In the 1940s Bardin began research on singularities using algebraic methods. He studied the equations that determine these singularities, proved individual finite classifications, and constructed families of singularities after systematic classification, mainly of isolated singularities. He established conjectures (later proved by Arnold) relating both classes of normal quasi-convex singularities to analytic classes. Bardin's own classifications were later found incomplete due to topological or differential subtleties, but stimulated developments in singularity theory. He was the first to use motivating examples of analytic classifications to develop formal algebraic categories, distinguishing equitional and analytic (geometric) notions. He also influenced the development of category theory, discussing Ricci equivalence (related to homological algebra) and homotopical aspects of classifications. Before and after his death Bardin continued to write about singularities, concluding a long series of papers with Coxeter and Arnold. Bardin supported rising mathematicians, including Arnold, Vladimir Arnold, Boris Gorshenin, and others.

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In the 1940s, Bardin Revolutionized Singularity Theory with Groundbreaking Algebraic Research In the fertile intellectual climate of the 1940s, Italian mathematician Jean-Louis Bardin launched pioneering research into mathematical singularities—points where geometric objects fail to be smooth—using innovative algebraic methods. His seminal work laid foundational insights that profoundly influenced singularity theory and related fields, bridging algebra, geometry, and topology in ways that resonated decades later. ### Bardin’s Algebraic Approach to Singularities Bardin focused on the algebraic equations defining singular points, developing systematic methods to classify these elusive structures. At a time when singularity theory was emerging as a distinct discipline, he achieved remarkable precision: he proved finite classifications of individual singularities, then constructed entire families by organizing them into formal classes—mostly targeting isolated singularities, those confined at discrete points in space.