We offer the ultimate birdtracker guide to exceptional Lie groups.

We define the Mordell exceptional locus Z(V ) for affine varieties V ⊂ Ga with respect to the action of a product of Drinfeld modules on the coordinates of Ga. We show that Z(V ) is a closed subset of V . We also show that there are finitely many maximal algebraic φmodules whose translates lie in V . Our results are motivated by DenisMordell-Lang conjecture for Drinfeld modules.

We find a full strongly exceptional collection for the Cayley plane OP, the simplest rational homogeneous space of the exceptional group E6. This collection, closely related to the one given by the second author in [Man11], consists of 27 vector bundles which are homogeneous for the group E6, and is a Lefschetz collection with respect to the minimal equivariant embedding of OP.

Let M be a four-holed sphere and Γ the mapping class group of M fixing ∂M . The group Γ acts on the space MB(SU(2)) of SU(2)-gauge equivalence classes of flat SU(2)-connections on M with fixed holonomy on ∂M . We give examples of flat SU(2)connections whose holonomy groups are dense in SU(2), but whose Γ-orbits are discrete in MB(SU(2)). This phenomenon does not occur for surfaces with genus gr...

Simple finite dimensional Kantor triple systems are classified in terms of Satake diagrams. We prove that every simple linearly compact Kantor triple system is finite dimensional and provide an explicit presentation of all the classical and exceptional systems.

In family unification models, all three families of quarks and leptons are grouped together into an irreducible representation of a simple gauge group, thus unifying the Standard Model gauge symmetries and a gauged family symmetry. Large orthogonal groups, and the exceptional groups E7 and E8 have been much studied for family unification. The main theoretical difficulty of family unification is...