Dean of Computing and Mathematical Sciences Professor Geoff Holmes looks at magnitudes of scholarship and the Millennium Prize Problems.
In recent years, Hollywood has shown a great deal of interest in the Millennium Prizes. These prizes are for the solution of seven problems in mathematics that have evaded proof by some of the finest minds in history. Each prize is worth one million US dollars but even with this level of incentive only one of the prizes has thus far been claimed (for a proof of the Poincaré Conjecture). The “P vs NP” problem has been covered by an episode of Elementary (Sherlock Holmes, no relation) where the solver was unfortunately murdered; the film “Gifted” covered the Navier-Stokes equation (implying that a proof had been produced by the dead mother of the focus of the film—her gifted daughter). Unfortunately, "Proof by Hollywood" is not really a recognised technique in mathematical research. Some other recent productions have been about famous mathematicians who have studied some of these problems: “The Man Who Knew Infinity” (Ramanujan) and “The Imitation Game” (Alan Turing).
One of the unsolved problems, posed in 1859 by Bernhard Riemann, is the Riemann Hypothesis (like Fermat I have no margin to explain this problem but you can read an explanation). Last December the Faculty of Computing and Mathematical Sciences celebrated the publication by Cambridge University Press of two volumes entitled "Equivalents of the Riemann Hypothesis", by Emeritus Professor Kevin Broughan. The volumes cover arithmetic and computational equivalents. Equivalents to the problem are problems that if proved or disproved immediately prove or disprove the Riemann Hypothesis. Because there are so many equivalents, solving one of these solves all of the others instantaneously. The Riemann Hypothesis has links to theory about the distribution of the prime numbers, which is one of Kevin’s research areas of interest.
Most mathematicians believe that the Riemann Hypothesis is true. For example, it is known to be true computationally for the first trillion values that were tested. However, this approach to a mathematician, is about as rigorous as "Proof by Hollywood".
To provide a perspective on the gravity of this problem to mathematicians Kevin quotes David Hilbert, the renowned German mathematician, who was once asked, “If you were to die and be revived after five hundred years, what would you then do?” Hilbert replied that he would ask “Has someone proved the Riemann hypothesis?”
I mention the celebration of the publication of these books in particular because in the age of the Performance Based Research Fund it is rare to celebrate scholarship of this magnitude. For Kevin they represent just two of the four required Nominated Research Outputs. Research texts are certainly not common in the computing and mathematical sciences and any mathematician tackling one of these problems would certainly have a hard job convincing their head of department that it would be productive. Andrew Wiles, solver of Fermat's Theorem was famously hiding his work from colleagues at Princeton. The remaining unsolved problems not mentioned above are: the Hodge Conjecture, the Birch and Swinnerton-Dyer Conjecture and the Yang–Mills and Mass Gap. Interested readers may find more details about these famous unsolved problems in mathematics.