This is a question for mathematics instructors: How do you feel about having solutions available for the exercises in a math textbook? What if the solutions are available on the internet?

Some colleges and universities have guidelines for how instructors should treat the possibility of cheating on homework (e.g. looking for solutions on the web rather than working problems out). I taught at about five different colleges and universities. Four of them had honor codes, which gave the instructor the luxury of giving students their assignments and instructions and assuming that they complied honestly, whether they did or not. It was up to the administration to sort out egregious problems and the students’ own consciences to deal with mild ones.

My favorite tactic was to simply make homework count for only a small part of the grade, and place more emphasis on in-class tests and quizzes. The homework is useful for studying, students could work together or work with a solution guide as they pleased.

And there is a third option: to give homework problems whose solutions are not available or very difficult to access online or in books. Is this the high ground approach, or is it simply impractical and too much trouble considering questionable benefits?

What do you think? Should textbooks contain solutions to problems, or should the problems only be made available to instructors in a separate manual, or online accessible only by password?

Featured Book of the Day

Integers, Fractions and Arithmetic by Judith Sally and Paul Sally. This book was co-published by the AMS and MSRI as part of a Math Circles Program for K-8 teachers. The book consists of twelve interactive seminars, and gives a comprehensive and careful study of the fundamental topics of K–8 arithmetic. The guide aims to help teachers understand the mathematical foundations of number theory in order to strengthen and enrich their mathematics classes.

I would describe the approach to exercises in this book as: use very few and explain the solutions carefully and completely.

Those of you over a certain age may remember when searching for math resources meant going to the library and perusing the subject catalog, spending time in shelving sections devoted to a topic, or leafing through heavy volumes of math reviews. Along the way, you found things you did not expect, leading to further trails and discoveries. You might remember musty smells, occasional hushed sounds, dark lighting, and the heft and texture of the volumes. There was a feeling of timelessness and escape from the real world in a library.

Today libraries seem quite different. Science and math libraries have well-lit, large spaces filled with tables and power outlets, conducive to sitting with a laptop and connecting to web-based resources with hardly a book in sight. Many hard copy books are available only by special order, and browsing is now almost completely digital.

What do you feel are the pros and cons of new library designs? How do today’s libraries affect the way you browse for books?

This book started as a picture book by mathematician and artist Lisl Gaal for her children and grandchildren. The illustrations depict whimsical creatures and settings juxtaposed with simple yet far-reaching mathematical ideas that appeal to every age group. Readers are encouraged to explore and understand at their own level and pace. A child of any age could read and re-read this book for years, picking up new insights each time.

(Supplementary text is included for educators and advanced readers.)

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!

Jennifer Schultens is Professor of Mathematics at University of California, Davis. Her book Introduction to 3-Manifoldsguides beginning graduate students through the foundations of low-dimensional topology to specialized topics such as triangulations of 3-manifolds, normal surface theory and Heegaard splittings. A fundamental challenge to building up the language of low-dimensional topology is to connect mathematical rigor with geometric and topological visualization. In her book, Jennifer Schultens does this seamlessly, melding clear exposition with lots of intuition-building examples, illustrations and exercises.

Question: What made you decide to write the book? Was there a gap in the literature you were trying to fill? Did you use existing notes from teaching?

While in the process of guiding graduate students through the basics of understanding 3-manifolds, I often wished for a reference containing the knowledge that first opened my eyes to the beauty of the subject. The subject had grown and flourished since the publication of the books that I had read as a novice. I especially remember one conversation, probably in the early 2000s, with Aaron Abrams and Saul Schleimer, concerning the need for more current books on the topic of 3-manifolds. The curve complex, rapidly emerging as a central tool in the study of 3-manifolds, needed to be added to the standard repertoire of a 3-manifold topologist. As it happened, I had the opportunity in the Spring of 2003 to teach a course on 3-manifolds to a group of highly motivated graduate students at Emory University. In this course I described the subjects near and dear to my heart. I recorded my lectures in a rather terse set of notes. Over the next 10 years, these notes grew into a book.

Question: How did you decide on the format and style of the book?

I treasure the traditional mathematical style of writing: definitions, theorems, proofs. As a topologist I also find illustrations indispensable. Today’s technology, most notably LaTeX, xfig and AMS style files provide for easy typesetting. Definitions, theorems, proofs and illustrations constitute the skeleton of my book. However, the life of the book derives from the knowledge verbally passed around among low-dimensional topologists that I incorporated into the text, often informally. For instance, I included a proof of the Schoenflies Theorem given in a lecture series by Andrew Casson in China in 2002, but not otherwise in the literature. I did not attend the lectures myself, the lecture notes had been given to me by Yoav Moriah.

Question: What was the writing process like? Did you write everyday on a set schedule, or did you have periods of setting it aside?

I started out with a set of notes recorded during a course I taught in Spring of 2003. In 2006 I spoke with Sergei Gelfand who encouraged me to turn the notes into a book. Most of the writing occurred during four bouts of productivity: Summer and Fall of 2006 at the Max-Planck-Institute for Mathematics in Bonn, Germany; Spring and Summer of 2008 as I grew ever heavier during my pregnancy; Fall of 2010; and Summer and Fall of 2013, again at the Max-Planck-Institute. I found the fallow periods, the months and years when I did not think about the book, indispensable to the maturation of the project.

Question: What did you focus on the most when writing? What was the most challenging aspect? What came easily?

I focused on my vision of the subject and an imaginary reader, either a graduate student or a well known colleague, reading the book. I tried to include all the background material necessary to understand the discussion. Occasionally, I got overwhelmed by the amount of background material still needed.

Question: What were the positives and negatives of the experience? Did anything about the experience surprise you?

Looking up and (re)familiarizing myself with references proved more time consuming than I had imagined. MathSciNet, developed by the AMS, proved indispensable in the process. Through this type of diligence, I learned more about the subjects being exposited and the people involved. The book gained more depth.

Question: How did you choose a publisher? What was important to you when you made the choice?

After speaking with Sergei Gelfand in 2006, I realized that publishing with the AMS meant that the book would be in good hands. Naturally, the AMS looks after the professional interests of mathematicians. In addition, the AMS has an excellent track record regarding publishing at fair prices.

Question: Was your writing influenced by other books? Which ones?

I enjoy reading. Fiction or non-fiction, classical or modern, formal or informal, short or long, I enjoy a well-crafted piece of writing. The first mathematical text that really ‘grabbed’ me was Walter Rudin’s “Principles of Mathematical Analysis” and later his “Real and Complex Analysis.” The topology courses at UC Santa Barbara teemed with good literature: “Topology” by Munkres, “Differential Topology” by Guillemin and Pollack (also “Topology from a Differential Viewpoint” by John Milnor from which Guillemin and Pollack’s book is derived), the books on 3-manifolds by Hempel and Jaco which to me are inseparable from their interpretation by Cooper, Long and Scharlemann. Then there was Rolfsen, not just a book, but an experience. Working one’s way through Rolfsen’s “Knots and Links,” rediscovering so many of the beautiful constructions in knot theory, was a rite of passage for low-dimensional topologists of my generation. Later, I thoroughly enjoyed Silvio Levy’s digest of Thurston’s Lecture Notes and Allen Hatcher’s books.

Question: Did you find ways to get feedback while writing your book or was it a solitary effort?

My writing tends to be introspective. However, my husband, Misha Kapovich, proofread many parts of the book. His feedback helped fill in background information on several subjects, especially the final chapter of the book. This increased the time it took to complete the book, but added depth. It certainly improved the quality of the book.

Question: Did time pressure or other responsibilities help or hurt your writing?

Being an academic involves many types of activities, opportunities and responsibilities. My writing tends to be introspective. Sabbaticals are indispensable to my work on larger projects. However, the busy times, filled with teaching, attending lectures, working on committees, taking care of family, fill my thoughts in a way that informs the quieter periods during which I write.

Question: What kind of feedback did you get after the book came out?

Friends from far and near wrote to tell me that they enjoyed my book. It was nice to (re)connect. Of course, there were also some corrections. I am happy to report that the AMS maintains a web page where corrections are easily posted.

Question: What advice would you give to new authors?

Write about the things you love in a style that suits you.

Recently, the phrase “free as in beer vs. free as in speech” caught my attention. It was the first I had heard of this way of distinguishing two English meanings of free, and how it particularly applies to what is accessible on the internet. The explanation I found is here.

The new AMS Open Math Notes site is definitely “free as in beer.” There is no cost to users who browse, upload, or download Notes. It is a place for Notes that are, for a variety of reasons, not published (typically because they are in unpolished form but are at a stage where they can benefit from public view). Notes are submitted by authors and vetted by AMS editors and advisory board members on the basis of completeness, readability, and interest to students, teachers, and researchers in mathematics.

So what about “free as in speech”? Users may only download PDF files for free, but they may not freely change and then redistribute the contents. Authors who post on AMS Open Math Notes retain all rights to their Notes. Of course, they may make their source files freely available on their own webpages or on other free posting sites. (If an author decides to “publish” the Notes commercially, self-publishing or otherwise, then they must withdraw their Notes from the AMS site.)

Let us know what you think of the “free as in beer” vs. “free as in speech” dichotomy for Open Math Notes — examples of both types of posting sites exist on the web. What do you feel are the pros and cons?

This book is a popular item in the AMS Student Math Library Series, a series dedicated to non-standard math topics accessible to advanced and/or extra curious undergraduates. The book takes students on a tour of a range of advanced topics in mathematics as it displays the far-reaching consequences of the concept of winding number.

Hossein Hosseini Giv is an Assistant Professor of Mathematics at the University of Sistan and Baluchestan in Zahedan, Iran. The AMS Bookstore’s description of his book Mathematical Analysis and its Inherent Nature begins, “Mathematical analysis is often referred to as generalized calculus. But it is much more than that. This book has been written in the belief that emphasizing the inherent nature of a mathematical discipline helps students to understand it better. ” This point of view sets the stage for a friendly and engaging textbook that is sure to appeal to students who are curious about the underlying mathematics behind single-variable calculus.

What made you decide to write the book? Teaching is my favorite mathematical activity, and I consider writing books as some kind of teaching. This has a simple reason: When you teach at a class, you are working with a limited number of students, but when you write a book, you have the opportunity to train a large number of students, some of whom may never meet you.

When I was asked to teach the foundations of mathematical analysis for the first time, I thought of the ways I could make the material accessible to my students. Mathematical analysis is one of the first courses in which students deal with abstract ideas seriously, and it is therefore absolutely essential to help students in finding some intuition. When I was thinking about the abstract parts of the course, I noticed that some parts are not of an abstract flavor, and they aim instead to complete the students’ calculus-based knowledge theoretically. This was against what I heard previously about undergraduate mathematical analysis, namely, its description as generalized calculus.

For this reason, I first tried to talk about the essence of any issue I was teaching. After I used this approach successfully several times in my classes, I tried to popularize it by presenting it within a book. In writing the book I did my best not to present the material in the usual definition-example-theorem style of mathematics books. I did this by spending more time on the interpretation of results and talking about the essence of the issues.

How did you decide on the format and style of the book? The format and style of the book was determined by the essence of undergraduate mathematical analysis, that is, what it has to do to the calculus-based knowledge. In fact, since analysis generalizes some aspects of calculus to wider frameworks and completes some others theoretically, I decided to present the material in two parts. These parts, which concentrate on the completion and abstraction of our calculus-based knowledge, respectively, allow students to go from concrete arguments to abstract ones. For example, Chapter 3 of the book presents some important aspects of the metric space theory within the classical space $\mathbb{R}$ equipped with the Euclidean metric. Most of the concepts and results of this chapter are then generalized to the abstract context of metric spaces in the second part of the book.

One reason I considered this style of presentation was to help students to understand which concepts and results are generalizable to the abstract setting and which ones are not. This approach helps students to understand the way abstract theories are developed.

What did you focus on the most when writing? What was the most challenging aspect? What came easily? My main focus was on the clarity of exposition. The most challenging aspect of this work was to develop mathematically rigorous ideas within an expository framework. A further challenge was to present enough material with sufficient interpretations and justifications in a book of reasonable size. All that said about my challenges, I should say that the writing itself was very easy for me. More precisely, although I was concerned with the choice of material and its volume, I have never had difficulty with how to write something. I think writing is my best ability and I enjoy writing expository texts.

What was the writing process like? I wrote more than one half of the book within the summer vacations of 2014 and 2015, when I had enough time for both writing and recreation. The remaining parts of the book were written when I was busy with my classes, and I sometimes had to write on weekends.

Generally speaking, I write whenever I can. I think writing is like playing a musical instrument, and one is able to write an influential text when he/she is mentally and physically at a good situation. For this reason I have never had a daily schedule for writing. Sometimes I even write on midnights. This is because on midnights there are no disturbing sounds and thoughts. Sometimes I stop writing one chapter to start or continue some other. This happens when I feel that working on some argument is easier than some other in a particular situation. Of course, I never work on more than two chapters simultaneously. Also, to remain focused and motivated, I never stop writing for more than one week.

What were the positives and negatives of the experience? Did anything about the experience surprise you? I think that writing a book for the AMS cannot involve any negative aspects. It is a great opportunity which has many positive aspects. One important positive point of publishing with the AMS is that you work with a society of mathematicians. This is very helpful, because they are able to evaluate your book idea quickly. Moreover, authors receive unfailing support from the AMS personnel. They are all strongly committed to work with authors and to help them effectively. I was really surprised when I found that professor Gerald Folland is the chair of the editorial committee of the AMSTEXT. I always admired him as a great author, and I learned measure theory and harmonic analysis from his excellent books. I have taught measure theory several times using his Real Analysis book.

Did you have a special place where you liked to write? How did you stay motivated and focused? I write in both my university office and my home. More generally, I can write in any quiet place where I have access to a pen and enough sheets of paper. Of course, access to appropriate books and an internet connection is sometimes necessary for me to write. When you sign a contract with a great publisher like the AMS, motivation and focus come automatically to your assistance!

Did time pressure or other responsibilities help or hurt your writing? Deadlines allow us to work more efficiently. I always consider deadlines for my projects, even if I am not asked to do so. No matter a deadline is proposed to me or it is considered by myself, I do my best to finish my task on or before that date.

What advice would you give to new authors? Writing books is a nice experience in which one is able to help students worldwide by making some parts of mathematics accessible to them. It is also a great opportunity for those who wish to share their teaching plans and approaches with other instructors. So, if you think you have hot ideas for book writing, please try to work hard on your ideas and help the mathematics community by writing influential books!

What process of writing works for you? There are two opposite approaches to writing, which I associate with Charlotte Bronte and Jane Austen, and the AMS is now providing a third with the help of its new website Open Math Notes.

The first traditional method is to wait until the entirety of what you want to write fills you, and then to write it in one fell swoop (I have heard that Jane Eyre was written in one night in final form). This goes along with the beautiful idea (due, I believe, to Jean-Pierre Serre) that it is best to write an article fresh from start to finish enough times so that version n equals version n+1, that being the sign that the paper is ready to submit. For some people, it seems, it is possible, after some preliminary unseen process, to hold a large and complex idea in the mind and set it down in words in one go.

The second traditional approach is to whittle. Somewhere I read that Jane Austen explained her writing in this way, though I cannot find the reference: first write a draft and then carefully rework it detail by detail until it takes a perfect shape. Perhaps this is like sculpting wood as opposed to painting with ink on rice paper. Karen Vogtmann has made the lovely quote: first you write it down, then you write it up.

AMS is now in the process of launching a new website called Open Math Notes which facilitates a third way to write a book, appropriate in this age of sharing and collaboration. Authors are invited to submit their lecture notes and other mathematical works in progress to make them available for free download. The author is still in charge, but readers can weigh in and make suggestions, from general comments on content and structure to specific comments on arguments and exposition. The website provides a browseable and searchable collection of Notes, which are all freely downloadable for use in teaching and research.

Open Math Notes was officially launched in the beginning of January at the Joint Math Meetings in Atlanta.

Our featured book of the day is inspired more by the recent season of family and giving than by the topic of this blog post. Enjoy!

Featured Book of the Day

I Have a Photographic Memory, by Paul R. Halmos

A great book for this time of year. Reading the book is like looking at a family photo album from the 60s through 80s (if your relatives and close friends happened to include 600 mathematicians!) The photographs exude warm friendships and collegiality, and the captions are both informative and wonderfully witty.

Dale Rolfsen is an expert in low-dimensional topology and knot theory, and is co-author of the AMS books Ordering Braids (with Dehornoy, Dynnikov and Wiest), and Ordered Groups and Topology (with Clay). His seminal work Knots and Links helped to popularize the study of low-dimensional topology in the 1970s and continues to be a standard reference today.

Knots and Links, in the AMS Chelsea Series

What made you decide to write the book? Was there a gap in the literature you were trying to fill? Did you use existing notes from teaching?
It was Mike Spivak who encouraged me to write Knots and Links. We had met at the IAS, where we became friends. That was the late 1960’s. Mike had established Publish or Perish, Inc. and encouraged me to write a book and publish with him if the muse ever struck me. A few years later I taught a graduate course on knot theory at UBC, where I was a young professor. It was based partly on notes from a course I had taken as a graduate student myself in Madison, Wisconsin, taught by Joe Martin. I also referred to several books in teaching that course, especially the Crowell and Fox classic, Introduction to Knot Theory. While in Princeton, I also had the pleasure of meeting Ralph Fox and frequently attended his weekly seminar, which further kindled my interest in the subject.
Regarding a gap in the literature… It was more a gap in my knowledge, and a bit my allergy to abstract algebra which was more severe than it is now. I was particularly mystified by the Alexander polynomial, which was treated quite formally and algebraically in all the sources I knew. How to make sense of a process involving putting dots at certain places near the crossing of a picture of a knot, using that to construct a matrix with polynomial entries, delete one column of the matrix and then take the determinant? It was a mystery to me for years. One day I read Milnor’s beautiful paper Infinite Cyclic Coverings and I had a real epiphany when I realized that the Alexander polynomial was actually describing the homology of a certain (infinite cyclic) covering of the knot’s complement. It was this more geometric understanding that I tried to emphasize in the book. In fact it’s fair to say that was one of the reasons I decided to write a book on knots.

How did you decide on the format and style of the book?
It was written in the ‘70s. Well before TeX. The state of the art at the time in preparing mathematical manuscripts was the IBM Selectric typewriter. It had a removable typing ball, so you could change fonts, type Greek letters, etc. Also, I like to draw, and luckily the subject calls for lots of pictures. So the book has illustrations and text interwoven, which is easier to do with cut and paste techniques than it would have been with LaTeX (even if it had existed at the time). I had a lot of fun drawing, cutting and pasting, sometimes using plastic overlays of cross-hatching, dots, etc.
I find I learn new concepts much more deeply by looking at examples and explicit calculations, rather than the formal, axiomatic approach. My experience as an often struggling student motivated me to introduce and motivate concepts by concrete, and hopefully interesting, examples. For both pedagogical and aesthetic reasons, I tried to emphasize the visual and geometric aspects of ideas and convey some of the excitement I felt toward the subject.

What was the writing process like? Did you write every day on a set schedule, or did you have periods of setting it aside? I started out with a sketchy first draft based on the notes from the course I gave in 1974. But the writing started in earnest when I had a sabbatical in 1975, which I mostly spent on a remote island off the west coast of British Columbia. No phone or electricity but a lot of peace and tranquility. I had a friend on a nearby island where there was electricity, and my rented IBM Selectric was there. So every week or so, I’d take a boat over to visit her and type a couple of pages. I didn’t have a schedule or explicit deadline, though I wanted to finish it before my teaching responsibilities started again. Eventually I heeded Mary-Ellen Rudin’s advice to me, “Don’t try to put everything into that book!”

What did you focus on the most when writing? What was the most challenging aspect? What came easily?
Although no doubt there would be many types of readers of my imagined book, I tried to write to a person a bit like myself as a student – a person who is easily confused, but also keen to see new ideas take shape. Also, it’s actually possible to build a little suspense into a mathematical narrative by hints, examples and questions which are then clarified later by theorems.
Knots are beautiful objects which had inspired some amazing new mathematical ideas – that’s even more true today than when I wrote the book. But to be honest, I don’t really like many aspects of knot theory. My bias is to be more interested in applications of knot theory to understand 3-manifolds, via surgery, branched coverings, etc., than in the program of classifying knots, for example. I was especially excited about surgery when writing the book. Rob Kirby was developing his surgery calculus at the same time. Rob later told me that he was surprised to see that some of his ideas were explored independently in Knots and Links. His work went much deeper, but he considered only integral surgery, whereas the more general rational surgery was developed in the book.
One of my great challenges in writing was that I didn’t have access to a library while on the island, so I did a bad job regarding the history of the subject and attributing names to ideas. I was also, at the time, totally ignorant of the theory of braid groups. Joan Birman’s seminal book Braids, Links and Mapping Class Groups was being written at the same time. But we were unaware of each other’s efforts. Later, Joan and I became, and remain, good friends.

What were the positives and negatives of the experience? Did anything about the experience surprise you?
The positive was that I learned a lot of the subject as I went along. Many of the examples and calculations had to be done from scratch. That was really fun. As I said, my library access was very limited. That was often frustrating, but forced me to work mostly independently of the literature. And of course there was no internet or even email back then.

How did you choose a publisher? What was important to you when you made the choice?
I chose Publish or Perish, Inc. for several reasons. Mike Spivak was my friend, a writer I admired, and he was very encouraging about the book. He promised to keep the price reasonable, an important issue with me, since students (and libraries) had to pay, even then, outrageous prices for textbooks, monographs and journals. Also I liked the name – it was so cheeky! I thought of my book as being a bit quirky, too. It seemed a good match.
The book went through several printings with P or P over a couple of decades, but at one point Mike told me that his publishing company was having financial difficulties and he couldn’t afford another printing run after his existing stock ran out. He suggested that I find another publisher, as the book was still selling well. I decided to go with the AMS, rather than a commercial publishing house. Another choice I considered was MSP, which is a non-profit mathematical publisher run by mathematicians. However, they wanted it redone in LaTeX, which was more work than I was prepared to engage in! The AMS edition uses the original format, reproduced photographically, with a few corrections put in. It was a challenge making those corrections – I prepared them in TeX using a font that looks like the dirty old IBM Selectric and then pasted them in. I consider it a great honor to be included in the AMS Chelsea series.

Was your writing influenced by other books? Which ones?
Two great influences on my writing (and thinking) were Edwin Abbot’s Flatland and John Milnor’s Topology from the Differential Viewpoint. I read Flatland as a kid, and that’s what got me thinking about higher dimensions and eventually into studying topology. Also, its mixture of mathematical ideas with humor, and even satire, inspired me to understand mathematics as an adventure that can really be fun. Spivak’s famous calculus book was also a guiding light. I admired the clarity of his writing, and was glad to have such a fine writer as my publisher. My ideal would be to write like Milnor, who manages to get right to the heart of the ideas he’s discussing. As I said, his paper on infinite cyclic coverings was a huge influence. Also his work with Fox, Singularities of 2-spheres in 4-space and cobordism of knots, was particularly appealing. It also hearkened back to my Flatland experience that sometimes an extra dimension clarifies things. For example, adding knots by connected sum is only a semigroup operation in 3-space, but if one considers cobordism classes of knots – a four-dimensional point of view – they form a group, and indeed a very complicated and interesting group which is still an active topic of research.

Did you find ways to get feedback while writing your book or was it a solitary effort?
It was mostly a very solitary effort, though of course I got feedback from my students when I taught the course. However, I did ask various people to look at it after the first draft was written, including Cameron Gordon and Andrew Casson, and of course Mike Spivak. They made some very useful suggestions. Also, my grad student, Jim Bailey, was very helpful in helping me edit the book. Crucially, Jim prepared the knot tables at the end of the book, a huge job. The source of those tables was Conway’s paper An enumeration of knots and links, and some of their algebraic properties which was published in 1970. Jim translated Conway’s notation to pencil drawings of the knots and I hired Ali Roth, his girlfriend at the time, to make the nice pen and ink drawings that appear in the tables. I think I paid Ali two dollars per knot! As I said, tabulation is not my strong suit, so I’m grateful to Jim for taking care of that. I always feel a bit guilty when I hear people refer to the “Rolfsen tables.” They’re not really mine. More accurately they should be called the Conway tables as interpreted by Bailey and drawn by Roth.

Did you have a special place where you liked to write? How did you stay motivated and focused?
My special place was a little cabin on the island. I had a little table right next to a wood stove on one side and a window on the other. Perhaps I stayed focused because there wasn’t much else to do there in the winter, other than the eternal problem of collecting firewood dry enough to burn. I was not well-prepared! Being in an isolated place without much distraction helped me concentrate and devote daydreams to issues concerning the book, rather than other responsibilities.

Did time pressure or other responsibilities help or hurt your writing?
Let me turn that around. I think that my writing hurt my other responsibilities. As I recall, after the book came out I found it difficult to get down to research on new things for a couple of years. Perhaps I was just a bit exhausted.

What kind of feedback did you get after the book came out?
I’ve gotten a lot of positive feedback from people who used the book when they were students, and still do now and then. That’s really gratifying to me, as I was specifically writing with the student audience in mind. But I’ve also gotten some negative feedback. For example, a book review in the AMS Bulletin criticized the scanty “kudology” in the book. That criticism is completely valid, my scholarship was pretty low-level.
Other feedback was the discovery by Kenneth Perko that two of the knots in the table, which were listed as distinct, are really the same knot. They’re now known as the “Perko pair.” Ken had been a student of Fox and later practiced law in New York for many years. I met him only recently, and we have a sort of correspondence. Recently Perko told me that I could have saved two bucks if only I had known! There were also some errors in some of the link polynomials in the P or P edition, pointed out to me by Nathan Dunfield, who checked them all by computer. They’re fixed in the AMS edition.

What advice would you give to new authors?
Have fun with your writing!

When should mathematics students begin publishing in research journals?

The publication resumes of many graduate students in recent years are impressive. It is not unusual for a student to have several refereed journal articles before getting a PhD, and these days even undergraduates face pressures to publish in refereed journals. At the same time, and possibly to support this productivity, eff0rts at research collaboration are starting at an increasingly early stage. There are many positives to the trend. The idea of getting to research frontiers quickly attracts more students to mathematics, there is something very satisfying about an end product like a journal article, and publications are important for rising in the mathematics profession. Added to this, doing mathematics research together in a supportive group is a fun activity in its own right.

Are there downsides to an emphasis on early exposure and quick tangible results? In today’s culture many students feel that fast tracks toward research and publications are the only way to succeed in math, and that this goes hand in hand with the need to build large networks of potential collaborations starting at an early stage. The popular Research Experience for Undergraduates (REU) programs are well-funded and an opportunity for undergraduate math majors to expose themselves to mathematics beyond the usual curricula. On the other hand, getting into a good REU and publishing as a result has become the first defining hurdle for many students (in their self-perceptions as well as in the practical reality of graduate school applications). Aside from the obvious potential dangers of vetting students according to a fixed list of established criteria at such an early stage, there is a need to consider how these judgements are shaping student conceptions of mathematics and their potential place in it.

What is the best way to nurture mathematicians? Opportunities for young mathematicians to learn and do projects together can be healthy, if done in a thoughtful way, but the emphasis on publishing is not always the best way to nurture mathematical potential, especially when one looks, for comparison, at the careers of successful mathematicians. The number of papers published (which can vary immensely) does not determine the influence and importance of a mathematician, and breadth of knowledge developed at a young age aids greater mathematical productivity in the long-term. Is the time-honored tradition of students pouring over books and doing every exercise fading away in the rush to find and solve a problem at the frontiers of knowledge?

Of course, beyond social pressures there are practical issues that affect success as a mathematician. Will I get into a better graduate school if I have a published paper? Will I get a better chance at a postdoc? How many papers do I need to get a tenure-track position? What do I need to do for tenure? Though these questions loom large for many, and do need to be addressed, they are by no means everything. As individuals, mathematicians have choices, mathematics is a personal journey, and ultimately by its very nature mathematics is an endeavor where being an independent thinker is an asset.

Your comments and suggestions are welcome as always!

Moduli problems in algebraic geometry date back to Riemann’s famous count of the 3g-3 parameters needed to determine a curve of genus g.

In this book, Zariski studies the moduli space of curves of the same equisingularity class. After setting up and reviewing the basic material, Zariski devotes one chapter to the topology of the moduli space, including an explicit determination of the rare cases when the space is compact. Chapter V looks at specific examples where the dimension of the generic component can be determined through rather concrete methods. Zariski’s last chapter concerns the application of deformation theory to the moduli problem, including the determination of the dimension of the generic component for a particular family of curves.

An appendix by Bernard Teissier reconsiders the moduli problem from the point of view of deformation theory. He gives new proofs of some of Zariski’s results, as well as a natural construction of a compactification of the moduli space.

Richard Evan Schwartz has written math books for a range of audiences: for university students, researchers, and several for children. His distinctive approach brings a touch of childlike freedom to his high-level research monographs, and mathematical depth to his whimsical children’s books.

(Added Note: Richard Schwartz has just received an honorable mention at the 2017 Prose Awards for Gallery of the Infinite. He has previously received the 2015 MSRI Mathical Book for Kids from Tots To Teens Award for Really Big Numbers.)

Here are Schwartz’s responses to a list of questions we posed by email.

What made you decide to write the book?

In the case of my first book, “You Can Count on Monsters”, I wrote the first version specifically to teach my then 6-year-old daughter Lucy about prime numbers and factoring. I wrote later versions of that first book because I thought that other children would like them as well. I wrote my second book, “Really Big Numbers” because I wanted to share my fascination with really huge numbers. Even though there are many counting books in the literature, there are none which explore the kind of insanely huge numbers that mathematicians occasionally think about. I thought that kids might like to see this.

Was there a gap in the literature you were trying to fill?

Yes, this is definitely true, especially of my last two picture books, “Gallery of the Infinite” and “Life on the Infinite Farm”. These books attempt to explain some concepts of infinity to audiences of various ages. I don’t think that there is anything like it available. In these cases, I wanted to write something really novel.

How did you decide on the format and style of the book?

I think that my limited artistic range dictated the format for me. I like to draw pictures using the computer drawing programs xfig and inkscape. I used those programs to illustrate my books because I was comfortable with them. For a long time, I had been obsessed with trying to find an illustrator who could illustrate my books, and this actually held me back for quite some time. I kept waiting for this magical person to come along who could bring my ideas to life. It was extremely difficult for me even to get in touch with illustrators, let alone find one who would be a good fit, and so eventually I just improved my own drawing to the point where (more or less) I could do it myself.

What did you focus on the most when writing?

I think that my focus varies depending on the stage of the writing. In the beginning, I focus on the overall shape of the book, the big picture. Once I have a good idea of the scope of the project, I focus on trying to produce as many pictures as fast as I can. I find that my projects often die if I can’t get going fast enough in the beginning, so I like to get up a big head of steam.

Once the project gets going, I don’t have too much problem with motivation and focus. What keeps me going is the strong desire to see the finished project. Once the ideas are (to my mind) fully formed, I can’t wait to get them all out on the page. On the other hand, it is often very hard for me to start a new project. In that case, I find that it is very hard to force myself to work. I have to get an idea I’m really excited about, and those don’t come along so often.

After I have something rough banged out, or at least many pages done, I focus quite a bit on revision and improvement. I try to make the pictures as clean and simple as possible, the color schemes harmonious and beautiful, and the writing sharp and graceful and interesting. At the very end, I focus almost entirely on eliminating typos and glitches.

What were the positives and negatives of the experience?

The most positive part of the experience is when someone sends me email or otherwise tells me that they have enjoyed the book. I have gotten a fair amount of email like this for my first two books, and it really makes me happy. I am delighted when some little kid says that he or she loves my books. Another positive part is when I actually get to hold the final product in my hands. For as long as I can remember, I wanted to write books. Often when I am holding one of my books, my mind goes back to those dark days in middle school or high school when I was slogging through boring work and dreaming of doing something bigger.

The most negative part of the experience is when I find out about a mistake in the published version. Like many mathematicians, I hate making mistakes. This doesn’t happen too often in the picture books, but I have found many little mistakes — mostly typos or notation errors — in my published works and this really sickens me. Another negative experience is getting bad or luke-warm reviews of my books online, especially if they complain about the artwork. One of my unpublished books, “Life on the Infinite Farm” somehow got onto tumblr, and about 20,000 people looked at it. There were lots of comments to the effect that the book was horrifying or monstrous or otherwise terrible, and these made me pretty unhappy.

Probably the most surprising thing is that my book, “You Can Count on Monsters” shot to number 1 on amazon.com after Keith Devlin reviewed it on NPR. It only stayed number 1 for about a weekend, but still it was a totally surreal experience.

Was your writing influenced by other books? Which ones?

I’m not sure how much other books have directly influenced my writing, but there are certainly many books I admire and try to live up to when I write. For comics and animation, I love the beautifully and simply animated television series “Justice League”. My drawing style is a lot like “Justice League”, but of course not nearly as good. For mathematical exposition, I love the book “Journey through Genius”. My books aren’t exactly like
“Journey through Genius”, but I love the great expository style of that book, as well as the great choice of topics. I also love the bright, primary-colored geometric sculptures of Calder. My drawing style is somewhat like Calder’s sculpture style but, again, not nearly as good.

Did you find ways to get feedback while writing your book or was it a solitary effort?

The writing process for me is mostly a solitary process, except that sometimes I will ask my wife or daughters (who are all good artists) for critiques of the pictures I have drawn. I am generally open to their criticism, and will implement suggestions they make. Usually the suggestions are of the form, “You drew the arms too big” or “That red and green don’t go together well.” Once the book is mostly written, and all the main ideas are in place, I send it out to friends and colleagues, asking for comments. I am generally very grateful for criticism and happy to implement suggestions I get.

What advice would you give to new authors?

I would say that the most important thing is just to jump in and WRITE IT! A lot of times people dither around with a good idea and never quite get around to bringing it to fruition. I used to listen to my dad talk for years about this great board game he was going to develop — it really was a good idea — but he just never got around to doing it.

Another piece of advice is that you shouldn’t solicit too much feedback early on. If you hear from lots of people before you get going very far on your idea, you will be daunted by all the different kinds things you hear. You may feel as if you have to satisfy all kinds of constraints and expectations. It is better to just write a bunch of stuff on your own first and then see what people think of an idea that is already well in progress.

On the other hand, do solicit a lot of feedback eventually. Once you have written a large part of your book, by all means solicit lots of feedback. I have found that other people’s suggestions have made my books much better. A lot of times you develop blind spots in isolation, and think that something will be clear to other people just because it is clear in your mind. By finding out what people actually experience when reading your book, you can adjust things so that they make sense to the outside world.

One last thing: protect your time and try to block out large free periods. Most people (including me) have a lot of demands on their time, but it is important to compartmentalize these tasks so that you are not working on trivia all day long. I try to concentrate the daily tasks so that I do them in short bursts, leaving myself free time. For instance, I set aside certain days for refereeing a paper or writing letters of recommendation or meeting with students, and I will do everything on those special days.

What kind of feedback did you get after the book came out?

I’ve gotten all kinds of feedback. Here are some examples

— Occasionally people have asked me to lecture about the books, or (on a very small scale) participate in book signings.

— I got one of the 2015 Mathical Prizes for my book “Really Big Numbers”. That involved me going down to an award ceremony in Washington D.C.

–I get a fair amount of emails about the books, from parents of children who like them. I like this feedback the best.

— I’ve had a few people tell me that they’re used the book as part of lectures or teaching presentations.

— I once got an email from an art school telling me that they were using my book, “You Can Count on Monsters” as the basis for an art project.

— A Korean computer scientist developed a video game based on “You can Count on Monsters”.

The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

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