Algebraic Morava $K$-theories: gamma filtration and Chern classes

“In between” Grothendieck $K$-theory and Chow groups there exist orientable cohomology theories which are called algebraic Morava $K$-theories. We will explain some of their properties (e.g. existence of the gamma filtration and of Chern classes to some orientable theories) which make Morava $K$-theories appear similar to $K$-theory. We will also provide some applications to the study of Chow groups of quadrics over algebraically non-closed fields.