Author Interview: Hossein Giv

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 madehosseini-givs-photo-1-2 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 hamstext-25-covelp 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!

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Open Math Notes: the Road in Between

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

51dclvuqskl-_sx356_bo1204203200_ 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.

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Author Interview: Dale Rolfsen

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

getfileattachmentWhat 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?
gsm-88-colorI 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 1limited. 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 anklcd 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.getfileattachment-1
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!

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Culture, Conventions and Publishing in Math

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!

Featured Book of the Day

ulect-39-1-jpgThe Moduli Problem for Plane Branches by Oscar Zariski

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.

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Author Interview: Richard Evan Schwartz

dsc03509Richard 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. mbk-84-cov 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 mbk-97-covyou 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?stml

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.

survThe 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 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?mbk-90-cov

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 dsc00030for 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.

stmlOn 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”.

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Author Interview: Tamara Lakins

This is the first in a series of author interviews.  Enjoy!

Suggestions for further mathematics author interviews can be made via the comments or email to

The amstext-26-covTools of Mathematical Reasoning by Tamara Lakins was published in the AMS Undergraduate Textbooks Series earlier this year.   In the few months since it has appeared, the book has already received 22 desk copy requests, and has been acclaimed for its “crystal clear exposition and abundance of exercises” and its careful attention to the language of mathematics.   Here are the author’s responses to a list of questions we posed to her by email.


Tamara Lakins

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?

I imagine that my motivation behind writing my book was similar to that of many authors. I had been teaching Allegheny’s “Foundations of Mathematics” (introduction to proofs) course for many years, without being able to find a textbook that I was completely happy with. My experience teaching from various textbooks helped clarify in my mind what I wanted in a textbook, such as a quick path to proofs and an emphasis on the process of finding a proof. So, I began by converting my existing notes from teaching into a very early draft of the textbook.

What was the writing process like? Did you write every day on a set schedule, or did you have periods of setting it aside? Did you find ways to get feedback while writing your book or was it a solitary effort?

I did my writing in “spurts” and “sprints”. I began converting my teaching notes into textbook form during a sabbatical about 8 years ago, with the goal of using that early draft as a textbook when I taught the course in the following fall. I updated the draft often during that fall semester, as I discovered what parts of the book were working well for students, and what parts weren’t. After that, I sent the manuscript to several colleagues at other institutions for their feedback, which was encouraging. I was also fortunate that, over the next several years, several members of my department used my manuscript as their textbook for the introduction to proofs class. I benefitted greatly from my colleagues’ feedback. The feedback from my colleagues, both at Allegheny and at other institutions, is what gave me the courage to proceed with my plan to try to publish the textbook. Except for reacting to feedback and making corrections as I and my colleagues continued to use my manuscript as a textbook, I essentially set the manuscript aside to await my next sabbatical. I returned to the manuscript about a year before my next sabbatical, to prepare it for submission to a publisher. When the manuscript was accepted by the AMS, I worked on it (thankfully while on sabbatical) for many hours a day, almost every day, until it was due, which was about three months later.

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

The early chapters of the textbook, on the introductory logic and discussion of proof techniques, came most easily to me because I had been thinking about how to best teach these concepts for about 15 years. I found the material on sizes of sets and the foundations of analysis (which I don’t have much time for in class) very difficult to write. I am a logician by training, but I didn’t want the material on sizes of sets to start sounding like a course in set theory; my goal was to focus on what the typical math major needed to know about sizes of sets. A similar tension existed in the chapter on the foundations of analysis, where I wanted to spend some time discussing the question “what is a real number?”.

What were the positives and negatives of the experience? Did anything about the experience surprise you? Did time pressure or other responsibilities help or hurt your writing?

When my manuscript was accepted by the AMS, I believed that it was essentially in its final form, with the exception of the chapters on sizes of sets and the foundations of analysis. Those two chapters were in very rough form (as I didn’t have much time in class to devote to this material), and I was expecting to have to spend a lot of time not only writing, but also thinking about the organization of, those chapters. But I was surprised at how much time I spent also carefully reviewing and revising the other chapters, partially in response to the reviewers’ comments, but also because this was my “last chance” to “get it right”.

I had been planning to work on the manuscript during an entire spring semester, and I was surprised at how much earlier the AMS suggested I set my deadline. In many respects, the time pressure helped keep me focused, although I did find that I made more typos and other errors when working late hours after my son was finally asleep!

One of the best aspects of completing the textbook was that it inspired many stimulating conversations with my husband (who is also a mathematician and who was also on sabbatical) about the intro to proofs course, teaching, and my vision for the textbook.

I was surprised at how much still needed to be done after my “deadline”, when I thought that my part of the process was complete. I was very fortunate that the anonymous reviewers carefully read that “final” draft, providing me with many valuable comments and suggestions for improving the exposition of my manuscript and the exercises. Expanding the exercises in the textbook beyond what I normally assign in class was surprisingly time consuming. So, I continued to work on my manuscript for several months after it was “due”.

I think of myself as a very careful writer and proofreader, but I learned that there are always typos that one misses each time one proofreads!

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

It seemed to me that the AMS Pure and Applied Undergraduate Texts, with its focus on post-calculus courses, was a good fit. It was also important to me that my textbook be affordable to students.

What advice would you give to new authors?

Write what you are passionate about. Teaching the introduction to proofs class is one of my passions, and I believe that passion was essential to all phases (beginning, development, and completion) of writing my manuscript.

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Mathematical sign-posts

When you think about mathematics, what are your markers?   How do you organize in your mind the development of mathematical reasoning and ideas?  How do you integrate your historical, social, and personal perspectives?   Day-to-day, as a teacher or as a researcher, you may have very clear practical and narrow goals: to effectively convey important and useful knowledge and methodology,  or to reduce an open-ended problem to a well-defined and solvable setting.   But all this is informed by a larger view of mathematics and its essential interest and importance.  What are the landmarks that guide you, and how did they come to be a part of your landscape?   What do you try to pass on to your students?   How do you pass these on?

I look forward to your comments!

In the meantime, here is my featured book for this post.

Featured Book of the Day

1Mathematical Omnibus: Thirty Lectures on Classic Mathematics

By Dmitry Fuchs: University of California, Davis, CA,
Serge Tabachnikov: Pennsylvania State University, University Park, PA

(Paraphrasing the Bookstore.) The book consists of thirty lectures on diverse topics, covering a broad area of the mathematical landscape. The common thread in the selected subjects is their illustration of the unity and beauty of mathematics. A special feature of the book is an abundance of drawings (more than four hundred), artwork by an award-winning artist, and about a hundred portraits of mathematicians. Almost every lecture contains surprises for even the seasoned researcher.


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Math books for children

In this post we ask: what makes a good math book for children?

Is it more important that a child be left with knowledge that they can understand and retain, or a new awareness that keeps them thinking and wondering?   Is mathematics a world that one can enter and join in, or is mathematics a personal journey?    Of course both sides are important, but how much weight should be put on one side or the other?

In a recent family conversation someone asked: why do we learn history at school? The standard answers came up: “so that we don’t repeat past mistakes” and “to learn a way of thinking”.  Math instruction has similar taglines: “math is everywhere”, and “math is a stepping stone to good jobs”.  The underlying idea behind these reasonable sounding slogans is that mandatory education should fundamentally 1) help us understand our world; and 2) teach useful skills to work and function in society.

But there is a third fundamental reason to learn things at school, which people forget to mention. It is learning for learning’s sake.  There are people who simply are driven to learn. Who like to turn ideas around in their heads, and who are grateful for avenues to new horizons. When one is lucky enough to have a teacher who encourages curiosity and appreciation, something beyond practical skills is gained.   Without this aspect of education, knowledge would not progress, society would stagnate, and, personally speaking, life would be less fun.

Similar questions come up with children’s books.   Is the goal to teach skills or to excite wonder and appreciation? Is it possible to do both? Many countries such as Japan, Russia and Hungary have come up with systems for teaching mathematics to children that are highly effective in producing students with strong problem solving skills. Not only that, these methods are fun, and incorporate a nice balance between group learning and competition that works well for a broad range of children and abilities, and lets the top few excel quickly.  Lacking wide spread systems like this, US mathematics graduate programs typically see fewer qualified applicants educated in the US compared to those educated in Europe and Asia.

So how is it that some of the greatest mathematicians in the world were born and educated in the United States with little or no extra instruction from family or teachers as children? For some people, it seems that the only encouragement that is needed is the tiniest of childhood triggers. Many successful mathematicians (American and otherwise) were primarily self-taught, before they began studying math more formally at college or university. (Are you one of these people? If so, I encourage you to contribute a comment or blog post explaining how you were introduced to mathematics.)

If there is room in mathematics for people who find their own way to math, then I believe there is also interest in describing the journey in a way that resonates with children. Instead of a single-minded focus on learning a subject and technique properly – that can come from individual hard work once the motivation is there – an alternative approach is to illustrate a few simple but deep ideas in a new and personal way.

Featured Books of the Day



mcl-8-covA Moscow Math Circle, by Sergey Dorichenko is a collection of problem sets for eighth graders written by mathematics faculty at the Moscow State University.   The problems are organized around weekly lessons at a magnet program called Math Circles run by the University.  The program, begun at Moscow State University, is designed to engage students and give them a sense of the continuity between new concepts and ones previously mastered.   Since its inception, Math Circles has spread to many mathematics departments  around the world, including in the United States.  The book contains an explanation by Dorichenko of how the original Moscow Math Circle was run, and includes translations of the problem sets into English.





You can Count on Monsters, by Rich Schwartz is an imaginative depiction of numbers as monsters drawn using fun geometric shapes and colors.  The prime numbers are individual monsters, and the composites are made by interactions of the primes that divide them.  Thus, the personality of each number is a carefully arranged conglomeration of the personalities of its factors.   Meanwhile, a lot of mathematics is suggested by the way the shapes interact, including fundamental concepts from algebra and geometry.  For this reason, the book can be appreciated at lots of different levels, and will bring a smile to children of any age.




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Comments on e-books

There were several responses to my e-books blog post, so I will share them here.   Thanks all!


  • Matilde Marcolli says:

    Certainly the current formats of ebooks are not suitable for mathematical content: apart from some horrible debacles with completely unreadable formulas on kindle (and other) ebook readers (on hugely expensive ebooks), it is not just the problem of properly displaying readable formulas and diagrams. Mathematics book have an important 3-dimensional nature, where one continuously needs to jump back and forth between a lot of different points in the book: something extremely easy to do with a print copy and an impossible nightmare on an ebook that is designed to linearly read a novel from first to last page. I don’t think print format will have a better replacement soon for mathematical content. That said, I personally often accompany the print copy I have of a book with a simple (and cheap) PDF file that is searchable, so I can search on the file and then I read and work back and forth on the paper copy.

  • Steve Ferry says:

    I like e books. Being an old guy, the backlighting, high contrast, and ability to adjust the type size are important to me. It is helpful if the reader allows one to page quickly back and forth through the text. Being able to print a few pages at a time would also be a big help. I don’t see the need for video widgetry in advanced texts.

  • David Fisher says:

    I do read math books in electronic format sometimes, but always as pdf or djvu on my ipad and not on my kindle. I agree with Matilde that none of this quite reproduces the high dimensional experience of reading a physical book, but the ipad is so much closer than the kindle. But I don’t know how singular this is to math books as opposed to academic reading more generally. Certainly I have heard many horrified and disdainful conversations about kindles and ebooks among English professors.

    But unless mathematicians stop travelling so much, ebooks are a part of our present and our future. I don’t have the talent for it, but I hope someone who does designs a better math ebook. I imagine a revolution of reading mathematics of roughly the same order as TeX was a revolution for the writing of it. I can’t imagine specifics, but I hope it comes soon.

  • Barbara says:

    I agree with Matilde that the current ebook technology still cannot compare with an actual book, for the reasons listed; at the same time ebooks have changed our lives for the better since we’re no longer limited to what we can carry on our backs when traveling.
    It is my hope that future ebooks will be available which allow one to easily keep notes and switch back-and forth. I also think computer-like screens aren’t good for reading; what we need is something kindle-like but 4 times larger area wise (it’s no use being able to enlarge the character if then you have to squint at one quarter of the commutative diagram at a time).

    As for interaction, I think we’ll move to being able to add notes and put them online, so that people who are confused (something that happens to me a lot!) can click and see what others have to say. This is something I found useful already as a student, when the same effect was achieved by pencil remarks on the margin of the library copy.

    I also think that David Fisher’s comparison with TeX is true in another important sense: mathematicians use books like no one else. If we want a system which is tailored to our needs, we will have to build it ourselves. I think the AMS and its sisters societies can play a very important role in this.

  •  Robert Ghrist says:

    i have been working for two years on how to make a math e-book that is readable, since, as you note, kindle math books aren’t really working.

    i’ve come up with an approach that i think works. the good news is that it takes full advantage of the color space, form factor, and dynamics possible on a phone/tablet. the bad news is that it’s not compiled, and requires a lot of fine-detail manual positioning. also, it took a year just to get the fonts right, and they are as yet far from perfect.

    i’m 1200+ pages into this project, resulting in three e-books on multivariable calculus. you can see the latest entry here at the kindle store & use the “look inside feature”. but, really, it looks so much crisper on a good phone.

    rob ghrist, math, upenn

    ps: i’ve talked with folks at amazon about how to adapt the kindle platform more to the needs of math e-books. they were polite, but not really ready to move into that space yet from what i can tell.


    Curtis McMullen says:

    September 10, 2016 at 7:26 pm (Edit)

    E-books can be very useful to undergraduate students. I am currently teaching a course on “Sets, Groups and Knots”, and there was homework during “shopping period”, while students were still deciding which courses to take. E-books allowed them both to get an idea of the course and to do the first reading and homework without having to buy the books.



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Personal webpages as publications

Thanks to those of you who commented on my previous post on ebooks. This conversation brings home to me that what will drive substantial math ebook innovation (apart from simply making existing math books readable on electronic devices) will be a joint work between mathematicians, programmers (and possibly publishers to egg them on) that anticipates accurately the way math students and researchers will want to write and read.  It may be the kind of change that will happen solely from within mathematics, perhaps in small steps.

In my memory, the steps leading to the advent of email and tex were rapid.  Some of you  remember the 1980s when email and tex were just coming into common use. I had friends whose primary job was technical typing for mathematics faculty, and I still wrote handwritten letters to friends and family, something I have not done in decades. It seems that before we knew it email and tex became commonplace in mathematics, and as I understand it both  developed rather organically with mathematicians playing a large role.

A similarly extraordinary, though much more subtle, transformation in our way of lives is happening (has happened?) as a result of personal webpages.  Although universities have standardized faculty webpages, most math faculty have their own personal ones, and they have become increasingly significant and indispensable to the way mathematicians function socially, culturally, and practically.

Personal webpages have become vital sources of open access information that is for the most part non-monitored and free form; and because there are few rules governing them, they are also expressions of a mathematician’s individuality.  If one wants to know something about a particular person, one can glean a lot from their webpage and what and how they choose to present.  The page is at once personal and public, and I cannot think of anything that existed pre-internet that played the same role.   Some are stark and minimal, while others have rather highly developed narratives; some include images, moving gifs, links, and computer programs in addition to the invaluable list of publications, vita and teaching information. Personal webpages have become a sort of publication that combines the fun of personality and self-expression with the ability to present a swath of convenient and useful content connected to the author’s activities, interests, and work.

The use of personal webpages by mathematicians has grown to the point that they have become regular “go to” resources for mathematicians to find lecture notes, teaching ideas, theorems, and software.  Contrast this to the  80s (my personal reference point) when students and researchers spent a lot more time in libraries than we do now.  As older readers of this blog will remember: we used to search through indexes of heavy volumes of Mathematical Reviews to find out what was known about a given topic and where to find the articles; then we’d search through the stacks and shelves of identically bound journals to find the correct year and volume number; and once we found the article we would carry the heavy book over to a desk and read awkwardly over the lumps made on the open page by the huge binding, or if we wanted to make the contents portable, we would head to the copy machine hoping we remembered to bring coins.   During my last move, I threw away almost two filing cabinets worth of xeroxed papers, that I previously did not have the heart to dispose of though most of the articles are now a snap to find online.

I see the good and bad sides of the ways of the past and now.  The time it took to do what is so easy now was also time to think slowly, which is usually a good thing.   Now, if one knows who the author is, one can usually simply go to their webpage.  From there one can look at references of their papers and search on to other webpages.    In the meantime, one may make unexpected discoveries, or if one is not careful, one might even stray onto a new tangent.  The journey now is potentially quicker but is more sedentary.  One interacts with fewer people, but is exposed to more personal expression.  What we have lost in terms of walking and lifting, we have gained in  a better passing acquaintance with the people we are citing.

It would seem that the electronic book question is really part of a much bigger question about the relation between technology and information , and creativity and expression. One of the exciting things about mathematics is that at the frontiers of the subject, there is a continual rethinking and creating of “tools” and “pathways” for communication. Unlike in experimental science, to “show” someone what one has discovered, one can only rely on language and symbols, and a feature of mathematicians is not only that they solve deep and difficult problems, but that they develop ways to take others there, by expressing complicated ideas in enlightening and simple ways.  It is natural that mathematicians will use all the means of communicating ideas at their disposal: through teaching directly, through writing, through programing, and consciously or unconsciously through their personal webpages.

Grothendieck-Serre Correspondence, Bilingual Edition.   2001

cgs-covGetting a view into the personalities of mathematicians can be fascinating and fun, and two of the most interesting personalities in mathematics are Grothendieck and Serre.  This book contains letters sent between these two mathematical giants from 1955 to 1987 in the original French with a translation into English.





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