Related papers: Microlocal analysis and beyond
The purpose of this article is to review the developments related to the notion of local fractional derivative introduced in 1996. We consider its definition, properties, implications and possible applications. This involves the local…
We extend to Gaussian distributions a result providing smoothed analysis estimates for condition numbers given as relativized distances to illposedness. We also introduce a notion of local analysis meant to capture the behavior of these…
Localic and realizability toposes are two central classes of toposes in categorical logic, both arising through the Hyland-Johnstone-Pitts tripos-to-topos construction. We investigate their shared geometric features by providing an…
The utilization of statistical methods an their applications within the new field of study known as Topological Data Analysis has has tremendous potential for broadening our exploration and understanding of complex, high-dimensional data…
In 20th century mathematics, the field of topology, which concerns the properties of geometric objects under continuous transformation, has proved surprisingly useful in application to the study of discrete mathematics, such as…
We extend the theory of fields/distributions developed the paper "A Feigin-Frenkel theorem with n singularities" to a general base scheme. In order to do so we introduce suitable notions of topological sheaves on schemes and study their…
In the first half of twentieth century the theory of complex analytic functions and of their zerosets was fully developed. The definition of holomorphic function has a local nature. Germs of holomorphic functions form a distinguished…
This text provides an introduction to distributed local algorithms -- an area at the intersection of theoretical computer science and discrete mathematics. We collect recent results in the area and demonstrate how they lead to a clean…
We introduce a sheaf-theoretic characterization of task solvability in general distributed computing models, unifying distinct approaches to message-passing models. We establish cellular sheaves as a natural mathematical framework for…
We introduce a topology, which we call the regional topology, on the space of all real functions on a given locally compact metric space. Next we obtain a new versions of Schauder's fixed point theorem and Ascoli's theorem. We use these…
We develop a microlocal and derived-geometric framework for index theory and analytic torsion of nonlinear PDEs. By integrating Spencer hypercohomology, microlocal sheaf theory, and factorization algebras, we establish new connections…
Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix integrals, and many problems of enumerative…
This document develops general concepts useful for extracting knowledge embedded in large graphs or datasets that have pair-wise relationships, such as cause-effect-type relations. Almost no underlying assumptions are made, other than that…
We classify the prelocalizing subcategories of the category of quasi-coherent sheaves on a locally noetherian scheme. In order to give the classification, we introduce the notion of a local filter of subobjects of the structure sheaf. The…
We present an auxiliary space theory that provides a unified framework for analyzing various iterative methods for solving linear systems that may be semidefinite. By interpreting a given iterative method for the original system as an…
These are lecture notes from my talks at the "Current Developments in Mathematics" conference (Harvard, 2006). They cover a variety of topics involving symplectic cohomology. In particular, a discussion of (algorithmic) classification…
Local Fourier analysis is a strong and well-established tool for analyzing the convergence of numerical methods for partial differential equations. The key idea of local Fourier analysis is to represent the occurring functions in terms of a…
We define a class of spaces on which one may generalise the notion of compactness following motivating examples from higher-dimensional number theory. We establish analogues of several well-known topological results (such as Tychonoff's…
The authors study the method of scaling in the context of the study of automorphism groups of complex domains in multiple dimensions. Various types of scaling techniques are compared and contrasted. Applications are given in a number of…
Exploring the shape of point configurations has been a key driver in the evolution of TDA (short for topological data analysis) since its infancy. This survey illustrates the recent efforts to broaden these ideas to model spatial…