## Summary

I am a Pure Mathematics PhD student. My research interests lie in the interplay between noncommutative geometry and harmonic analysis, representation theory, spectral theory, operator algebras, quantum groups and manifolds, deformations of Lie groups and Lie algebras.

To be more specific, I study smooth structure, a Dirac operator and the Dirac operator, Fourier multipliers on topological groups, quantum groups and spectral triples, My most recent research (joint with professor Majid and professor Ruzhansky) is focused on developing harmonic analysis, theory of distributions and pseudo-differential calculus on spectral triples. Interesting applications are to elliptic and hyperbolic equations, their geometric analysis and further applications of these properties to nonlinear equations.

Fundamental problem in representation theory is to "describe" the unitary dual of a topological group. In another project (with Alexis Arnaudon), I investigate how certain degenerations of Lie group structure are reflected on the unitary dual. The main tool is geometric quantisation. As an application, we obtain transference-type results linking harmonic analysis on Lie groups.

## Publications

### Journals

Akylzhanov R, Majid S, Ruzhansky M, Smooth dense subalgebras and Fourier multipliers on compact quantum groups

Akylzhanov R, Nursultanov E, Ruzhansky M, 2015, Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and Lp-Lq Fourier multipliers on compact homogeneous manifolds

Akylzhanov R, Ruzhansky M, $L^p$-$L^q$ multipliers on locally compact groups

Akylzhanov R, Ruzhansky M, 2016, Fourier multipliers and group von Neumann algebras, *Comptes Rendus Mathematique*, Vol:354, ISSN:1631-073X, Pages:766-770