Maria Aparecida Soares Ruas (ICMC-Universidade de São Paulo)

Wednesday, May 2nd, 16:20, Room 3-010

**Title: The extra-nice dimensions**

**Abstract:** We define the extra-nice dimensions and prove that the subset of stable 1-parameter families in $C^{\infty}(N\times[0,1],P)$, also known as pseudo-isotopies, is dense if and only if the pair of dimensions $(\dim N, \dim P)$ is in the extra-nice dimensions. This result is parallel to Mather's characterization of the nice dimensions as the pairs $(n,p)$ for which stable maps are dense.

The extra-nice dimensions are characterized by the property that discriminants of stable germs in one dimension higher have $\mathcal A_e$-codimension 1 hyperplane sections. They are also related to the simplicity of $\mathcal A_e$-codimension 2 germs. We give a sufficient condition for any $\mathscr A_e$-codimension 2 germ to be simple and give an example

of a corank 2 codimension 2 germ in the nice dimensions which is not simple.

Then we establish the boundary of the extra-nice dimensions.