Abstract: In 1985, Gromov proved the remarkable Non-Squeezing Theorem: a symplectic embedding of a closed ball of radius r into the symplectic cylinder $Z^{2n}(R)=B^2(R)\times \mathbb{R}^{(2n-2)}$ exists if and only if $r\le R$. This show that in general, symplectic embeddings have more obstructions than volume constraints. One of the obstructions is known as the Gromov’s capacity. In this thesis, we study the problem of symplectic embeddings of five disjoint closed balls of Gromov’s capacities $c_1,…,c_5$ into the complex projective space $\mathbb{C}P^2$. By investigation of the action of the group of symplectomorphisms of $\mathbb{C}P^2$ on the space of symplectic embeddings, the homotopy type of the space of symplectic embeddings can be computed: it is homotopy equivalent to a union of strata in the configuration space of five points on $\mathbb{C}P^2$, with the precise strata determined by the chosen capacities. Moreover, the homotopy type of the space of symplectic embeddings of five balls remains constant as the capacities vary within any given stability chamber of capacities. The complete set of stability chambers is also determined.

Abstract: Synthetic algebraic geometry (SAG) is an extension of homotopy type theory that provides a language for internal reasoning about the big Zariski topos. In SAG, we postulate the existence of a generic local ring R with some additional properties. Schemes over R are not defined by giving the underlying space a structure sheaf; rather, they are defined by a property of the space itself. Sheaves on a scheme are then expressed as bundles over the scheme, and on the sheaves themselves we have many of the usual operations, such as taking cohomology.

However, algebraic geometry often looks different from this internal point of view, compared to the classical external one. For instance, we can show that the generic local ring R is not Noetherian, and so the category of finitely presented R-modules is not abelian. In particular, the cohomology groups of sheaves of finitely presented R-modules may no longer be finitely presented. In this talk, we shall study the abelian closure of the finitely presented R-modules in the category of all R-modules, which we call the finitely adequate R-modules. We will characterize the finitely adequate R-modules which are injective and projective in this subcategory. Then, we prove that finitely adequate R-modules are closed under extensions. We hope that the category of finitely adequate R-modules gives us a suitable replacement for the category of finitely presented modules, so that the cohomology groups of finitely adequate sheaves are finitely adequate.