**Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.**

Given \$d\$ and \$q\$ the topological Tverberg problem asks for the minimal \$n\$ such that any continuous map from the \$n\$-dimensional simplex to \$\mathbb R^d\$ identifies \$q\$ points from pairwise disjoint faces. For \$q\$ a prime power \$n\$ is \$(q-1)(d+1)\$. The lower bound follows from a general position argument, the upper bound from equivariant topological methods. It was shown recently that for \$q\$ with at least two distinct prime divisors the lower bound may be improved. For those \$q\$ non-trivial upper bounds had been elusive. I will show that \$n\$ is at most \$q(d+1)-1\$ for all \$q\$. I had previously conjectured this to be optimal unless \$q\$ is a prime power. This is joint work with Pablo Soberón.