We conjecture that the classical geometric 2-designs $PG_d(n,q)$ formed by the points and $d$-dimensional subspaces of the projective space of dimension $n$ over the field with $q$ elements, where $2 \leq d \leq n-1$, are characterized among all designs with the same parameters as those having line size $q+1$. This conjecture is known to hold for the case $d=n-1$ (the Dembowski-Wagner theorem) and also for $d=2$ (a recent result jointly established with Tonchev). Here we extend these results to the cases $d=3$ and $d=4$. The general case remains open and appears to be rather difficult; in this direction, we will discuss some partial results (and problems) for the case $d=n-2$. Moreover, we will see how one may approach the quite different problem of characterizing the classical point-line designs $PG_1(n,q)$.

The necessary background will be explained in detail, so that the talk
should be accessible to a general mathematical audience.

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