Projections of self-affine fractals

I will discuss the extension of Falconer's landmark 1988 result -- on the Hausdorff dimension of typical self-affine fractals -- to linear projections of these fractals. The result uncovers an algebraic structure on the exceptional sets of projections in the sense of Marstand projection theorem. Furthermore, the results comes with various examples of new phenomena that I will mention. These include

• existence of equilibrium states having non-exact dimensional linear projections (equilibrium states themselves are exact dimensional by Feng);

• existence of self-affine fractals in dimensions at least 4, whose set of exceptional projections contains higher degree algebraic varieties in Grassmannians (such constructions are not possible even in Borel category in dimension 3 by the solution of a conjecture of Fässler-Orponen by Gan et.al., nor in any dimension if the linear parts of affinities acts strongly irreducibly on all exterior powers, by Rapaport);

• existence of self-affine fractals whose sumsets have lower than expected dimension without satisfying an arithmetic resonance (impossible in dimension 1 by Hochman, Shmerkin, Peres and in dimension 2 by Pyörälä).

This is joint work with Ian D. Morris.