School of Mathematics
by Steve Awodey and Thierry Coquand
In 2012–13, the Institute’s School of Mathematics hosted a special year devoted to the topic “Univalent Foundations of Mathematics,” organized by Steve Awodey, Professor at Carnegie Mellon University, Thierry Coquand, Professor at the University of Gothenburg, and Vladimir Voevodsky, Professor in the School of Mathematics. This research program was centered on developing new foundations of mathematics that are well suited to the use of computerized proof assistants as an aid in formalizing mathematics. Such proof systems can be used to verify the correctness of individual mathematical proofs and can also allow a community of mathematicians to build shared, searchable libraries of formalized definitions, theorems, and proofs, facilitating the large-scale formalization of mathematics.
The possibility of such computational tools is based ultimately on the idea of logical foundations of mathematics, a philosophically fascinating development that, since its beginnings in the nineteenth century, has, however, had little practical influence on everyday mathematics. But advances in computer formalizations in the last decade have increased the practical utility of logical foundations of mathematics. Univalent foundations is the next step in this development: a new foundation based on a logical system called type theory that is well suited both to human mathematical practice and to computer formalization. It draws moreover on new insights from homotopy theory—the branch of mathematics devoted to the study of continuous deformations in space. This is a particularly surprising source, since the field is generally seen as far distant from foundations.
On May 21, King Harald V of Norway presented the Abel Prize of the Norwegian Academy of Science and Letters to Pierre Deligne, Professor Emeritus in the School of Mathematics. In his acceptance speech, published here, Deligne articulates the essential role of freedom and curiosity in research––as the source of most of the important applications of sciences and as a powerful incentive to do the best work possible. The Institute is deeply grateful to Deligne, who has donated a portion of his monetary prize to support curiosity-driven research in the Institute’s School of Mathematics. His gift will be matched by the Simons/Simonyi $100 million challenge grant, which is the basis of the $200 million Campaign for the Institute.
Your Majesty, Minister, Excellencies, colleagues, family, friends, and guests,
I am very honored that this Abel Prize associates me with the luminaries who received it before me, amongst whom are my teachers and mentors Jacques Tits and Jean-Pierre Serre.
The past century has been a golden century for mathematics. When I look back, I am amazed at all the questions that in my youth seemed inaccessible, but which have now been solved. The last half-century has also been a golden time for mathematicians, but I worry that the prospects for young people are now far from being as good.
Throughout my life, I have received crucial help from many people and institutions. This for me is an occasion to give thanks.
In early April 1972, Hugh Montgomery, who had been a Member in the School of Mathematics the previous year, stopped by the Institute to share a new result with Atle Selberg, a Professor in the School. The discussion between Montgomery and Selberg involved Montgomery’s work on the zeros of the Riemann zeta function, which is connected to the pattern of the prime numbers in number theory. Generations of mathematicians at the Institute and elsewhere have tried to prove the Riemann Hypothesis, which conjectures that the non-trivial zeros (those that are not easy to find) of the Riemann zeta function lie on the critical line with real part equal to 1⁄2.
Montgomery had found that the statistical distribution of the zeros on the critical line of the Riemann zeta function has a certain property, now called Montgomery’s pair correlation conjecture. He explained that the zeros tend to repel between neighboring levels. At teatime, Montgomery mentioned his result to Freeman Dyson, Professor in the School of Natural Sciences.
In the 1960s, Dyson had worked on random matrix theory, which was proposed by physicist Eugene Wigner in 1951 to describe nuclear physics. The quantum mechanics of a heavy nucleus is complex and poorly understood. Wigner made a bold conjecture that the statistics of the energy levels could be captured by random matrices. Because of Dyson’s work on random matrices, the distribution or the statistical behavior of the eigenvalues of these matrices has been understood since the 1960s.
By D. Kotschick
In many different ways, 1953 was an exciting year. In February–March, at the Cavendish Lab in Cambridge, England, James Watson and Francis Crick discovered the structure of DNA molecules and its double-helix geometry. This discovery was announced in Nature in April and hit the news during the second half of May. Several newspaper articles in the English and American press at the time celebrated the work of Watson and Crick as closing in on the secret of life. This enthusiasm was vindicated by later developments, so that the publication of Watson and Crick’s paper1 is now regarded as one of the most important scientific events of the twentieth century.
At the same time, a British expedition was on the slopes of Mount Everest attempting the first climb to the top of the highest mountain on earth, which had, until then, defeated all challengers. The expedition set up base camp in March and then worked its way up the mountain. After a failed summit attempt by a different pair of climbers, Edmund Hillary and Tenzing Norgay finally conquered Mount Everest on May 29, 1953. News of their success reached the Western world on June 2 and was a front-page story. It so happened that June 2 was also Coronation Day in England, when Queen Elizabeth II was crowned after having ascended to the throne upon the death the previous year of her father, King George VI. The New York Times called the news of Hillary and Tenzing’s exploit a "coronation gift."
In that spring of 1953, Friedrich Hirzebruch was hard at work at the Institute in Princeton. He had arrived from Germany the previous summer for what would become a two-year stay and had immediately immersed himself in learning sheaf theory, algebraic geometry, and characteristic classes under the guidance of Kunihiko Kodaira and Don Spencer of Princeton University. By the spring of 1953, Hirzebruch was trying to solve what he thought of as the Riemann-Roch problem: to formulate and prove a far-reaching generalization of the nineteenth-century Riemann-Roch theorem, extending it from algebraic curves to algebraic varieties of arbitrary dimensions. The setting was sheaf cohomology and the goal was to find a formula for certain Euler characteristics in sheaf cohomology in terms of Chern classes.
By Mina Teicher
It is known that mathematicians see beauty in mathematics. Many mathematicians are motivated to find the most beautiful proof, and often they refer to mathematics as a form of art. They are apt to say “What a beautiful theorem,” “Such an elegant proof.” In this article, I will not elaborate on the beauty of mathematics, but rather the mathematics of beauty, i.e., the mathematics behind beauty, and how mathematical notions can be used to express beauty—the beauty of manmade creations, as well as the beauty of nature.
I will give four examples of beautiful objects and will discuss the mathematics behind them. Can the beautiful object be created as a solution of a mathematical formula or question? Moreover, I shall explore the general question of whether visual experience and beauty can be formulated with mathematical notions.
I will start with a classical example from architecture dating back to the Renaissance, move to mosaic art, then to crystals in nature, then to an example from my line of research on braids, and conclude with the essence of visual experience.
The shape of a perfect room was defined by the architects of the Renaissance to be a rectangular-shaped room that has a certain ratio among its walls—they called it the “golden section.” A rectangular room with the golden-section ratio also has the property that the ratio between the sum of the lengths of its two walls (the longer one and the shorter one) to the length of its longer wall is also the golden section, 1 plus the square root of 5 over 2. Architects today still believe that the most harmonious rooms have a golden-section ratio. This number appears in many mathematical phenomena and constructions (e.g., the limit of the Fibonacci sequence). Leonardo da Vinci observed the golden section in well-proportioned human bodies and faces—
in Western culture and in some other civilizations the golden-section ratio of a well-proportioned human body resides between the upper part (above the navel) and the lower part (below the navel).