At his desk at KTH Royal Institute of Technology, Brändén prefers to work with pen and paper as he – step by step – devises new solutions to problems in his field of mathematics. In his capacity as a Wallenberg Scholar, he has set his sights on problems regarding negative dependence.

Negative dependence

In physics, negative dependence may mean that two magnitudes or phenomena have a special type of relationship to one another: when one magnitude increases the other decreases. A simple example of negative dependence in motion is a teeter-totter in a playground. As the child at one end goes up, the child at the other end goes down.

“In medicine the term ‘correlation’ is often used, i.e., that being overweight increases the risk of cardiovascular disease, for example. This represents a positive correlation or a positive dependence. An example of negative dependence is then that regular exercise is negatively correlated to dying of cardiovascular disease,” says Brändén.

In recent years negative dependence has also become an increasingly important concept in mathematics. Finding ways of proving negative dependence is of great importance in a number of mathematical fields, such as combinatorics, algebraic geometry, mathematical statistics and computer science.

But although many have tried, it has proven difficult to formulate an effective mathematical theory of negative dependence. Two successful theories have been developed over the past few years, however. Brändén is the pioneer behind one of them.

“Successful theories of positive dependence in the fields of statistical physics and probability theory were developed in the 1970s. But no such advances for negative dependence were made until we realized that properties of zeroes of polynomials can be used to prove negative dependence. This led to development of a new theory.”

A polynomial is an algebraic expression consisting of coefficients and variables, along with mathematical operations such as addition, subtraction and multiplication. The zeroes are the inputs for which the polynomial evaluates to zero.

Two become one

After Brändén had published his method, the American mathematician June Huh worked on certain problems involving negative dependence. He used other fields of mathematics to approach the problems, however. In particular, he used algebraic geometry and methods from Hodge theory. Hence there were

 two parallel theories on negative dependence. In the summer of 2018 Brändén and Huh both lectured at a summer school at Nordfjordeid in Norway. As they attended each other’s lectures, they realized striking similarities between the two theories.

“Shortly afterward we began to collaborate and things went pretty quickly from there. Together, we developed the Lorentzian polynomials method, which was a great success and soon became widely recognized.”

A scientific article describing Lorentzian polynomials was published in 2019. The method has been found to be a useful way of addressing a number of problems in different fields. One of the more classical applications is the “Traveling Salesman Problem” (TSP) in computer science and mathematical optimization. Simply put, it describes the task of finding the shortest route for a traveling salesman between a number of destinations.

“There are multiple examples of research in the overlap between Lorentzian polynomials and computer science. One goal is to analyze and improve various algorithms.”

But the method also has some limitations. Although it can be used for multiple problems, Lorentzian polynomials are not as powerful a method as the one earlier developed by Brändén. He has therefore taken on the task of refining the methods so they can be used to address more problems in the field of negative dependence.

No pop star

It was not until he went to university that Brändén really discovered the attraction of mathematics. At high school he was more interested in music and literature.

“But I gradually realized that I wasn’t going to be a pop star and enrolled on the engineering physics program at Chalmers University of Technology in Gothenburg.”

In Gothenburg he also studied theoretical philosophy. He thinks that mathematics and philosophy have much in common.

I have always seen the link between mathematics and philosophy. After all, mathematics involves logic, which is a part of philosophy. In both these disciplines the concept of truth is central.

“At that age it’s common to be attracted to subjects like philosophy or mathematics that enable you to embark on a quest to find answers to the big questions in life. And that was what initially inspired me. Later on I discovered that I was good at solving problems.”

Nowadays he is driven by a desire to develop a better understanding of the mathematics he is worked on. This he usually does by spending long hours at his desk. But sometimes he arrives at insights more unexpectedly. One such breakthrough came to him during a lunchtime walk when he was a PhD student.

“The right approach came to me in a flash, and after twenty minutes at the computer I had managed to disprove a certain mathematical conjecture. At that point I had been working on the problem for two years to no avail.”

Alongside his research he teaches at both undergraduate and doctoral level.

“Research alone would be too monotonous. It’s good to intersperse it with teaching, where there is less pressure. And it’s also very rewarding to be able to teach others what you know.”

Text Magnus Trogen Pahlén
Translation Maxwell Arding
Photo Magnus Bergström