This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson–Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces. They correspond to Yukawa couplings and correlation functions in M-theory. We expect that the Yukawa coupling characterizes (co-)associative fibrations on these manifolds. We discuss the Fourier transformation along such fibrations and the analog of the Strominger–Yau–Zaslow mirror conjecture for G2-manifolds. We also discuss similar structures and transformations for Spin(7)- manifolds.

We determine all helix surfaces with parallel mean curvature vector field which are not minimal or pseudo-umbilical in spaces of type $M^{n}(c)\times\mathbb{R}$ , where$M$^{$n$}($c$) is a simply connected$n$-dimensional manifold with constant sectional curvature$c$.

We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer--Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich--Soibelman, Gross--Siebert and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer--Cartan elements, and therefore consistent scattering diagrams, in terms of the refined counting of tropical disks. We also describe theta functions, in both their tropical and Hall algebraic settings, in terms of distinguished flat sections of the Maurer--Cartan-deformed differential. In particular, this allows us to give a combinatorial description of Hall algebra theta functions for acyclic quivers with non-degenerate skew-symmetrized Euler forms.

Let (M, ω) be a compact symplectic 4-manifold with a compatible almost complex structure J. The problem of finding a J-compatible symplectic form with prescribed volume form is an almost-K¨ahler analogue of Yau’s theorem and is connected to a programme in symplectic topology proposed by Donaldson. We call the corresponding equation for the symplectic form the Calabi- Yau equation. Solutions are unique in their cohomology class. It is shown in this paper that a solution to this equation exists if the Nijenhuis tensor is small in a certain sense. Without this assumption, it is shown that the problem of existence can be reduced to obtaining a C0 bound on a scalar potential function.

We establish an analytic interpretation for the index of certain transversally elliptic symbols on non-compact manifolds. By using this interpretation, we establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a non-compact symplectic manifold with proper moment map. In particular, we present a solution to a conjecture of Michèle Vergne in her ICM 2006 plenary lecture.