Instanton counting and wall-crossing in Donaldson invariants

Donaldson invariants of 4-manifolds were originally introduced via the moduli spaces of anti-self-dual connections, but can be approached also by moduli spaces of stable bundles when the underlying manifolds are complex projective surfaces. Based on this approach, we compute the wall-crossing terms of Donaldson invariants, describing what happens when the ample line bundle is changed. This is an old subject, but a recent advance, the `instanton counting', enables us to compute them in the torus equivariant setting. Then the terms are naturally expressed in terms of the so-called Seiberg-Witten curves. Finally, we express the blow-up formula of Donaldson invariants, as a wall-crossing formula via moduli spaces of perverse coherent sheaves on the blow-up. Talks are based on series of joint works with Kota Yoshioka, Lothar G\"ottsche, and Takuro Mochizuki.