Author Interview: Jennifer Schultens

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Question: What advice would you give to new authors?

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

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