Related papers: Some problems at the interface of approximation th…

This article is an interdisciplinary review and an on-going progress report over the last few years made by myself and collaborators in certain fundamental subjects on two major theoretic branches in mathematics and theoretical physics:…

We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration…

This paper presents the computational challenge on differential geometry and topology that was hosted within the ICLR 2022 workshop ``Geometric and Topological Representation Learning". The competition asked participants to provide…

The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this…

These are the expanded and detailed notes of the lectures given by the authors during the school and workshop entitled "Liaison and Related Topics," held at the Politecnico di Torino during the period October 1-5, 2001. In these notes we…

In this paper we provide an approximation \`a la Ambrosio-Tortorelli of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two. We introduce…

Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…

In recent years a series of remarkable advances in tropical geometry and in non-archimedean geometry have brought new insights to the moduli theory of algebraic curves and their Jacobians. The goal of this survey, an expanded version of my…

Open problems of the conference "Automorphisms of Affine Spaces" held July 6-10 2009 at the Radboud University, Nijmegen, The Netherlands.

Consider the following toy problem. There are $m$ rectangles and $n$ points on the plane. Each rectangle $R$ is a consumer with budget $B_R$, who is interested in purchasing the cheapest item (point) inside R, given that she has enough…

This is an introduction to topology of complement to plane curves and hypersurfaces in the projective space and is based on the lectures given in Lumini in February and in ICTP (Trieste) in August of 2005. We discuss key problems concerning…

This is a survey of open problems in different parts of combinatorial and additive number theory. The paper is based on lectures at the Centre de Recerca Matematica in Barcelona on January 23 and January 25, 2008.

We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation, and model selection. These are all optimization problems, well-known to be challenging due to…

This text is an introduction to algebraic enumerative geometry and to applications of tropical geometry to classical geometry, based on a course given during the X-UPS mathematical days, 2008 May 14th and 15th. The aim of this text is to be…

We survey recent applications of topology and singularity theory in the study of the algebraic complexity of concrete optimization problems in applied algebraic geometry and algebraic statistics.

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings,…

This article is devoted to propose some lower and upper bounds for the coupled-tasks scheduling problem in presence of compatibility constraints according to classical complexity hypothesis ($\mathcal{P} \neq \mathcal{NP}$,…

This is a slightly edited version of the transparencies for a seminar at UCL, May 7, 2003. It is intended to give a quick view of background, ideas, and some calculations, in the applicatioon of some non commutative methods to algebraic…

A list of the problems presented on August 12, 2002 at the open-problem session of the 14th Canadian Conference on Computational Geometry held in Lethbridge, Alberta, Canada.