Here is a brief overview of my current research and key publications on each topic. Whilst I try to keep this page up to date, the best way to see what I am doing at the moment is by checking my Google Scholar page.

Functional analysis of neural networks and neural operators
Universal approximation properties of various types of neural networks have been known since the late 1980’s. However, it has also been shown that the approximation rates in terms of the number of neurons scale exponentially with the dimension of the input space. However, certain types of functions can be approximated with dimension-independent Monte-Carlo rates.

When neural networks are used in inherently infinite-dimensional applications such as inverse problems and imaging, they need to be considered as nonlinear operators between infinite-dimensional spaces, rather than functions between Euclidean spaces (even if these are high-dimensional). The generalisation from high but finite dimensions to infinite dimensions is far from trivial and requires advanced functional-analytic techniques.

  • Bartolucci, F., Carioni, M., Iglesias, J. A., Korolev, Y., Naldi, E., and Vigogna, S. (2026). A Lipschitz Spaces View of Infinitely Wide Shallow Neural Networks. SIAM Journal on Mathematical Analysis, 58(3), 2786–2828. https://doi.org/10.1137/24M1718615
  • Furuya, T., Korolev, Y., and Yaguchi, T. (2026). Approximation of Maximally Monotone Operators: A Graph Convergence Perspective. Advances in Neural Information Processing Systems.
  • Korolev, Y. (2022). Two-layer neural networks with values in a Banach space. SIAM Journal on Mathematical Analysis, 54(6), 6358–6389.

Mathematics of transformers
Transformers have become the backbone of many modern AI systems. A series of recent works have demonstrated that they can be understood mathematically as transformations of measures, hence they inherently have an infinite-dimensional domain and range. Under some assumptions on the weights, the propagation of a measure through the transformer follows a gradient flow in the space of probability measures on the unit sphere under a variant of the Wasserstein metric with a non-local mobility term. This perspective allows one to investigate the emergence of either clusters or absolutely continuous measures in the large-time limit and to characterise them as stationary points of an interaction energy.

  • Burger, M., Kabri, S., Korolev, Y., Roith, T., and Weigand, L. (2025). Analysis of mean-field models arising from self-attention dynamics in transformer architectures with layer normalization. Philosophical Transactions of the Royal Society A, 383(2298), 20240233 (48 pages). https://doi.org/10.1098/rsta.2024.0233

Image reconstruction in light-sheet microscopy
In light-sheet microscopy the data are corrupted by spatially varying blur and a combination of Poisson and Gaussian noise. The spatial variation of the point spread function (PSF) of a light-sheet microscope is determined by the interaction between the excitation sheet and the detection objective PSF. This requires careful modelling of the image formation process as well as the measurmeent noise.

  • Toader, B., Boulanger, J., Korolev, Y., Lenz, M. O., Manton, J., Schönlieb, C.-B., and Mureşan, L. (2022). Image Reconstruction in Light-Sheet Microscopy: Spatially Varying Deconvolution and Mixed Noise. Journal of Mathematical Imaging and Vision, 64(9), 968–992. https://doi.org/10.1007/s10851-022-01100-3

L-infinity variational problems
I am interested in minimisers of Rayleigh quotients involving $L^\infty$ type norms such as the $W^{1,\infty}$ Sobolev norm:

\begin{equation} \label{eq:quotient} \min_u \frac{\|u\|_{W^{1,\infty}}}{\|u\|_{L^p}}, \end{equation}

where $p \leq \infty$. For $p=\infty,$ many global minimisers exist: ground states of the $\infty$‑Laplacian $\Delta_\infty$ (solutions of $\min(|\nabla u|- \lambda u,-\Delta_\infty u)=0$ with an appropriate constant $\lambda$), $\infty$‑harmonic functions (solutions of $\Delta_\infty u = 0$) and distance functions are all minimisers of \eqref{eq:quotient} for $p=\infty$. For any $p < \infty$ the distance function is the only global minimiser and the only positive local minimiser, which motivates an efficient numerical algorithm for computing the distance function using a gradient flow with guaranteed convergence from any positive initialisation. For $p = \infty$ we develop a convex duality approach to the L-infinty variational problem (\ref{eq:quotient}) and characterise all minismisers as solutions of a divergence-type PDE.

  • Bungert, L., and Korolev, Y. (2022). Eigenvalue Problems in L^∞: Optimality Conditions, Duality, and Relations with Optimal Transport. Communications of the American Mathematical Society, 2, 345–373.
  • Bungert, L., Korolev, Y., and Burger, M. (2020). Structural analysis of an L-infinity variational problem and relations to distance functions. Pure and Applied Analysis, 2(3), 703–738.