Noncommutative harmonic analysis & spectral theory · Mathematics of deep learning · Almaty, Kazakhstan
Endorsed by the Royal Society under the UK Global Talent route (Exceptional Promise, April 2026)
My research is about how the spectrum of an operator controls analysis where classical Fourier analysis is not available: non-abelian and non-unimodular groups, compact quantum groups, and general von Neumann algebras. Multipliers are the operators affiliated with the algebra, and the decay of the spectral projections of one Dirac- or Laplace-type operator is enough to prove sharp Lp–Lq estimates, Hardy–Littlewood–Paley inequalities and Hörmander–Mikhlin multiplier theorems. Key collaborations include work with Michael Ruzhansky on spectral multipliers and with Alexis Arnaudon on contractions of group representations via geometric quantization.
I am developing two linked programmes, set out in grant proposals prepared with Michael Ruzhansky: (i) regularity of spectral and Fourier multipliers in affiliated von Neumann algebras, with heat and Klein–Gordon decay estimates governed by an intrinsic spectral dimension; and (ii) a global pseudo-differential calculus directly on von Neumann algebras, combining compatible Fourier structures, quantum Riemannian geometry and Safarov's connection-based calculus, with Lp regularity, Gårding-type inequalities and applications to nonlinear evolution equations. The test cases are quantum tori, quantum Euclidean spaces, quantum groups and symmetric spaces.
A second strand applies the same spectral and geometric toolkit to the mathematics of deep learning: the geometry of loss landscapes, degenerate minima and the training dynamics of neural networks, and the structure of learned representations. As one probe, I use log-signature representations of sequences to study how transformers compress and organise contextual information. Recent work explores bridging byte-latent transformers with Qwen bodies through representation re-alignment, combining subword tokenization with continuous latent representations.
Alongside research I work in industry. I am currently at Kaspi Bank (AI strategy and statistical modelling), and before that worked on tabular foundation models and LLM evaluation at HighSky (2024–2026).
Note: Log-signatures as structured compression and probes of representation geometry (March 2026)
12 peer-reviewed articles, 2 preprints; 220+ citations; h-index 7 (Google Scholar, 2026). For live metrics see Google Scholar.
| 1. |
Hörmander–Mikhlin type theorem on non-commutative spaces
arXiv preprint arXiv:2503.01240
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| 2. |
Norms of certain functions of a distinguished Laplacian on the ax+b groups
Mathematische Zeitschrift, 302(4):2327–2352
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| 3. |
Contractions of group representations via geometric quantization
Letters in Mathematical Physics, 110(1):43–59
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| 4. |
Lp–Lq multipliers on locally compact groups
Journal of Functional Analysis, 278(3):108324
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| 5. |
Re-expansions on compact Lie groups
Analysis and Mathematical Physics, 10(3):33
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| 6. |
Hardy–Littlewood, Hausdorff–Young–Paley inequalities, and Lp–Lq Fourier multipliers on compact homogeneous manifolds
Journal of Mathematical Analysis and Applications, 479(2):1519–1548
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| 7. |
Smooth dense subalgebras and Fourier multipliers on compact quantum groups
Communications in Mathematical Physics, 362(3):761–799
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| 8. |
Net spaces on lattices, Hardy–Littlewood type inequalities, and their converses
Eurasian Mathematical Journal, 8(3):10–27
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| 9. |
Hausdorff–Young–Paley inequalities and Lp–Lq Fourier multipliers on locally compact groups
arXiv preprint arXiv:1510.06321
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| 10. |
Fourier multipliers and group von Neumann algebras
Comptes Rendus Mathématique, 354(8):766–770
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| 11. |
Hardy–Littlewood–Paley-type inequalities on compact Lie groups
Matematicheskie Zametki, 100(2):287–290
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| 12. |
Hardy–Littlewood–Paley inequalities and Fourier multipliers on SU(2)
Studia Mathematica, 234(1):1–29
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| 13. |
Well-posed solvability of functional-differential equations with unbounded operator coefficients
Differential Equations, 50(9):1161–1172
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| 14. |
On an inequality for the non-increasing rearrangement of a function
Eurasian Mathematical Journal, no. 3, 34–36
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Prof. Michael Ruzhansky (Imperial / Ghent); Prof. Shahn Majid (QMUL); Prof. Yulia Kuznetsova (Franche-Comté); Prof. Terry Lyons FRS (Oxford); Dr Alexis Arnaudon (Imperial); Prof. Fedor Sukochev (UNSW, planned).