Who was Nevanlinna and what is his prize?

BLOG: Heidelberg Laureate Forum

Laureates of mathematics and computer science meet the next generation
Heidelberg Laureate Forum

Mathematics and computer science do not have a Nobel Prize. Instead, these areas have their own prizes to honor outstanding achievement. Mathematics has the Fields Medal and the Abel Prize. Computer science has the Turing Award. And for those straddling mathematics and computer science, there is the Rolf Nevanlinna Prize

for outstanding contributions in mathematical aspects of information sciences including all mathematical aspects of computer science… scientific computing and numerical analysis, computational aspects of optimization and control theory, computer algebra.

The Nevanlinna Prize was created by the Executive Committee of the International Mathematical Union in 1981 to honor Rolf Nevanlinna (1885–1980). As with the Fields Medal, the Nevanlinna Prize is only awarded to mathematicians who are younger than 40 at the beginning of the year the prize is awarded.

This year four winners of the Nevanlinna Prize, half of all winners since the prize was first awarded in 1982, will be attending the Heidelberg Leadership Forum.

Rolf Nevanlinna was one of the most famous Finnish mathematicians. His father, Otto Neovius-Nevanlinna (1867–1927), and his grandfather, Edavard Engelbert Neovius (1823–88), were mathematicians. His mathematical genealogy traces back through his advisor Ernst Leonard Lindelöf (1870–1946) and Lindelöf’s advisor Robert Hjalmar Mellin (1854–1933). Nevanlinna, Lindelöf, and Mellin all made contributions to the area of mathematics known as complex analysis and their names have become mathematical adjectives. Nevanlinna is remembered for Nevanlinna theory discussed below. Lindelöf’ is known for Lindelöf’s theorem, the Phragmén-Lindêlöf theorem, and Lindelöf topological spaces. Mellin is known for the Mellin transform.

Nevanlinna studied meromorphic functions. The simplest definition of meromorphic functions is functions that can be written as the ratio of two entire functions, where an entire function is a complex valued function that is complex-differentiable everywhere. Entire functions cannot have any singularities, places where the function becomes infinite, but meromorphic functions can, and so meromorphic functions can be more challenging to work with. Part of Nevanlinna’s achievement was to generalize theorems about entire functions to counterparts for meromorphic functions.

Many of Nevanlinna’s results, now grouped under the heading of Nevanlinna theory, are quite technical and require a fairly lengthy sequence of definitions to state. However, some of his results, such as the one below, are simpler to express and can be quite surprising.

Suppose f(z) and g(z) are two functions meromorphic in the plane. Suppose also that there are five distinct numbers a1, …, a5 such that the solution sets {z : f(z) = ai} and {z : g(z) = ai} are equal. Then either f(z) and g(z) are equal everywhere or they are both constant.

Like many theorems in complex analysis, the conclusion of this theorem seems out of proportion to its hypotheses. Being meromorphic in the complex plane is not a terribly restrictive condition, and yet two such functions that coincide on five solution sets must coincide everywhere, unless both are constant.

Incidentally, one connection between the Nevanlinna Prize and the Fields Medal is that Nevanlinna’s student Lars Alfors was one of two people awarded the first Fields Medals.

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