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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Books: Hard Copy or E-?

How important are e-books for math?  I always start a blog with one question, and end up asking many more.

People according to their personality, their intentions, limitations and environments read differently.   Even when there were only hard copy books, I remember noting the different ways my fellow math graduate students read math books.  Some had a habit of reaching for a math book for bedtime reading.  To me, truly reading meant sitting at a desk or table, notebook and pen in hand, scribbling and drawing as I read. Others I knew sat perfectly still while reading, hardly moving a muscle for hours.  Some read while listening to music, and one person I knew even played piano while reading.

By now, most avid readers (particularly of non-technical books) are familiar with kindles and other e-readers. They are about the same size and weight of a paperback; you can turn pages with roughly the same movement; you can resize fonts, look up words you don’t know with an easy click;  many have their own light source; and most of all there is no weight difference between the data of one book and that of hundreds or thousands of them.   These qualities alone attract even diehard proponents of paper books, especially those who are frequent travelers.

But the popularity and availability of e-books and e-readers for mathematics lags behind.  What more can and should E-books offer, particularly in math?   Are we taking enough advantage of current technological capabilities?  Is there a need to specifically treat the particular nature of mathematics exposition?  Should we have moving graphics, and built-in software that help illustrate the material, perhaps with interactive feature?  What about making it possible for classes and reading groups to share comments online while reading a text?

Books are a medium for packaging and communicating ideas.  Assuming that there will always be a need to record and deliver mathematical ideas using some sort of print medium, do you think math e-books are here to stay, if so in what form, and how will they affect teaching, research and individual reading habits in the years to come?

Moving Things Around by Kerins, Young, Cuoco, Stevens, and Pilgrim.sstp-5-cov

Increasingly I see books that make me wonder…what would this look like if…?  One example is a new series of books produced by the Park City Summer Program  for secondary school teachers.   These sequenced collections of problems are carefully chosen to progress future teachers toward a deeper understanding of a subject through exploration, discussion and active learning: in this particular case the topic is permutations, symmetries and numbers.  Users of the book are encouraged to experiment with computer software and to work in teams.  Could electronic media help to implement the goals of these books on a wider scale, connecting people who are unable to attend sessions like the one at Park City?


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Books that have left their mark

Below are the noteworthy books suggested in comments to my last post:  “Books with Longevity”.  (I could not find good photos of Grothendieck’s EGA and SGA, published by publications IHES)   Thanks for sharing them!

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The intention of any worthy math book is to communicate a collective understanding of a subject by experts to potential future practitioners, but is it just me, or is there sometimes something more personal that happens between author and reader?  Some books seem to “talk” to you.  They can make you smile for the beauty that they reveal (examples for me are David Mumford’s Lectures on Curves on an Algebraic Surface, or Emil Artin’s Galois Theory), or they can egg you on with challenging problems leading you to deeper understanding (Attiyah and MacDonald’s Introduction to Commutative Algebra comes to mind).

On a more pragmatic level, as one comment pointed out,  it is also important for books to be useful for teaching.  That will be a topic for a future post.

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Math books with longevity

What gives a math book (textbook or otherwise) longevity?   Is there more to a book than just a record of current knowledge, or an aid and reference for a class? Some books continue to be read and reread, and used over and over from one generation to another defying loss of novelty and fashion. What makes a math book great? Is it the subject matter, the presentation, the author’s personal touch or something else that keeps a book relevant over generations?

The answers to these questions may be  personal.  They could be tied up with memories of a favorite spot in the library; the music that was playing as you worked; or the people who were around you. Or perhaps there are universal qualities that make a book great.  Style, elegance, care, quirkiness, beauty, originality…what resonates most with you?

One book that continues to be relevant despite the passage of time is Knots and Links by Dale Rolfsen (AMS Chelsea Series), an introduction to knot theory and low-dimensional topology that was first published in 1976 (Publish or Perish press).   One reason for its lasting significance is that   `Rolfsen’s knot and link table’ is still commonly used to quickly identify knots and links with low crossing numbers.   But what really distinguishes the book, especially for its time, is that it facilitates (in a very effective way) active learning by emphasizing well-chosen hand-drawn illustrations and exercises over long explanations and proofs.


What book was a game-changer for you as a student?  What made/makes it special? Are there any out of print math books you would like to see republished?  Please enter your comments, and mention your favorite book!

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