**Jiahao Hu,
Stony Brook University**

**Title:**
Algebraic properties of the four components of the exterior differential d on almost
complex manifolds

**Abstract:** The exterior differential d on complex-valued
differential forms of complex manifolds decomposes into the
Cauchy-Riemann operator and its complex conjugate. Meanwhile on almost
complex manifolds, the exterior differential d in general has two
extra components, thus decomposes into four operators. In this talk, I
will introduce these operators and discuss the structure of the
(graded) associative algebra generated by these four components of d,
subject to relations deduced from d squaring to zero. Then I will
compare this algebra to the corresponding one in the complex
(i.e. integrable) case, we shall see they are very different strictly
speaking but similar in a weak sense (quasi-isomorphic). This is based
on joint work with Shamuel Aueyung and Jin-Cheng Guu.