This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Khlerian complex structures on six-dimensional nilpotent algebras whose differential Gerstenhaber algebra is quasi-isomorphic to that of the symplectic structure. In a weak sense of mirror symmetry, this gives a classication of pseudo-Khler structures on six-dimensional nilpotent algebras whose mirror images are themselves.

We establish the second part of Milnor's conjecture on the volume of simplexes in hyperbolic and spherical spaces. A characterization of the closure of the space of the angle Gram matrices of simplexes is also obtained.

For two nearby disjoint coassociative submanifolds C and C′ in a G2-manifold, we construct thin instantons with boundaries lying on C and C′ from regular J-holomorphic curves in C. We explain their relationship with the Seiberg-Witten invariants for C.

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