Georgia Institute of Technology College of Sciences

I will discuss some results from our ongoing work with Ian D. Morris which aims at a systematic study of projections of self-affine fractals.
After explaining the extension of classical results of Falconer to the projections of self-affine fractals, I will discuss:

• the existence of equilibrium states having non-exact dimensional linear projections (equilibrium states themselves are exact dimensional by Feng);
• the existence of self-affine fractals in dimensions at least 4, whose set of exceptional projections in the sense of Marstand Projection Theorem 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 Fassler-Orponen by Gan et.al., neither in any dimension if the linear parts of affinities acts strongly irreducibly on all exterior powers, by Barany, Hochman, Rapaport);
• the 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 Pyorala).