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We discuss issues of problem formulation for algorithms in real algebraic geometry, focussing on quantifier elimination by cylindrical algebraic decomposition. We recall how the variable ordering used can have a profound effect on both…
This text is a support for different courses of the master of Mechanics of the University Paris-Saclay. The content of this text is an introduction, for graduate students, to tensor algebra and analysis. Far from being exhaustive, the text…
The focus of this paper is two fold. Firstly, we present a logical approach to graph modification problems such as minimum node deletion, edge deletion, edge augmentation problems by expressing them as an expression in first order (FO)…
Trimming is a core technique in geometric modeling. Unfortunately, the resulting objects do not take the requirements of numerical simulations into account and yield various problems. This paper outlines principal issues of trimmed models…
This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel…
This paper introduces DOoM, a new open-source benchmark designed to assess the capabilities of language models in solving mathematics and physics problems in Russian. The benchmark includes problems of varying difficulty, ranging from…
This is an attempt to present axioms for Euclidean geometry, aiming at the following goals: to work with geometric notions (thus not merely identify points with pairs of numbers, giving a special status to a particular coordinate system);…
Some established and also novel techniques in the field of applications of algorithmic (Kolmogorov) complexity currently co-exist for the first time and are here reviewed, ranging from dominant ones such as statistical lossless compression…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
We present a formalization of geometric instruments that considers separately geometric and arithmetic aspects of them. We introduce the concept of tool, which formalizes a physical instrument as a set of axioms representing its geometric…
We use SMT technology to address a class of problems involving uninterpreted functions and nonlinear real arithmetic. In particular, we focus on problems commonly found in mathematical competitions, such as the International Mathematical…
We propose a gradient descent method for solving optimization problems arising in settings of tropical geometry - a variant of algebraic geometry that has attracted growing interest in applications such as computational biology, economics,…
We will present several examples in which ideas from ergodic theory can be useful to study some problems in arithmetic and algebraic geometry.
Our aim is to give a friendly introduction for students to systolic inequalities. We will stress the relationships between the classical formulation for Riemannian metrics and more recent developments related to symplectic measurements and…
Mathematical understanding is built in many ways. Among these, illustration has been a companion and tool for research for as long as research has taken place. We use the term illustration to encompass any way one might bring a mathematical…
Problems for the graduate students who want to improve problem-solving skills in geometry. Every problem has a short elegant solution -- this gives a hint which was not available when the problem was discovered.
We give an introduction to Tropical Geometry and prove some results in Tropical Intersection Theory. The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of…
Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm…
Geometry is essentially a global language, which is fully understood in different times, countries and cultures. The proof of a geometric theorem (e.g. the Pythagorean Theorem) or a geometric construction (e.g. the construction of an…
These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM.