  In the present paper, we prove the existence of universal polynomials which
express multi-singularity loci classes of prescribed types for proper morphisms
between smooth schemes over an algebraically closed field of characteristic
zero -- we call them Thom polynomials for multi-singularity types of maps. It
has been referred to as the Thom-Kazarian principle and unsolved for a long
time. This result solidifies the foundation for a general enumerative theory of
singularities of maps which is applicable to a broad range of problems in
classical and modern algebraic geometry. In particular, it would contribute to
a satisfactory answer to the rest of (an advanced form of) Hilbert's 15th
problem and connect such classics to recent new interests in enumerations
inspired by mathematical physics and other fields. A main feature of our proof
is a striking use of algebro-geometric cohomology operations. Somewhat
surprisingly, when trying to grasp a full perspective of classical enumerative
geometry, we will inevitably encounter algebraic cobordism and motivic
cohomology.
