Marion Lefevre 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 field of applied mathematics to statistics and data science, information geometry emerges as a central discipline, where probability modeling exceeds simple calculations and enters a spatial dimension. In 2025, a deep understanding of the geometric structures associated with probability distributions not only sheds light on the foundations of the metrics used but also … Read more
The understanding of quantum groups has revolutionized the field of applied mathematics in physics. These mathematical objects, which extend the classical notion of symmetry groups, allow for an understanding of the profound transformations inherent in quantum systems. In 2025, the study of these groups offers a modern view of deformed symmetries, surpassing classical frameworks and … Read more
The theory of distributions, also known as the theory of generalized functions, revolutionizes the way mathematicians approach concepts of function and differentiation. By transcending the limits imposed by classical differentiability, this theory creates a robust framework for extending differentiation to much more general mathematical objects, such as locally integrable functions and beyond. Since its foundations … 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