Orthogonal polynomials are at the heart of many advancements in numerical analysis and functional approximation. These mathematical tools allow for the precise representation of complex functions while controlling the error associated with this approximation. More than just a theoretical concept, they find practical applications in the calculation of numerical integrals, the solving of linear systems, … Read more
Subriemannian geometry stands out as a fundamental field in the study of spaces where movements are subject to non-trivial constraints. These spaces, often referred to as Carnot spaces, are characterized by the presence of non-holonomic distributions that limit accessible directions. In this context, classical notions of distance and metrics evolve into adapted concepts, such as … Read more
Contemporary research in mathematics is engaged in unveiling the deepest foundations that unite arithmetic, geometry, and topology. At the heart of this quest, motif theory stands out as an ambitious answer, aiming to decipher a universal language behind the diversity of cohomological invariants arising from algebraic varieties. This perspective, sketched by Grothendieck in the mid-1960s, … Read more
In the contemporary mathematical landscape, non-commutative harmonic analysis stands out as a major discipline, breaking the boundaries of classical harmonic analysis developed since the 18th century. This branch refines its sophistication by directing its tools towards algebraic structures where commutativity is absent, thus offering unprecedented perspectives in quantum mechanics and beyond. The quantum Fourier transform … Read more
In the landscape of contemporary mathematics, convex analysis occupies a central place, particularly due to its extensive applications in optimization and economics. It focuses on the properties and behaviors of convex functions as well as convex sets, which constitute the essential foundations of this theory. Convexity ensures a rigid and exploitable structure that allows for … Read more
Arithmetic geometry, at the crossroads of algebra and number theory, redefines the classical understanding of mathematical structures. This discipline uses algebraic geometry to approach fundamental problems related to rational numbers and algebraic curves. More than just a simple extension of traditional methods, it offers a sophisticated framework to interpret complex arithmetic phenomena through algebraic varieties, … Read more
Moduli spaces are renowned mathematical objects that allow for the classification and study of entire families of geometric structures through finely defined parameters. They serve as an essential interface between geometry, topology, and the theory of varieties, offering new perspectives on the understanding and deformation of complex shapes. These spaces help organize the infinite diversity … Read more
Floer’s homology has now established itself as a fundamental bridge between topology and the in-depth study of differential equations. Originating from the developments in symplectic geometry at the end of the 20th century, this theory has decisively changed the understanding of manifolds, particularly those of infinite dimension. Designed as an analogue of Morse theory in … Read more
The theory of deformations occupies a central place in the study of changes in shape within geometry and physics. It sheds light on how often complex structures can evolve based on notions of continuous families, topology, and differential geometry. The study of continuous families of structures allows for a better understanding of the gradual variations, … Read more
The ergodic theory opens fascinating perspectives beyond the traditional boundaries of mathematics and physics. This discipline focuses on the study of dynamical systems and how, over time, the trajectories of a system explore its entire state space. In particular, the key concepts of invariant measures, mixing, and ergodicity provide essential tools for understanding how seemingly … Read more