**Title**: On the Reeb spaces of definable maps.

**Speaker**: Saugata Basu (Purdue University)

**Title**: We prove that the Reeb space of a proper definable map in an arbitrary o-minimal expansion of the reals is realizable as a proper definable quotient. We also show that the Betti numbers of the Reeb space of a map $f$ can be arbitrarily large compared to those of $X$, unlike in the special case of Reeb graphs of manifolds. Nevertheless, in the special case when $f:X \rightarrow Y$ is a semi-algebraic map and $X$ is closed and bounded, we prove a singly exponential upper bound on the Betti numbers of the Reeb space of $f$ in terms of the number and degrees of the polynomials defining $X,Y$ and $f$.

(Joint work with Nathanael Cox and Sarah Percival).

**Seminar URL**: https://tgda.osu.edu/activities/tdga-seminar/