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We interpret some results of persistent homology and barcodes (in any dimension) with the language of microlocal sheaf theory. For that purpose we study the derived category of sheaves on a real finite-dimensional vector space V. By using…
The goal of this note is to present some recent results of our research concerning multiplier ideal sheaves on complex spaces and singularities of plurisubharmonic functions. We firstly introduce multiplier ideal sheaves on complex spaces…
Based on the methods developed in [Kashiwara-Rouquier], we consider microlocalization of the rational Cherednik algebra of type $\Z/l\Z$. Our goal is to construct the irreducible modules and standard modules of the rational Cherednik…
The main objective of this work is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Such…
The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on $n$-dimensional smooth projective varieties $X$ with an $n$-block collection $\cB $ which generates…
Given a projection $f$ of a product of real analytic manifolds onto one factor, let us say, $S$, and a subanalytic sheaf $\mathcal{F}$ on the associated subanalytic site, we give a natural construction of the (subanalytic) relative sheaf…
We consider time-frequency localization operators $A_a^{\varphi_1,\varphi_2}$ with symbols $a$ in the wide weighted modulation space $ M^\infty_{w}(\mathbb{R}^{2d})$, and windows $ \varphi_1, \varphi_2 $ in the Gelfand-Shilov space…
We establish several results combining discrete Morse theory and microlocal sheaf theory in the setting of finite posets and simplicial complexes. Our primary tool is a computationally tractable description of the bounded derived category…
In this work we classify the at-point regularities of set-valued mappings into two categories and then we analyze their relationship through several implications and examples. After this theoretical tour, we use the subregularity properties…
The Stokes equation with the varying viscosity is considered in a thin tube structure, i.e. in a connected union of thin rectangles with heights of order $\varepsilon<<1 $ and with bases of order 1 with smoothened boundary. An asymptotic…
These notes constitute a survey on the geometric properties of globally subanalytic sets. We start with their definition and some fundamental results such as Gabrielov's Complement Theorem or existence of cell decompositions. We then give…
We consider a combination of local and nonlocal $p$-Laplace equations and discuss several regularity properties of weak solutions. More precisely, we establish local boundedness of weak subsolutions, local H\"older continuity of weak…
We study approximation properties generated by highly regular scaling functions and orthonormal wavelets. These properties are conveniently described in the framework of Gelfand-Shilov spaces. Important examples of multiresolution analyses…
We study the multifractal analysis of self-similar measures arising from random homogeneous iterated function systems. Under the assumption of the uniform strong separation condition, we see that this analysis parallels that of the…
In the analysis of PDEs, regularity of often measured in terms of Sobolev, H{\"o}lder, Besov or Lipschitz spaces, etc. However, sometimes a gain of regularity can also be expressed just in terms of Lebesgue spaces, by passing from a…
We prove the existence of fine moduli spaces of simple coherent sheaves on families of irreducible curves. Our proof is based on the existence of a universal upper bound of the Castelnuovo-Mumford regularity of such sheaves, which we…
We consider the stationary solutions for a class of Schroedinger equations with a symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassical limit we prove that the reduction to a finite-mode approximation give…
Soft-walled microchannels arise in many applications, ranging from organ-on-a-chip platforms to soft-robotic actuators. However, despite extensive research on their static and dynamic response, the potential failure of these devices has not…
We consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues. In the sequel to Ammari, Waters, Zhang (2015), we use pseudodifferential methods for the problem of…
The concept of ``multi-microlocalization'' was introduced to extend the usual microlocal sheaf theory to a more general scope. This paper aims to further extend this theory by exploring advanced topics. One is a stalk formula for…