## Summary

I obtained my PhD in January 2013, at the university Paris 6 under the supervision of Raphaël Krikorian. The title of my thesis is *Global **aspects of the reducibility of quasi-periodic cocycles in semisi**mple compact Li**e groups*, and therein I obtained a generalization of R. Krikorian's theorem on the global density of reducible cocycles in TxSU(2) in the C-infinity topology. I also established a Differentiable rigidity theorem for cocycles reducible to constant ones satisfying a relevant Diophantine condition.

I continued my research on the subject during my first post-doc at the university Paris 7, under the supervision of Artur Avila. I worked mainly on obtaining a version of a theorem of Avila-Fayad-Kocsard on the existence of Distributionally Uniquely Ergodic skew-product diffeomorphisms on TxG (G a homogeneous space of the compact type) with fixed (and moreover Diophantine) frequency. I also showed that in the space of cocycles in TxSU(2), under a Recurrent Diophantine condition on the frequency, there exist no counter-examples to a conjecture by Anatole Katok on Cohomologically Rigid diffeomorphisms.

During my second post-doc at UFF, supervised by Alejandro Kocsard, I started working more actively on Katok's Conjecture. I also continued my work on the dynamics of cocycles in T^{d}xSU(2) and established a spectral dichotomy between measurable reducibility (and thus pure-point spectrum) and weak mixing in the fibers (and thus singular continuous spectrum in an appropriate subspace of L^{2}(T^{d}xSU(2)).

I am currently working on the generalization of Denjoy theory in higer-dimensional tori, under the supervision of Sebastian van Strien

## Publications

### Journals

Karaliolios N, Global aspects of the reducibility of quasiperiodic cocycles in semisimple compact Lie groups, *M\'emoires De La Soci\'et\'e Math\'ematique Fran\c{c}aise, 146 (2016)*

Karaliolios N, Cohomological rigidity and the Anosov-Katok construction

Karaliolios N, 2018, Local Rigidity of Diophantine Translations in Higher-dimensional Tori, *Regular & Chaotic Dynamics*, Vol:23, ISSN:1560-3547, Pages:12-25