Russian Mathematician Ivan Remizov Derives Universal Formula for Long-Unsolvable Differential Equations

The global mathematical community is acknowledging a significant theoretical advance following the January 27, 2026, announcement in Almaty, Kazakhstan, where Russian mathematician Ivan Remizov presented a universal analytical formula for a specific class of differential equations. This achievement effectively resolves a problem that had been considered analytically intractable since the seminal work of Joseph Liouville in 1834, marking a turning point in the science of complex computing.

Remizov’s solution targets second-order differential equations characterized by variable coefficients, which are fundamental to modeling dynamic phenomena ranging from the trajectories of celestial bodies to the propagation of electrical signals. For nearly two centuries, the prevailing consensus, rooted in Liouville’s 1834 findings, suggested that a general analytical solution for this equation type using standard mathematical operations was unattainable. Liouville’s theorem, which restricts antiderivatives expressible as elementary functions, formed a theoretical barrier.

Remizov’s innovation reportedly circumvents these historical limitations by employing a novel methodology: utilizing the Laplace transform to reconstruct a static solution, referred to as the resolvent, from time-evolving 'slices' of the problem. This conceptual leap, published in the Vladikavkaz Mathematical Journal, provides a new avenue for tackling problems previously deemed unsolvable in elementary terms, such as the Gaussian integral or the logarithmic integral.

Ivan Remizov is affiliated with prominent Russian scientific institutions, holding a senior research fellow position at the Higher School of Economics (HSE) University–Nizhny Novgorod and the Institute for Information Transmission Problems of the Russian Academy of Sciences (IPPI RAN). The IPPI RAN, founded in late 1961, focuses on information theory and complex dynamical systems, while the HSE Faculty of Mathematics, established in 2008, is a key center for mathematical research.

The significance of this result extends beyond pure mathematics, as these differential equations govern many physical systems. Remizov’s universal formula is noted for its potential to simplify the mathematical modeling of complex, chaotic systems, including the chaotic movement of elementary particles and the predictable orbits of planets. This unification of laws governing both microscopic quantum phenomena and macroscopic celestial mechanics into a single formula is generating interest in astrophysics and engineering sectors.

Furthermore, a related achievement by Remizov and Oleg Galkin in 2025 involved solving a 57-year-old problem concerning the convergence speed for Chernoff approximations of operator semigroups, suggesting a sustained period of high-level productivity from the researchers involved. This work is situated within the research tradition of the HSE University–Nizhny Novgorod Department of Fundamental Mathematics, which actively explores the combination of topology and algebraic geometry to solve complex analytical problems.