The Art of Doing Mathematics

To admirers, mathematics is part science and part art. The science part is easy to understand. But it is also art in that proving a theorem often requires thinking intuitively in pictures (or “spaces” to use math jargon). But it is also an art form in a more literal sense – the scribbles mathematicians leave on chalkboards and whiteboards often resemble the work of a graffiti artist or the work of an abstract painter, like Jean-Michel Basquiat. These scribbles are cryptic, mysterious, and endlessly fascinating. The love of blackboards doesn’t end with mathematicians. Theoretical physicists, from Albert Einstein and Niels Bohr, from Werner Heisenberg and Paul Dirac to Richard Feynman, all joined in the fun as you will see in the following photo gallery.

The mathematical scribbles of Rice University professor David Damanik exploring advanced mathematical fields such as spectral theory, dynamical systems, and aperiodic order via board work.

Chalkboard calculations and ruminations can appear bafflingly complex. Clockwise from top, these boards show the work of graduate student Zhongyi Zhang at Columbia University in New York City, graduate student Shuai Wang at Columbia University, graduate student Boya Song at MIT in Cambridge, Massachusetts, and graduate student Sahar Khan at Columbia University.

At the Instituto Nacional de Matemática Pura e Aplicada in Rio de Janeiro, Brazil, Jacob Palis sees green at the chalkboard and out the classroom windows. A professor of mathematics, Palis does research in the areas of dynamical systems and ergodic theory.

At Stanford University in California, mathematics professor Tadashi Tokieda does research in applied mathematics and macroscopic physics—though this day’s board work also shows off his French.

Sir Michael Atiyah (1929 – 2019), Fields Medallist and perhaps the most important mathematician in the second half of the 20th century, pictured in 2014 at the King’s Buildings, University of Edinburgh, where he was made an honorary professor in 1997. It was said of Professor Sir Michael Atiyah was “an irrepressible, unstoppable force of nature — almost literally a ball of energy, with a tendency to bounce into rooms and ping unpredictably from conversation to conversation.” The Atiyah-Singer Theorem, proved in 1963 was his most well-known result.

Paul Dirac (1902 – 1984) made fundamental contributions to mathematical physics, especially quantum mechanics, which led him to win the Nobel Prize for Physics in . In 1927, he was made a Fellow of St. John’s College, Cambirdge, and in 1932, the Lucasian Professor of Mathematics, a chair once held by Sir Issac Newton.

Richard Feynman (1918 – 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, as well as his work in particle physics. For contributions to the development of quantum electrodynamics, Feynman received the Nobel Prize in Physics in 1965 jointly with Julian Schwinger and Shin’ichirō Tomonaga.

The great Albert Einstein (1879-1955), deriving special relativity, for an audience, in 1934.

Einstein’s desk at the Institute of Advanced Study in Princeton. Einstein was a faculy member at the IAS from 1933 till the time of his death in 1955. Photo: Alan W. Richards.

British theoretical physicist, Stephen Hawking (1942 – 2018), who was Lucasian Professor of Mathematics at the University of Cambridge between 1979 and 2009, with scribbles of his final paper, now published posthumously on the Internet. Titled, “Black Hole Entropy and Soft Hair”, the paper is a technical dive into one of the greatest unresolved questions in physics: Can matter that into a black hole truly disappear, even though the laws of physics say that should be impossible? The paper also serves as a kind of bookend to Hawking’s career, collecting some of his final work on the quantum structure of black holes — a topic that Hawking pursued throughout the last 40 years.

Cédric Patrice Thierry Villani (b. 1973) is a French mathematician working primarily on partial differential equations, Riemannian geometry and mathematical physics. He was awarded the Fields Medal in 2010 and he was the director of Sorbonne University’s Institut Henri Poincaré from 2009 to 2017.

In 2014, the Iranian mathematician Maryam Mirzakhani (1977 – 2017) became the first female Fields Medalist for “her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces”.

Edward Witten, Professor in the School of Natural Sciences at the Institute for Advanced Study, Princeton, lecturing on string theory in 2008.

Among the most brilliant theorists cloistered in the IAS, Witten stands out as a kind of high priest. The sole physicist ever to the win the Fields Medal in mathematics, he is also known for discovering M-theory, the leading candidate for a unified “theory of everything”. He is currently the Charles Simonyi Professor in the School of Natural Sciences at the Institute for Advanced Study.

“I like blackboards … There is something about how easy it is to change things. I think that’s it. You put something out and you can modify as much more easily. If you draw something, when you can draw it in pencil and keep rubbing it up but then a little while your paper rips and that doesn’t work. But on the blackboard, you can very easily sketch something up, use your hand, wipe it off. See that’s what superior to whiteboard.”

~ Sir Roger Penrose, co-winner of the 2020 Nobel Prize for Physics for his “for the discovery that black hole formation is a robust prediction of the general theory of relativity.” This photo shows Sir Roger at Oxford in 1980.

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