# Ivan Panin

### Quaternionic Grassmannians and Pontryagin classes in algebraic geometry: II

**Abstract**. It is well-known in our days what an oriented cohomology theory is. It is a cohomology theory equipped with Chern classes. However, such theories as Balmer-Witt theory and Grothendieck-Witt theories are not oriented. That is why it is more difficult to compute them. It turns out that the latter two theories are equipped with Pontryagin classes for symplectic vector bundles. This leads, in particular, to nice computations of W(HP^n) and GW(HP^n). Moreover, jointly with Ch. Walter, we develope a nice theory of symplectically oriented cohomology theories, which is analogous to the theory of oriented cohomology theories. A top of this theory is an explicit expression of GW(X) using symplectic algebraic cobordisms MSp.