# Algebraic cobordism: applications and perspectives

# Algebraic cobordism: applications and perspectives

We will survey our theory, with F. Morel, of algebraic cobordism. This is the algebraic analog of complex cobordism, and may be viewed as a refinement of the Chow ring, replacing algebraic cycles with algebraic manifolds. We will discuss its relation with the Chow ring and the Grothendieck group of coherent sheaves, with applications to Riemann-Roch and degree formulas (used in the proof of the Bloch-Kato conjecture). With R. Pandharipande, we have given a simple description of the relations defining algebraic cobordism, the so-called double point cobordism; we will discuss applications this has had to Donaldson-Thomas theory. Finally, we will discuss the relation of our geometric theory with a more sophisticated version defined using motivic homotopy theory.