method Developed a monitoring procedure to determine when to retrain machine learning algorithms.
result Monitor-based retraining produces accurate forecasts compared to benchmarks.
FreDN separates trends and periodicities in non-stationary time series forecasts.
problem Spectral entanglement and computational burden in frequency-domain methods for non-stationary time series.
method FreDN introduces a learnable Frequency Disentangler module to separate trend and periodic components directly in the frequency domain, and uses a ReIm Block to reduce complexity.
result FreDN outperforms state-of-the-art methods by up to 10% on long-term forecasting benchmarks.
Study examines the geometry of a group of Fourier-integral operators related to Diff(S^1).
problem Investigating the geometry of a specific group of Fourier-integral operators.
method Defined a right-invariant pseudo-Riemannian metric on the group using renormalized traces of pseudo-differential operators.
result Extended the Hilbert-Schmidt Riemannian metric to the group.
Study on colored Jones polynomial of figure-eight knot for complex parameters.
problem Asymptotic behavior of colored Jones polynomial for figure-eight knot.
method Analyzing the asymptotic growth rate of the polynomial for complex parameters with small imaginary part.
result Growth rate of polynomial is related to the Chern-Simons invariant for large real part of the parameter and to the reciprocal of Alexander polynomial for small real part.
A new method for real-time anomaly detection in flight data.
problem Challenges in clustering dynamically growing flight data for anomaly detection.
method Incremental Gaussian Mixture Model (GMM) using EM algorithm.
result Significantly reduced processing time and memory usage compared to offline methods.
Study reduces complexity and uncertainty in human atrial cell models.
problem Uncertainty in parameter estimates from gating kinetics models.
method Approximate Bayesian computation to re-calibrate models, investigate two approaches: more complete datasets and less complex formulations.
result Less complex model with fewer parameters gives better fit and lower uncertainty.
SteinGen generates diverse graph samples from a single example.
problem Generating graphs with characteristic structures and diversity from a single example.
method Combines Stein's method and MCMC with Glauber dynamics and re-estimation of the Stein operator.
result High distributional similarity to the original data, combined with high sample diversity.
Study eigenvalues of Laplace operator on specific 3D manifolds under Ricci flow.
problem Analyze eigenvalues of Laplace operator with potential under backward Ricci flow.
method Use backward Ricci flow on locally homogeneous 3-manifolds, derive bounds and convergence results.
result Eigenvalue λ+(t) approaches zero as flow converges to sub-Riemannian geometry. The paper studies a special Grassmannian space and shows it's an orbit of a unitary group.
problem Investigating a specific Grassmannian space of infinite-dimensional subspaces.
method Analyzing the restricted p-Schatten class Grassmannian and showing it's an affine coadjoint orbit of a unitary group. result The restricted p-Schatten class Grassmannian is shown to be an affine coadjoint orbit of an infinite-dimensional restricted unitary group. This paper re-examines Bregman functions and their divergences, introducing new properties and functions.
problem Exploring properties and applications of Bregman functions and divergences.
method Re-examination of existing Bregman functions and introduction of new ones, providing sufficient conditions for construction.
result Several known Bregman functions are reclassified, and new Bregman functions are introduced.
We propose a new proximal, path-following framework for a class of constrained convex problems. We consider settings where the nonlinear---and possibly non-smooth---objective part is endowed with a proximity operator, and the constraint set is equipped with a self-concordant barrier. Our approach relies on the followin…
Speed up neural networks by 2x with 5% mAP loss.
problem High computational cost of neural network forward passes.
method Deep Learning Approximation: lossless and lossy optimizations.
result 2x speedup in network forward pass with 5% mAP drop.
First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept o…
For a Dirac operator Dgˉ over a spin compact Riemannian manifold with boundary (Xˉ,gˉ), we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of harmonic spinors on Xˉ, and we analyze their Schwartz kernels. Our approach is based on th…
Scalable NAS by factorizing operators into subspaces.
problem Scaling up NAS search space while avoiding operator competition.
method Factorizing a large set of candidate operators into smaller subspaces.
result Achieved state-of-the-art performance on CIFAR10 and ImageNet.