Wants to develop mathematical theory for negative dependence

As a Wallenberg Scholar, Petter Brändén wants to develop the mathematical theory of negative dependence.

Negative dependence is an umbrella term for mathematical modelling of repelling particles in statistical physics, and negative dependent events in statistics. It is manifested in inequalities such as negative correlation, negative association and log-concavity. In recent years, it has been made evident that negative dependence is of utmost importance in several mathematical problems in various fields such as combinatorics, algebraic geometry, mathematical statistics and computer science.

Despite many attempts, it has long been difficult to develop a working mathematical theory of negative dependence. In recent years, however, two successful theories have been developed. Petter Brändén is one of the pioneers in developing a theory of negative dependence based on the geometry of the zeros of polynomials in several variables. A polynomial is an algebraic expression where several different terms have been combined by addition, subtraction and multiplication

Develop mathematical theory

Another successful theory uses methods from Hodge theory and algebraic geometry to explain negative dependence. Using these two theories, several open problems in mathematics have been solved.

June Huh and Petter Brändén recently brought together ideas from the two aforementioned theories to form a theory of so-called Lorentzian polynomials and used the theory to answer a famous 1972 problem of independence.

Several important problems on negative dependence remain unsolved and Petter Brändén now wants to develop the mathematical theory of negative dependence to try to solve these.