8 mar 2017 -- 16:00

Sala Riunioni, Dipartimento di Matematica, Pisa

**Abstract.**

Let k be a knot (i.e. an embedding of $\mathbb{S}^1$ into $\mathbb{S}^3$). Once $\mathbb{S}^3$ is seen as the boundary of $\mathbb{D}^4$, one can ask which kind of (properly) embedded surfaces in $\mathbb{D}^4$ have k as boundary. Finding the minimal genus of such a surfaces (called slice genus) is a central topic in low dimensional topology.

In this talk I wish to describe some inequalities, arising from contact topology and quantum homologies. These inequalities, called Bennequin-type inequalities, can be used to estimate the slice genus of a knot in term of other invariants.

The seminar will be organized as follows; first, I will give a brief overview of the history of this problem, and indicate some motivations to study it. Then, I will describe some known results that may help to determine the slice genus of a knot. In doing so, I will introduce some basic contact topology. Finally, I will describe how we can use some ''contact'' knot invariant to estimate the slice genus.