We prove that for $n\geq2$ there exists a compact subset $X$ of the closed ball in $\mathbb{R}^{2n}$ of radius $\sqrt{2}$, such that $X$ has Hausdorff dimension $n$ and does not symplectically embed into the standard open symplectic cylinder. The second main result is a lower bound on the $d$th regular coisotropic capacity, which is sharp up to a factor of $3$. For an open subset of a geometrically bounded, aspherical symplectic manifold, this capacity is a lower bound on its displacement energy. The proofs of the results involve a certain Lagrangian submanifold of linear space, which was considered by Audin and Polterovich.

We prove that for $n\geq2$ there exists a compact subset $X$ of the closed ball in $\mathbb{R}^{2n}$ of radius $\sqrt{2}$, such that $X$ has Hausdorff dimension $n$ and does not symplectically embed into the standard open symplectic cylinder. The second main result is a lower bound on the $d$th regular coisotropic capacity, which is sharp up to a factor of $3$. For an open subset of a geometrically bounded, aspherical symplectic manifold, this capacity is a lower bound on its displacement energy. The proofs of the results involve a certain Lagrangian submanifold of linear space, which was considered by Audin and Polterovich.

In Contact 3-manifolds, twenty years since J. Martinet’s work, Eliashberg proved that two overtwisted contact structures on a closed oriented 3-manifold are isotopic through contact structures if and only if they are homotopic as 2-plane fields. We provide an alternative proof of this theorem using the convex surface theory and bypasses.

We study the half-form Kähler quantization of a smooth symplectic toric manifold $(X,\omega)$, such that $[ \omega/ 2\pi]- c_{1}(X)/2 \in H^{2}(X,{\mathbb{Z}} )$ and is non-negative. We define the half-form corrected quantization of $(X,\omega)$ to be given by holomorphic sections of a certain Hermitian line bundle $L\to X$ with Chern class $[ \omega/ 2\pi]- c_{1}(X)/2$. These sections then correspond to integral points of a "corrected" polytope $P_{L}$ with integral vertices. For a suitably translated moment polytope $P_{X}$ for $(X,\omega)$, we have that $P_{L}\subset P_{X}$ is obtained from $P_{X}$ by a one-half inward-pointing normal shift along the boundary.

We use our results on the half-form corrected Kähler quantization to motivate a definition of half-form corrected quantization in the singular real toric polarization. Using families of complex structures studied in Baier-Florentino-Mourão-Nunes, which include the degeneration of Kähler polarizations to the vertical polarization, we show that, under this degeneration, the half-form corrected $L^{2}$-normalized monomial holomorphic sections converge to Dirac-delta-distributional sections supported on the fibers over the integral points of $P_{L}$, which correspond to corrected Bohr-Sommerfeld fibers. This result and the limit of the corrected connection, with curvature singularities along the boundary of $P_X$, justifies the direct definition we give for the corrected quantization in the singular real toric polarization. We show that the space of quantum states for this definition coincides with the space obtained via degeneration of the Kähler quantization.

We also show that the BKS pairing between Kähler polarizations is not unitary in general. On the other hand, the unitary connection induced by this pairing is flat.

A new relation between homoclinic points and Lagrangian Floer homology is presented: in dimension two, we construct a Floer homology generated by primary homoclinic points.We compute two examples and prove an invariance theorem. Moreover, we establish a link to the (absolute) flux and growth of symplectomorphisms.