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In recent years, significant advances have been made in the design and analysis of fully dynamic algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are…
We develop the structure theory for transformations of weakly geometric rough paths of bounded $1 < p$-variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. We…
Over the last decade, implementations of several desingularization algorithms have appeared in various contexts. These differ as widely in their methods and in their practical efficiency as they differ in the situations in which they may be…
Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face is to find the discrete protoforms of…
Recently, continuous-time dynamical systems have proved useful in providing conceptual and quantitative insights into gradient-based optimization, widely used in modern machine learning and statistics. An important question that arises in…
Reconstructing a complete object from its parts is a fundamental problem in many scientific domains. The purpose of this article is to provide a systematic survey on this topic. The reassembly problem requires understanding the attributes…
In this survey we discuss a wide variety of aspects related to Lie group integrators. These numerical integration schemes for differential equations on manifolds have been studied in a general and systematic manner since the 1990s and the…
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…
We consider the proximal gradient method on Riemannian manifolds for functions that are possibly not geodesically convex. Starting from the forward-backward-splitting, we define an intrinsic variant of the proximal gradient method that uses…
We report on the following highlights from among the many discoveries made in Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the modular theory for noncommutative tori, 2) Advances on the Baum-Connes…
We survey the notion and history of error-correcting codes and the algorithms needed to make them effective in information transmission. We then give some basic as well as more modern constructions of, and algorithms for, error-correcting…
We present a concise survey of matrix completion methods and associated implementations of several fundamental algorithms. Our study covers both passive and adaptive strategies. We further illustrate the behavior of a simple adaptive…
In this survey paper we give an historical and at the same time thematical overview of the development of ring geometry from its origin to the current state of the art. A comprehensive up-to-date list of literature is added with articles…
We survey the recent applications and developments of sieve methods related to discrete groups, especially in the case of infinite index subgroups of arithmetic groups.
In the present paper, we propose a new discrete surface theory on 3-valent embedded graphs in the 3-dimensional Euclidean space which are not necessarily discretization or approximation of smooth surfaces. The Gauss curvature and the mean…
We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…
This is a draft of an article to appear in the October 2022 issue of the Notices of the AMS. In this survey article we explore a fascinating area called descriptive combinatorics and its recently discovered connections to distributed…
We produce algorithms to detect whether a complex affine variety computed and presented numerically by the machinery of numerical algebraic geometry corresponds to an associated component of a polynomial ideal.
We derive properties and a characterization of discrete composition matrices which are useful in the field of numerical computation of shape correspondences.
The aim of these notes is to present a self contained account of discrete weak KAM theory. Put aside the intrinsic elegance of this theory, it is also a toy model for classical weak KAM theory, where many technical difficulties disappear,…