# Complex analysis and convexity

​Neither ”complex” nor ”analysis” are words that are particularly uncommon in the ordinary language. But the mathematical meanings of these words are far more precise. Bo Berndtsson, Professor in Mathematics, work in the research field of Complex Analysis.
– Analysis in mathematics, that is when functions are analysed  using derivatives and integrals. Above all it was Newton that in the 1600s began to use  analysis to describe the physical reality and showed that it is possible to calculate and thus predict mechanical processes, everything from rifle bullets to planetary orbits. This is about  functions that depend on real numbers, which can be measured for example in time or in space. A great discovery in the 1800s was the use of the complex numbers, a mathematical construction where numbers are introduced that do not exist in “reality”. This was  initially a tool of mathematics, for example when trying to solve cubic equations, but turned out to also be very useful in physics. To get complex numbers, all the ordinary numbers are taken with the addition of the square root of -1, which abbreviated is called i (from imaginary). And when you have a function that depends on both real and imaginary numbers, then it is complex analysis.

During the 1900s, complex analysis was something of a Swedish specialty. The influential mathematician Mittag-Leffler researched in the area at the beginning of the century, and the tradition was carried on by many prominent researchers such as Carleman, Beurling and Carleson. Bo began his career as a “pure complex analyst” but followed a trend that took place in the field and went more and more towards geometry. A little more than ten years ago he was head of the department for three years, and when he returned to the research he felt like it would be a good opportunity to start with something new. More or less by chance, he found a connection between complex analysis and convexity.

A convex function is a function that can be illustrated with a curve that is “bowl shaped”. It has long been known that convex functions play a role in complex analysis, but in a rather superficial way. At the same time the field of convex geometry has emerged. The seemingly very simple concept of “convexity” has resulted in a rich theory with many deep and surprising results. One concept that emerged is called Brunn-Minkowski inequality, and Bo discovered a complex version of this.

– Much is happening when working with analogies. Many people believe that mathematics is about logic, that it is like a logical game with a set of rules for which conclusions one is allowed to draw. But the fun part is to find out in which direction that development can go and what conclusions are interesting, and for me, analogies taken from the convex geometry has given interesting things to study in complex geometry. Surprisingly, they have proven to be useful also in other mathematical areas, such as Kähler geometry and Algebraic geometry.

The research group, which is called Complex analysis in several variables and includes about ten people, is successful with its research and when it comes to getting grants and prizes. There is a good deal of collaboration within the group, but also subgroups whose research is not directly related to each other. Bo was recently awarded Eva and Lars Gårding’s Prize in mathematics, a prize that is every two years awarded in linguistics and in every other in mathematics. Lars Gårding was a well-known mathematics professor at the University of Lund, who died in 2014, and the prize is to encourage research and reward academic skills in mathematics.

– I have a number of different projects in progress and there are still many interesting directions left to investigate, but the low-hanging fruits of the field might be picked now, Bo concludes.

​Text: Setta Aspström
Photo: T. Sjöstedt: Bo Berndtsson at the statue The Oracle by Tilda Lovell
Picture: Brunn-Minkowski inequality and its complex counterpart

Page manager Published: Fri 26 Apr 2019.