**John Roe** studied with Michael Atiyah at Oxford, and his research has focused on the interaction of index theory and large scale or “coarse” geometry. After teaching at Oxford for twelve years he became Professor of Mathematics at Penn State in 1998. He has written a number of books, starting with Elliptic Operators, Topology and Asymptotic Methods in 1988 and including several published by the AMS. His most recent book, *Winding Around***,** was published by the AMS in 2015, and he has also added several lecture note sets to the **Open Math Notes** project. Mathematics for Sustainability, a text for a quantitative reasoning course for non-mathematicians, is expected in 2018 from Springer.

**What made you decide to write the book “Winding Around”?** The spark for Winding Around was lit when I was about nine. My dad drew an incredibly convoluted simple closed curve (something like Figure 4.3 in the book),

made a dot on the paper somewhere in the midst of the convolutions, and asked me, “Is that inside or outside the curve?” I knew about maze puzzles so naturally enough I picked up my pencil and drew a path starting at his

dot, staying between the lines and heading, so I hoped, towards the exterior. After several minutes of wrong turns and entanglements I was finally able to announce, “Outside”. Then he did something I did not expect. He took a straightedge and drew a line directly from his point to the exterior. “But it crosses the curve”, I complained. He didn’t respond directly but just started at the exterior of the *curve* and marked off O, I, O, I…“outside”, “inside”, “outside”, “inside”, changing at every crossing until he arrived at the original dot. I saw at once what was going on and I have never forgotten that “aha” moment.

**Wait, you had really been planning this book since the age of nine?** That would be a good story, wouldn’t it? And it’s absolutely true that that experience lit a fire for me. Another source of inspiration was something I learned from Atiyah rather later in my mathematical career: that exciting things happen when different branches of the mathematical family – analysis, topology, geometry, algebra – are made to interact in significant ways. That story is usually told in higher dimensions, as pa

rt of a grad student’s research training, but it can also be told in dimension 2. There’s a beautiful expository paper of Atiyah from the 1960s where he reviews how these different branches of mathematics approach the winding number and then goes on to say, look, if you take the correct higher-dimensional generalization of all this, you will get Bott periodicity. It had been kicking about in the back of my mind for some years that you could build an undergraduate course on that paper and when I had the chance to teach in the MASS program in fall 2013, I decided to give it a try. Winding Around was the result – a book which is centered on the many different definitions of the winding number and the ways they *interact*.

**Tell us more about MASS.** This is a unique program that has been held at Penn State for about twenty years. MASS gets a class of twenty to thirty very good students – half from Penn State, half from other institutions across the US and the world – and puts them together in this high-intensity math environment for a full semester. They are focused entirely on mathematics (as Oxford students would be, for instance) and because of that and the strong peer group they learn very fast. It’s a great context for trying the kind of experiment that produced *Winding Around*.

**Was there a gap in the literature that you were trying to fill?** To get me energized to write, a necessary condition is the sense that “no-one has ever said these things in exactly this way before, and this is how they need to be said.” Of course that can work out differently in different contexts. For myfirst book it was just, “I wish someone could have put all this together for me when I was starting my thesis”. For Winding Around, it was more “I wish undergraduate students could see all these different kinds of mathematics engaging with each other”. Of course there are plenty of books about complex analysis or plane topology, but I couldn’t find one that gave the sense of deep interconnectedness that I’ve tried to convey.

**Did you use existing notes from teaching?** For several books but not all of them. As I mentioned, Winding Around comes from a course in the MASS program, for which I prepared detailed notes. Lectures on Coarse Geometry comes from notes of a graduate course. Mathematics for Sustainability is based on a course that I developed for our undergraduate program. In recent years I’ve developed a very specific set of personal procedures for preparing slides and notes (in TeX) for each course I teach. But that doesn’t make it magically easier to produce a book once you have finished!

**Was your writing influenced by other books? Which ones?** Early in my career I was greatly influenced by Jean Dieudonné’s Foundations of Modern Analysis, which of course is very much in the Bourbaki style – all numbered paragraphs and subparagraphs, and no concessions to “intuition” such as might be suggested by (gasp) a diagram! But later, through reading Milnor I think, and also through listening to Atiyah and his colleagues explain things, I’ve moved away from that style towards something more conversational. In general I would say that exposition has played a vital part in my mathematical life. I am always “explaining” things, even if it is only to myself. I feel that if you really can explain something

clearly, you’re quite likely to discover something new about it. I suppose it is also quite likely that you’ll end up writing a lot of books ☺

**How did you decide on the format and style of the book?** I wanted Winding Around published in the Student Mathematical Library (as it eventually was) because I had always envisaged it as something to put in the hands of bright final-year undergraduates. But I had to fight for that a bit. Some of the AMS’s reviewers (of the first draft) wanted the book in a graduate series, with one saying something like “the book needs readers who already understand real and complex analysis, measure theory, topology and abstract algebra”. As though all these exciting ingredients have to be carefully synthesized in isolation – in laboratory conditions – before the trainee chef can be allowed to combine them! I’d rather we get cooking, and clean up the mess as we go along.

Anyhow, the compromise that (AMS chief editor) Sergei Gelfand and I arrived at was to leave the main structure of the book as it was but to add a bunch of appendices, A through G, giving capsule developments of these various items just to the extent needed in the main text. I guess this is an example of the influence of Dieudonné, who did something similar with the linear algebra he needed for his book. My ideal reader will more or less ignore the appendices – pushing through the main text, being content perhaps that some things are a little mysterious, and referring forward to an appendix only when mystery has accumulated so much as to impede progress. I wonder if this is how the book is actually read?

**What next?** *Mathematics for Sustainability*, out next year, is likely to be my last book. This is quite a departure from my previous works, both in terms of audience and content, but once again feels to me like something that has to be said. I’ve long felt that we mathematicians owe more than we presently offer to the thousands of students who take our pre-calculus courses simply to fulfil a ‘breadth’ or ‘general education’ requirement – and that we should take the opportunity, across our curriculum, to connect what we do with big challenges like climate change. Kaper and Engler’s Mathematics and Climate aims to do this at the graduate level. My co-authors and I are trying for the same connection, assuming nothing but high school algebra. It’s a tall order, but one I am very excited about!

What advice would you give to new authors? Books are magic. Is there a story that only you can tell, or tell right? Do you have the time for a long project and the discipline to add a little more each day, even when the end seems far off? Is this the right point in your career, and is your institution enlightened enough to value your work on a book appropriately? Yes, yes and yes? Go ahead and add to the magic – and good luck!

This book looks very interesting for me. I would definitely get and read it. It is also nice to learn its story and publishing process. Thank you for sharing such a great source.

I am reading Edward Frenkel’s book Love and Math this week. It gives me hope that I’ll begin to understand a glimmer of the Langlands Program. I’m keen to read your book because I bet I’ll understand the Langlands Program even better after going thought it. Thanks!

Thank you for your comment. We hope you enjoy the book.