Marion Lefevre

Enumerative combinatorics: advanced generating functions

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Enumerative combinatorics, a fundamental discipline of discrete mathematics, deploys sophisticated techniques to quantify and structure complex combinatorial objects. At the heart of these methods are generating functions, algebraic tools that transcend simple counting to explore the deep organization and asymptotic properties of the studied structures. By exploring advanced generating functions, notably algebraic, D-finite, or non-commutative … Read more

Geometric probabilities: random measures on spaces

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In short: Foundations and definitions of geometric probabilities in measurable spaces Geometric probabilities represent an essential branch of contemporary probabilistic theories, focusing on the measurement of events within continuous spaces. Unlike classical probabilities based on counting outcomes, they involve a rigorous measure of subsets in often vector or metric spaces, particularly within measurable spaces. This … Read more

Game theory of cooperative games: coalitions and solutions

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The theory of cooperative games represents a dynamic branch of game theory that focuses on analyzing strategic interactions between players capable of forming coalitions to optimize their common gains. It opens a fascinating perspective to understand how groups can collaborate, organize, and fairly share the benefits generated by their cooperation. This approach enriches the traditional … Read more

Model theory: logic and mathematical structures

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Model theory occupies a central place in mathematical logic by providing a rigorous framework to interpret axiomatic theories through concrete mathematical structures. This powerful branch explores how formal languages can be translated into coherent systems where concepts such as satisfiability, truth, or isomorphism gain precise meaning. It develops essential tools for analyzing and classifying complex … Read more

Hyperbolic geometry: negative curvature and geodesics

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Hyperbolic geometry intrigues as much as it challenges the intuitions anchored in Euclidean geometry. Diving into a universe where the classical rules of geometry no longer apply, it explores a space with negative curvature, a notion that upends the way we view lines, surfaces, and distances. At the beginning of the 21st century, this mathematical … Read more

Extremal Graph Theory: Combinatorial Optimization

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The theory of extremal graphs is a fascinating area of combinatorics, focused on analyzing the limits and optimal structures of graphs under certain constraints. Questions such as “what is the minimum number of edges needed to ensure a given property?” or “how can we characterize the graphs that reach these limits?” are central to this … Read more

Harmonic analysis: generalized Fourier transforms

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Harmonic analysis represents a fundamental advance in mathematics, offering a powerful key to understanding, decomposing, and studying complex functions and signals. Originating from Fourier series, this field has evolved to include generalized Fourier transforms, allowing for the analysis of not only periodic signals but also a vast array of non-periodic signals in many contexts. This … Read more

Algebraic topology: homotopy and cohomology

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Algebraic topology, a flagship discipline of mathematics, focuses on studying the properties of topological spaces through the lens of algebraic tools. Among its fundamental concepts, homotopy and cohomology play a central role by providing powerful invariants to classify and understand the deep structure of spaces. From loops on a circle to more complex spaces, these … Read more

Operator theory: analysis on infinite spaces

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The operator theory has today established itself as an essential pillar of functional analysis, particularly in the study of infinite spaces that extend well beyond familiar finite dimensions. In this general framework, Hilbert and Banach spaces stand out as the privileged environments for investigating linear operators. These spaces, equipped respectively with a complete inner product … Read more

Lie groups: continuous symmetries and geometry

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Lie groups represent a cornerstone in understanding continuous symmetries that shape both pure geometry and advanced physical theories. Their study allows one to grasp the deep nature of continuous transformations that preserve certain differentiable structures and offers a bridge between abstract mathematics and its concrete applications in physics and geometry. This field of research, heir … Read more