We study symplectic structures on Kähler surfaces with $p_g = 0$. We give an example of a projective surface which admits a symplectic structure which is not compatible with any Kähler metric.

We examine a moduli problem for real and quaternionic vector bundles on a smooth complex projective curve with a fixed real structure, and we give a gauge-theoretic construction of moduli spaces for semistable such bundles with fixed topological type. These spaces embed onto connected subsets of real points inside a complex projective variety. We relate our point of view to previous work by Biswas et al., and we use this to study the $\mathrm{Gal}(\mathbb{C}/\mathbb{R})$-action $[\mathcal{E}] \mapsto [\overline{\sigma^*E}]$ on moduli varieties of stable holomorphic bundles on a complex curve with given real structure $\sigma$. We show in particular a Harnack-type theorem, bounding the number of connected components of the fixed-point set of that action by $2^g + 1$, where $g$ is the genus of the curve. In fact, taking into account all the topological invariants of $\sigma$, we give an exact count of the number of connected components, thus generalizing to rank $r \gt 1$ the results of Gross and Harris on the Picard scheme of a real algebraic curve.

We introduce a notion of the noncommutative integrability within a framework of contact geometry.

We define a higher analogue of Dirac structures on a manifold $M$. Under a regularity assumption, higher Dirac structures can be described by a foliation and a (not necessarily closed, non-unique) differential form on $M$, and are equivalent to (and simpler to handle than) the multi-Dirac structures recently introduced in the context of field theory by Vankerschaver et al. We associate an $L_\infty$-algebra of observables to every higher Dirac structure, extending work of Baez et al. on multisymplectic forms. Further, applying a recent result of Getzler, we associate an $L_\infty$-algebra to any manifold endowed with a closed differential form $H$, via a higher analogue of split Courant algebroid twisted by $H$. Finally, we study the relations between the $L_\infty$-algebras appearing above.

We define a new variant of Rabinowitz Floer homology that is particularly well suited to studying the growth rate of leaf-wise intersections. We prove that for closed manifolds $M$ whose loop space $\Lambda M$ is "complicated", if $\Sigma \subseteq T^*M$Σ⊆ T*M is a non-degenerate fibrewise starshaped hypersurface and $\varphi \in \mathrm{Ham}_c (T^*M,\omega)$ is a generic Hamiltonian diffeomorphism then the number of leaf-wise intersection points of $\varphi$ in $\Sigma$ grows exponentially in time. Concrete examples of such manifolds are $(S^2 \times S^2)\#(S^2\#S^2)$, $\mathbb{T}^4\#\mathbb{C}P^2$, or any surface of genus greater than one.