Nizhny Novgorod Scientist Solves 200-Year-Old Math Problem

A scientist from the Nizhny Novgorod branch of HSE University, Ivan Remizov, has made a conceptual breakthrough in the theory of differential equations. He has derived a universal formula for solving problems that were considered unsolvable analytically for over 190 years. The obtained result radically changes the worldview in one of the oldest fields of mathematics, according to the university«s press service.

«Imagine that the sought solution to the equation is a large painting. It is very difficult to view it all at once. But mathematics excels at describing processes that evolve over time.

The result of the work is a theorem that allows «slicing» this process into many small, simple frames, and then, using the Laplace transform, assembling these frames into a single static picture—the solution to the complex equation, i.e., the resolvent. Simply put, instead of guessing what the painting looks like, the theorem allows restoring its appearance by quickly scrolling through the «film reel» of its creation,« explained Remizov.

The second-order differential equations in question are used, for example, for modeling real-world events and defining new functions that cannot be specified otherwise. These include the so-called Mathieu and Hill special functions, which are critically important for understanding how satellites move in orbit or protons in the Large Hadron Collider.

Previously, Ivan Remizov has already contributed to solving another «eternal» problem. In 1968, American mathematician Paul Chernoff proposed a method for approximately computing operator semigroups, which describe changes in complex systems. However, it long remained unclear how quickly this method leads to accurate results. This was pondered for over 50 years, until our mathematicians solved the problem.