## Olivetti Club

Let $f$ be a smooth real-valued function on a manifold $M$. Typically, the critical points of $f$ will be nondegenerate. If all critical points are nondegenerate, we say that $f$ is **Morse**, and if all of the critical values are distinct, we say that $f$ is **generic**. The Morse Lemma says that in the neighborhood of a nondegenerate critical point of $f$, a function can be reduced to its quadratic part for a suitable choice of local coordinate system. In this talk, I will give a proof of this lemma and talk about its implications. This lemma serves as a springboard for the study of the structure of the space of nongeneric functions. If time permits, I will talk about how the work of Jean Cerf, which builds on the work René Thom and John Mather allows one to deduce that the mapping class group of a closed, orientable surface is finitely presented as shown by Hatcher and Thurston in 1978.