Our research aims at advancing the frontiers of computer-aided proofs in mathematical analysis. This area of research is geared to deal with problems that cannot be solved by traditional mathematical methods alone.
Typically, such problems have a global component as well as a non-linear ingredient. Hard problems of this type have traditionally been studied through numerical computations alone, and therefore our knowledge of these lack the rigour demanded by a mathematical proof. Our research aims to bridge the gap between a numerically observed phenomenon, and its mathematical counterpart.
Bringing computers into the mathematical framework of a proof is achieved by developing a means of computing numerically yet with mathematical rigour – validated numerics. The underlying arithmetic is set-valued, and is based on the inclusion principle that enables us to enclose the image of a function or operator. By carefully designing the numerical method, we can ensure that the result of the original problem is enclosed in the set-valued output of the program.
These techniques, and their underlying principles, play an instrumental role as computer simulations become increasingly dominant in scientific research, gradually replacing the need for physical experimentation. Indeed, this is the only reasonable way to certify that a numerical computation meets required error tolerances. This sort of certification is important in control of industrial robots, in the manufacturing of new drugs, and vital in computer-aided proofs.
Principal investigator: Warwick Tucker
After receiving his doctoral degree in mathematics at Uppsala University
in 1998, proving that the Lorenz attractor exists, Tucker spent two years at IMPA (Rio de Janeiro, Brasil) as a postdoctoral fellow.
During the years 2000 to 2002, Tucker held a H.C. Wang assistant professorship at Cornell University (Ithaca, USA) teaching and doing research in mathematics. During this period, Tucker was awarded the Swedish Mathematical Society’s Wallenberg Prize, and the R.E. Moore Prize for Applications of Interval Analysis.
Upon his return to Sweden, Tucker was awarded a five-year research fellowship from the Swedish Royal Academy of Sciences.
In 2004, Tucker was awarded the European Mathematical Society’s Prize for distinguished contributions in Mathematics.
In 2007, Tucker formed the CAPA group at the Department of Mathematics at University of Bergen (Bergen, Norway). In 2009, the group moved to its current location at Uppsala University.
Tucker also holds an adjunct professorship at the University of Bergen, we he supervises a PhD student, and gives courses.