Ph.D. Public Lecture (Math) - Aftab Patel

Title: Equisingular Approximation of Real and Complex Analytic Germs

Abstract: In this talk we consider the problem of the approximation of the germ of a real or complex analytic space by germs that are Nash or algebraic, and which are equisingular with respect to the Hilbert-Samuel function. The Hilbert-Samuel function is a key measure of the singularity that features prominently in Hironaka's resolution of singularities. We show that a Cohen-Macaulay analytic germ can be arbitrarily closely approximated by Nash germs which are also Cohen-Macaulay and share the same Hilbert-Samuel function. Also, we obtain a result that states that every analytic germ is topologically equivalent to a Nash germ with the same Hilbert-Samuel function. Key ingredients in our approach are Hironaka's diagram of initial exponents and a generalization of Buchberger's criterion to the case of standard bases of power series due to T. Becker in 1990. This talk is based on the PhD thesis research of the speaker conducted under the supervision of Prof. Janusz Adamus at Western University.