Abstract: In the late 1970s, R. Hamilton initiated a program to extend the Kodaira-Spencer's elliptic deformation theory of complex structures to manifolds with boundary. The stable case can be stated as follows. Let D be a relatively compact domain in a complex manifold M with certain complex analytic geometry. Assume H1(D,T) = 0, where T is the holomorphic tangent bundle of M. Given a formally integrable almost complex structure X defined on the closure D, and provided that X is sufficiently close to the standard complex structure on M, does there exist a complex/holomorphic coordinate that is compatible with X? In other words, does there exist a diffeomorphism from D into M that transforms X into the complex structure on M? Locally near a point inside D, such a diffeomorphism always exists by the classical Newlander-Nirenberg theorem. Thus, we also call this problem the global or boundary Newlander-Nirenberg problem. 

In this talk I will present some recent progress on the existence of such diffeomorphism with almost sharp regularity, on a large class of domains with finite smooth boundaries and finite smooth almost complex structure. The talk is partially based on joint work with Xianghong Gong. 

Location and Zoom link: SCEN 406 and at https://uark.zoom.us/j/87503165233?pwd=S7QdLHdFaJYEkZL3at0OV4aE2rRPwh.1