# Algebraic cobordism and the cobordism ring of varieties

# Algebraic cobordism and the cobordism ring of varieties

Algebraic cobordism is expected to be the universal oriented cohomology theory in algebraic geometry, playing the role of complex cobordism from algebraic topology. Supposedly, it refines both the theory Chow rings and algebraic K-theory, making it a rather sophisticated invariant. I will discuss a particularly geometric approach to realizing such a theory, and explain the main results of this approach. Then, I will turn my attention to a rather concrete problem: namely, computing the algebraic cobordism ring of a field k, which boils down to classifying quasi-smooth projective derived k-schemes "up to cobordism". Over a field of characteristic 0, the result is known due to work of Levine--Morel and Lowrey--Schürg, and, over a field of characteristic p>0, the cobordism ring, with p inverted in the coefficients, is generated by the classes of regular projective k-varieties. I will then fail to classify these (in several ways, time allowing).