arXiv research

We compare various notions of weak subsolutions to degenerate complex Monge-Amp{è}re flows, showing that they all coincide. This allows us to show that the viscosity solution coincides with the envelope of pluripotential subsolutions. Dedicated to Duong Hong Phong on the occasion of his 65th birthday.

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

Study solves complex equation on specific types of manifolds.

The paper constructs invariant Calabi-Yau structures on complexified symmetric spaces.

We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.

We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log te…

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural ass…

We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a $G\times G$-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 …

Study complex Monge-Ampère flows on Kähler manifolds using Perron method.

N. V. Efimov \cite{Ef1} proved that there is no complete, smooth surface in $\R^3$ with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if $M^3$ has sectional curvature between two constants $K_2$ and $K_3$, then there exists $K_1 < \min(K_2, 0)$ such that $M…

Solves complex Monge-Ampère equations on Kähler manifolds.

We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, while the Hessian is approximated by automatic differentiation of the gradient…

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop K{ä}hler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin-Abreu-D…

We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the g…

The paper studies mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.

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-Amp$\grave{\rm{e}}$re equation …

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…

We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption …

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 $\R^p$, where $p$ is the dimensionality of the data, and show that a class of random matrices with indep…

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…

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…

Study of free particle's geometry and its perturbations using complex projective structures.

Improved DeepONets for PDE solution operators with adaptive re-weighting and new architecture.

Let $U_2({\cal H})$ be the Banach-Lie group of unitary operators in the Hilbert space ${\cal H}$ which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit $\{upu^*: u\in U_2({\cal H})\},$ of an infinite projection $p$ in ${\cal H}$. This orbit coincides with t…

A new framework for recycling Gaussian process approximations.

Adaptive ML learns complex time-varying systems without new data.

Paper shows re-solving heuristics have constant regret for price-based revenue management.

Model for dynamic pricing across multiple RE groups to maximize revenue.

Physics-informed neural networks and neural operators speed up solving parametric PDEs by orders of magnitude.

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.

A monitoring procedure improves machine learning forecasts for digital platforms.

Study examines the geometry of a group of Fourier-integral operators related to Diff(S^1).

FreDN separates trends and periodicities in non-stationary time series forecasts.

Study on colored Jones polynomial of figure-eight knot for complex parameters.

A new method for real-time anomaly detection in flight data.

Study reduces complexity and uncertainty in human atrial cell models.

The paper studies a special Grassmannian space and shows it's an orbit of a unitary group.

Study eigenvalues of Laplace operator on specific 3D manifolds under Ricci flow.

SteinGen generates diverse graph samples from a single example.

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…

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 $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$, we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of harmonic spinors on $\bar{X}$, and we analyze their Schwartz kernels. Our approach is based on th…

The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the "Bregman function"). Bregman functions and divergences have been extensively investigated during the last decades and have found applications in optimization, o…

This paper addresses detection of a reverse engineering (RE) attack targeting a deep neural network (DNN) image classifier; by querying, RE's aim is to discover the classifier's decision rule. RE can enable test-time evasion attacks, which require knowledge of the classifier. Recently, we proposed a quite effective app…

Neural Architecture Search (NAS) is an emerging topic in machine learning and computer vision. The fundamental ideology of NAS is using an automatic mechanism to replace manual designs for exploring powerful network architectures. One of the key factors of NAS is to scale-up the search space, e.g., increasing the numbe…