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

"Equisingular Approximation of Real and Complex Analytic Germs"

In this talk we consider the problem of the approximation of a real or complex analytic germ by germs of Nash or even algebraic sets 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 singularity can be arbitrarily closely approximated by germs of Nash sets which are also Cohen-Macaulay and share the same Hilbert-Samuel function. Also, we obtain a result that states that every analytic singularity is topologically equivalent to a Nash singularity with the same Hilbert-Samuel function. A key ingredient in our results is a generalization of Buchberger's criterion to 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.