Imperial College London

DrNikolaosKaraliolios

Faculty of Natural SciencesDepartment of Mathematics

Research Associate
 
 
 
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Contact

 

n.karaliolios CV

 
 
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Location

 

Huxley BuildingSouth Kensington Campus

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Summary

 

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 semisimple compact Lie 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 TdxSU(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 L2(TdxSU(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, Local Rigidity of Diophantine translations in higher dimensional tori

Karaliolios N, 2017, DIFFERENTIABLE RIGIDITY FOR QUASIPERIODIC COCYCLES IN COMPACT LIE GROUPS, Journal of Modern Dynamics, Vol:11, ISSN:1930-5311, Pages:125-142

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