Geometric realizations of noncommutative varieties and phantoms

In my talk I am going to discuss such phenomena as phantom and quasi-phantom categories which appear as admissible subcategories in derived categories of coherent sheaves on fake del Pezzo surfaces. They give examples of smooth and proper noncommutative varieties whose additive invariants are almost or completely trivial. I am also going to discuss general notions of noncommutative varieties and their geometric realizations.