Baer Colloquium
May 24, 2014 — Ghent, Belgium

The classical parallelism is the Clifford parallelism. In Res. Math. 2005 we gave the first non-classical parallelisms, and after this we found various construction methods and many examples. The collineation group of each topological parallelism is a Lie group and we use this Lie group (A) to study the known examples and (B) to start the classification of topological parallelisms.

We will give an introduction to axial algebras, a class of nonassociative algebras controlled by fusion rules for eigenvectors of idempotents. Axial algebras link topics in finite group theory and mathematical physics. We will present connections to Fischer spaces and lattices.

We characterize \( d \)-uple Veronese embeddings of finite-dimensional projective spaces. The easiest non-trivial instance of our theorem is the embedding of the projective plane in 5-dimensional projective space, a result obtained in 1901 by Severi when the underlying field is complex. (This is joint work with Jeroen Schillewaert.)

A word map on a group \( G \) is a map sending \( (g_1,\dots,g_k) \) to \( w(g_1,\dots,g_k) \), where \( w \) is a fixed word in \( k \) variables. For example, the commutator map and power maps are word maps. I shall discuss various questions about word maps, such as surjectivity and distribution of values, with particular emphasis on simple groups.