Scientific Scope
The main theme of this axis is the theory of singularities of algebraic varieties or real or complex analytic spaces, mappings between these spaces, and differential equations in these categories.
The theory of singularities has a long history dating back to the 19th century (Puiseux, Noether, and Halphen) and the early 20th century (Picard, Enriques, Zariski, Lefschetz). Since the 1950s, it has been recognized as a distinct branch of mathematics, following fundamental work by Zariski, Whitney, and Thom, then Arnold, Hironaka, Milnor, and Pham. France has played a significant role in this field from the beginning, as illustrated by this historical context and the many recent contributions, one of the starting points of which was the Singularities Congress held in Cargèse in 1972.
Since 2002, the former GDR Singularities has promoted contact and cohesion among various French teams working on this topic. The project is focused on a mix of classical topics in singularity theory, such as resolution of singularities, uniformization, and valuation theory; stratification theory; families of real and complex singularities in spaces and mappings; analytic arcs and motivic integration; along with modern themes and methods like D-modules and Hodge theory; b-functions and vanishing cycles; bi-Lipschitz geometry; moderate geometry, rings of rational continuous functions, and regulous functions; and the study of special classes of varieties (toric varieties, hyperplane arrangements, etc.).
The RT project also aims to incorporate new methods, some of which are highly promising for interaction with the GAGC axis. Indeed, the scientific committee has noted that the introduction of techniques from logarithmic geometry, non-Archimedean geometry, the minimal model program, tropical geometry, and Heegaard-Floer homology has led to striking advances in singularity theory. Finally, the scientific committee is also attentive to new connections arising from moderate geometry and differential geometry, two areas that will be developed alongside the EFI axis.
Genesis of the Singularities and Applications Theme
The Singularities and Applications theme continues the work of the GDR Singularities and Applications (GDR Singularities), created in 2006 by Michel Granger, which concluded on December 31, 2023. More information is available at this link, the website of the former GDR.