Our entire scientific understanding of nature is based on various types of systems of equations (linear, polynomial, analytic, differential) and their solutions. Singularities are points of local instability of these equations that can have tremendous impact on the global behavior of solutions. Thus, singularity theory is fundamentally important for mathematics and natural sciences.
For polynomial/analytic equations, the singular locus is that of failure of the manifold structure of the solution space and can be described in terms of differential forms. An main tool to study this phenomenon is desingularization which relates complicated singularities to simple ones - normal crossing divisors. Normalization is a step in this direction that removes singularities in codimension one.
Codim1Sing will ultimately lead to the first simple algebraic conditions characterizing normal crossing properties. To this end, a widely laid-out research project will be completed: Kyoji Saito's theory of logarithmic forms will be embedded into the theory of regular differential forms of Kersken and Kunz/Waldi and generalized beyond the hypersurface case, including the concept of free divisors crucial in singularity theory. In the process, Codim1Sing discovers the geometric meaning of deep algebraic conditions in terms of regular differential forms and (natural partial) normalizations. The project's innovative results include generalizations of the Le-Saito theorem, a proof of Faber's conjecture, as well as novel insights in the geometry of free divisors and in Vasconcelos' normalization algorithm.
Codim1Sing addresses fundamental constructions and objects in singularity theory and it advances long-term collaborations between experts in Europe and worldwide, notably North America. Consequently, it comprises knowledge transfer within and into the EU, the introduction of novel approaches and the sustained reintegration of a high-potential researcher into the European science community.
Call for proposal
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