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The article provides an introduction to infinite-dimensional differential calculus over topological fields and surveys some of its applications, notably in the areas of infinite-dimensional Lie groups and dynamical systems.
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We derive optimal estimates for the Bergman kernel and the Bergman metric for certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. Being unbounded models, these domains obey certain geometric constraints…
In this paper, by modifying significantly the Friedrichs-Gross mollifier technique and/or using the Lasry-Lions regularization technique together with some carefully chosen cut-off functions, for the first time we construct explicitly…
We give a new characterization of pseudoconvex point, and of finite type point, using analytic discs.
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Comparison and localization results for the Lempert function, the Carath\'eodory distance and their infinitesimal forms on strongly pseudoconvex domains are obtained. Related results for visible and strongly complete domains are proved.
We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and nonconvex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental…
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These notes briefly discuss Fourier transforms of finite measures and extensions of Fourier integrals to points in complex domains.
We give an explicit verifiable characterization of weakly pseudoconvex but locally nonconvexifiable hypersurfaces of finite type in dimension two. It is expressed in terms of a generalized model, which captures local geometry of the…
We treat the boundary problem for complex varieties (with isolated singularities) of dimension greater than one, which are contained in a suitable class of strictly pseudoconvex, unbounded domains of C^n.
We consider a class of discrete convex functionals which satisfy a (generalized) coarea formula, and study their limit in the continuum.
We give precise estimates of some holomorphically invariant infinitesimal metrics near a pseudoconcave points in a wide family of ``model'' domains for that situation in $\mathbb C^2$. This extends to metrics (rather distances) the authors'…
We give a description of complex geodesics and we study the structure of stationary discs in some non-convex domains for which complex geodesics are not unique.
We survey results arising from the study of domains in C^n with non-compact automorphism group. Beginning with a well-known characterization of the unit ball, we develop ideas toward a consideration of weakly pseudoconvex (and even…
This paper presents a general study of one-dimensional differentiability for functionals defined on convex domains that are not necessarily open. The local approximation is carried out using affine functionals, as opposed to linear…
We present some results in the analysis of non-compact differential equations on unbounded domains.
In the framework of shape constrained estimation, we review methods and works done in convex set estimation. These methods mostly build on stochastic and convex geometry, empirical process theory, functional analysis, linear programming,…