A complex projective variety is called rational if there is a Zariski-open subset on which it is isomorphic to a Zariski-open subset of projective space. There has been a huge amount of progress and activity in determining when varieties are rational. One the other hand, one can ask: given a projective variety whose nonrationality is known, can we measure how far it is from being rational?

Measures of irrationality provide an answer to the question above; they are birational invariants that offer an orthogonal viewpoint to questions concerning rationality. They have recently gained interest, in part due to work of Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery [BDELU] on hypersurfaces of large degree. In this dissertation, we make advances in the study of measures of irrationality on abelian surfaces and codimension two complete intersections, which answer a few questions posed in [BDELU].

The homotopy theory of rationalized simply connected spaces was shown by Quillen to be encoded algebraically in differential graded Lie algebras in his seminal work on rational homotopy theory. Motivated by this theory and Whitney's treatment of differential forms on arbitrary complexes, Sullivan later described a theory of computable algebraic models for rational homotopy types in terms of differential graded algebras of differential forms in his "Infinitesimal Computations in Topology". Following a problem posed therein, we give a characterization of the simply connected rational homotopy types, together with a choice of rational Chern classes and fundamental class, that are realized by closed almost complex manifolds in complex dimensions three and greater, with a caveat in complex dimensions congruent to two modulo four depending on the first Chern class. As a consequence, beyond demonstrating that rational homotopy types of closed almost complex manifolds are plenty, we observe that the realizability of a simply connected rational homotopy type by a closed almost complex manifold, of complex dimension not congruent to two modulo four, depends only on its cohomology ring.

In complex algebraic geometry, the decomposition theorem asserts that semisimple geometric objects remain semisimple after taking direct images under proper algebraic maps. This was conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a series of long papers via harmonic analysis and D-modules.

In this thesis defense, we would like to explain a simpler proof in the case of semisimple local systems, using a more geometric approach adapting de Cataldo- Migliorini. On the one hand, we complement Simpson’s theory of weights for local systems by proving a global invariant cycle theorem in the setting of local systems. On the other hand, we define a notion of polarization via Hermitian forms on pure twistor structures. This is partially based on joint work with Chuanhao Wei.