Seminar in Real and Complex Geometry

Thursday, December 22, 2022, 16:15-17:45, online via zoom

Dmitry Kerner
(Ben-Gurion University)

Unfolding theory, Stable maps and Mather-Yau/Gaffney-Hauser results in arbitrary characteristic


In 40's Whitney studied maps of C^\infty manifolds. When a map is not an immersion/submersion, one tries to deform it locally, in hope to make it 'generic'. This approach has led to the rich theory of stable maps, developed by Mather, Thom and many others. The main 'engine' was vector field integration. This chained the whole theory to the C^\infty, or R/C-analytic setting. I will present the purely algebraic approach, studying maps of germs of Noetherian schemes, in any characteristic. The relevant groups of equivalence admit 'good' tangent spaces. Submodules of the tangent spaces lead to submodules of the group orbits. Then goes the theory of unfoldings (triviality and versality). Then I will discuss the new results on stable maps and theorems of Mather-Yau/Gaffney-Hauser.