The unstable intersection asserts that any two maps from
compacta $X$ and $Y$ to an $n$-dimensional Euclidian space can be
arbitrarily closely approximated by maps with disjoint images if and
only if the product of $X$ and $Y$ is of dimension at most $n-1$. This
conjecture is proved in all the dimensions except $n=5$. We are planning
to present the proof for $n>5$. The proof heavily relies on Cohomological
dimension and Extension theory and in the first talk we will focus on
auxiliary results.