http://www-math.umd.edu/research/seminars.html http://www-math.umd.edu/research/seminars.html Wed, 15 Apr 2026 14:00:00 EDT Where: Kirwan Hall 3206
Speaker: Martin Andler (CNRS Versailles-Saint-Quentin) - https://lmv.math.cnrs.fr/en/laboratory/directory/martin-andler-english-version/
Abstract: (joint work with Siddhartha Sahi)

In the 1950s, topologists introduced the notion of equivariant cohomology
\(H_G(E)\) for a topological space \(E\) with an action by a compact group \(G\). If the action is free, the definition should yield \(H_G(E) \simeq H(E/G)\), and be computed using de Rham cohomology. In 1950, even before the concept of equivariant cohomology had been formulated, Henri Cartan introduced a complex of equivariant differential form for a compact Lie group acting on a differential manifold \(E\), and proved a result amounting to stating that the cohomology of that complex computes \(H_G(E)\). In 1999, Guillemin and Sternberg reformulated Cartan’s work in terms of a supersymmetric extension of the Lie algebra of \(G\).

Our aim is to reconsider such considerations, by replacing vector spaces by objects
in a \(k\)-linear symmetric abelian monoidal category, requiring that this category contain an ``odd unit" to account for the supersymmetric dimension plus some further properties, and considering modules for a rigid Lie algebra object in that category. In that context, we obtain a version of Koszul’s homotopy isomorphism theorem, and recover as a consequence some known results as the acyclicity of the Koszul resolution. This approach has an advantage to treat in a uniform way the three categories of vector spaces, vector superspaces and graded vector spaces, as well as more exotic tensor categories which have been considered by Deligne and others, or categories of sheaves.
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