Do Not Erase

Mathematicians and Their Chalkboards

Jessica Wynne

• Publisher: Princeton University Press
• Year: 2021
• ISBN:
  9780691222820
• Topic: Matemáticas
• Subtopic: Matemáticas general

A photographic exploration of mathematicians' chalkboards "A mathematician, like a painter or poet, is a maker of patterns, " wrote the British mathematician G. H. Hardy. In Do Not Erase, photographer Jessica Wynne presents remarkable examples of this idea through images of mathematicians' chalkboards. While other fields have replaced chalkboards with whiteboards and digital presentations, mathematicians remain loyal to chalk for puzzling out their ideas and communicating their research. Wynne offers more than one hundred stunning photographs of these chalkboards, gathered from a diverse group of mathematicians around the world. The photographs are accompanied by essays from each mathematician, reflecting on their work and processes. Together, pictures and words provide an illuminating meditation on the unique relationships among mathematics, art, and creativity.The mathematicians featured in this collection comprise exciting new voices alongside established figures, including Sun-Yung Alice Chang, Alain Connes, Misha Gromov, Andre Neves, Kasso Okoudjou, Peter Shor, Christina Sormani, Terence Tao, Claire Voisin, and many others. The companion essays give insights into how the chalkboard serves as a special medium for mathematical expression. The volume also includes an introduction by the author, an afterword by New Yorker writer Alec Wilkinson, and biographical information for each contributor. Do Not Erase is a testament to the myriad ways that mathematicians use their chalkboards to reveal the conceptual and visual beauty of their discipline—shapes, figures, formulas, and conjectures created through imagination, argument, and speculation.

CHALKBOARDS & REFLECTIONS

PHILIPPE MICHEL

Philippe Michel is head of the chair of Analytic Number Theory at the Swiss Institute of Technology, Lausanne, where he has been on the faculty since 2008. He obtained his PhD in 1995 from Université Paris XI and subsequently held several positions in France.

Blackboards are a fundamental component of a mathematical life. The very first thing I did when I arrived in my office in Lausanne ten years ago was to ask that the ugly whiteboard, with its smelly red pen, be replaced by a true chalkboard. I got a very good one. A blackboard has to be as tall as possible so that its boundaries do not interrupt the flow of thinking and writing. In this regard, the immense blackboard in this photograph (which covers an entire wall of the common room in the mathematics department at Columbia) far exceeds expectations.

For a smooth and seamless writing experience, high-quality chalk is also important. I was particularly moved when a postdoc, returning from Christmas break one year, offered me two boxes of the legendary Japanese “Hagamoro Full Touch” chalk. (No need to break into my office; the stock is unfortunately exhausted.)

The last—but not least—piece of the trinity is cleaning. After years of training, I became adept at the Swiss (or is it German?) way. Here is the recipe: clean the board by applying water with a large cloth wiper, and dry it with a large rubber wiper (a colleague of mine in Zurich is even capable of doing both at once). Use that time to clear your mind. With a clean board and a clear mind, you’ll be ready to start anew.

This photograph was taken during a discussion with Will Sawin (who is assistant professor of mathematics at Columbia) concerning a joint project in number theory. Although number theory aims to describe the properties and structures of whole numbers, it borrows methods and techniques from many parts of mathematics; this blackboard is a good illustration of this. On this board, you can find ideas related to analysis (modular forms) as well as algebra and geometry (cohomology and sheaves).

AMIE WILKINSON

Amie Wilkinson is a professor of mathematics at the University of Chicago working in ergodic theory and smooth dynamics. She received her undergraduate degree at Harvard in 1989 and her PhD at Berkeley in 1995. She held postdoctoral positions at Harvard and Northwestern and rose to the level of professor at Northwestern before moving to Chicago in 2011. Wilkinson’s research is concerned with the interplay between dynamics and other structures in pure mathematics—geometric, statistical, topological, and algebraic.

I work in an area called dynamics, which is the study of motion. The origins of the subject lie in physics; one of the earliest problems in dynamics was the stability of solar systems. My work focuses more on abstract mathematical spaces, but visualization and drawing pictures are key components of both understanding and discovery.

On this blackboard is a central argument in a paper I wrote with Keith Burns (professor of mathematics, Northwestern University) about a mechanism for chaotic dynamics. It depicts a sequence of shapes that, in a precise sense, are equivalent to each other, starting with a spherical ball and ending with something called a julienne, named for its resemblance to a thinly sliced vegetable. We are proud of this paper; as Keith likes to say, a good paper has one truly new idea, and this one has two.

Doing math on a blackboard is a tactile experience. You are constantly going back to change and modify things you have already written. Sometimes you write something and immediately realize you need to change it, and with a rub of your fist, you can erase the offending material and fix it. Or, when you are explaining something, you might draw an initial picture, write some formulas and explanations, and then go back to the original picture to add more—to explain in the picture what you just explained in words.

Chalk comes in many rich colors. You can shade things in with chalk, indicate depth and form. I love chalk, even though I hate the mess it makes and how it dries out my skin and hair. It is smooth and crisp on a clean blackboard and can be seen from a distance. It is one of my main tools of expression.

TERENCE TAO

Terence Tao was born in Adelaide, Australia, in 1975. He has been a professor of mathematics at UCLA since 1999, having completed his PhD at Princeton in 1996. Tao’s areas of research include harmonic analysis, PDE, combinatorics, and number theory. He has received a number of awards, including the Fields Medal in 2006, the Mac Arthur Fellowship in 2007, and the Breakthrough Prize in Mathematics in 2015. Terence Tao also currently holds the James and Carol Collins Chair in Mathematics at UCLA, and is a member of the National Academy of Sciences and the American Academy of Arts and Sciences.

I see mathematics as this wonderfully connected web of ideas and insights; a thriving ecosystem that spans pure abstract thought at one end of the spectrum, and concrete practical applications at the other.

I don’t know where this metaphor originates, but I have often heard mathematics described in terms of a landscape shrouded in fog and mist. At first there is nothing to see; then, as the mist lowers, you begin to observe isolated peaks (a peak of geometry, a peak of algebra, and so forth). A little later, when the mist is lower still, you start to see connections between the peaks (for instance, Descartes’s discovery of Cartesian coordinates makes it possible to link geometry and algebra, ultimately leading to the fertile valley of algebraic geometry). When you are experienced enough in mathematical research, the fog dissipates almost entirely, and you finally begin to see the cities and roads at the bottom of the landscape, in between all the peaks. That’s where some really interesting stuff happens.

The blackboard shown here depicts part of the web of ideas and results that surround one of my favourite unsolved problems, namely, the twin prime conjecture. This conjecture asserts that there are infinitely many pairs of prime numbers that are separated by a distance of two from one another. This problem has its roots in number theory, but all the progress towards solving it proceeds through analysis—in particular, through trying to establish upper and lower bounds on the count of twin primes, up to a given threshold. This problem remains intractable for a number of reasons, but one of them is the parity problem, which is the frustrating phenomenon that it is difficult to distinguish analytically between numbers with an even number of prime factors and numbers with an odd number of prime factors. But a close cousin of the twin prime conjecture, known as Chowla’s conjecture, has seen recent progress, even though it is also subject to the parity problem. Both conjectures are still open, but for the first time there seems to be hope of a breakthrough. The recent progress has connections to many other areas of mathematics: ergodic theory, additive combinatorics, information theory, graph theory . . . we are still trying to understand exactly how it all fits together, but the emerging landscape looks exciting. It certainly looks broader than what can fit on even the largest blackboard.

BENSON FARB

Benson Farb is professor of mathematics at the University of Chicago, where he has been since 1994. He obtained his PhD from Princeton University in 1994 under the direction of William Thurston.

Consider all the possible configurations of points moving around in a fixed space: think of the roughly 10,000 airplanes flying around as you read this, the quintillions of molecules moving around in a test tube, or the hundreds of robots moving around in an Amazon distribution center. To mathematicians, each of these systems is a special case of a common framework called a “configuration space.” To keep track of such a system—let’s take the airplanes as an example—we need to record the longitude, latitude, and height of each plane at any given time. In other words, we need to specify (4 × 10,000) = 40,000 numbers. We can think of the snapshot of all airplanes currently in the air as a single point in a 40,000-dimensional space. Since each plane is changing its position over time, the corresponding “configuration of planes,” represented by a single point in this 40,000-dimensional space, is likewise moving around over time.

The space of all possible configurations is extremely complicated, as flight paths (or molecules, or robots) can interweave with one another in complicated patterns over time. The crucial property of all of these systems is that the objects cannot occupy the same point in space at the same time. As a mathematician, if I discover something about a configuration space, it applies not only to airplanes, molecules, and robots but to any possible collection of objects moving around.

Such systems are too complicated for even the most powerful computers. To describe and understand such systems, and to make predictions about them, we can apply the powerful edifice of mathematics. While I can’t visualize 40,000-dimensional—or even 4-dimensional—space, I can reason about it. How do I actually do this? How can I communicate this reasoning to others, and they to me?

The main tool that most of us use to communicate our ideas is the chalkboard. (The term “blackboard” is used in Wikipedia, but I don’t like that term because not all chalkboards are black.) A computer doesn’t help much with 40,000 dimensions, but on a blackboard, I can work up a schematic of the situation, explaining it in real time to a student or collaborator. She can jump up and start writing on the board during my explanation, amending my computations, noting possible problems, unwrapping some equation into a flurry of computations of her own.

Doing this dance at the blackboard with someone is an intense, frustrating, energizing, and sometimes moving experience. It’s the kind of connection with the mind of another human being that is rare in everyday life. On a huge chalkboard, or even better, on two adjacent boards (or ...

Table of contents

• Cover Page
• A Photographic Exploration of Mathematicians’ Chalkboards
• Title Page
• Dedication Page
• Introduction
• Chalkboards & Reflections
• Afterword
• Acknowledgments
• The Mathematicians
• Copyright
• About the Author