We give a detailed proof of Siu’s theorem on extendibility of holomorphic vector bundles of rank larger than one, and prove a corresponding extension theorem for holomorphic sprays. We apply this result to study ellipticity properties of complements of compact subsets in Stein manifolds. In particular we show that the complement of a closed ball in $\mathbb{C}^{n}, n \geq3$ , is not subelliptic.

We prove that for any free ergodic probability measure-preserving action $${\mathbb{F}_n \curvearrowright (X, \mu)}$$ of a free group on$n$generators $${\mathbb{F}_n, 2\leq n \leq \infty}$$ , the associated group measure space II_{1}factor $${L^\infty (X)\rtimes \mathbb{F}_n}$$ has$L$^{∞}($X$) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II_{1}factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II_{1}factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products.

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We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics literature. In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regularity phenomena in the corresponding non-uniformly hyperbolic cocycles, which is also of interest as both the first natural example of some of those phenomena and, more generally, the first non-artificial model where non-regularity can be explicitly studied.

We investigate Sarnak's M\"obius Disjointness Conjecture through asymptotically periodic functions. It is shown that Sarnak's conjecture for rigid dynamical systems is equivalent to the disjointness of M\"obius from asymptotically periodic functions. We give sufficient conditions and a partial answer to the later one. As an application, we show that Sarnak's conjecture holds for a class of rigid dynamical systems, which improves an earlier result of Kanigowski-Lema{\'{n}}czyk-Radziwi{\l}{\l}.