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More than one math Ph.D. program I am applying to has a section where I am to list relevant coursework. An example is as follows:

"Enter under each category list the two most advanced courses or course sequences you have had or will definitely take before you come to (School)."

Under each class title there is a blank line where I am to list a text or textbooks. (No other instruction)

My question is this: In a few of my courses, the professor has listed a textbook for the course, but whose lectures have not adhered to the structure or level of rigor of the book. The homework also did not come from the course book. If I chose to use another textbook to study out of should I list this one or the one listed in the course syllabus? My reasons for wanting to list the book I used are:

  1. The book on the syllabus is not a good indication of the level of rigor at which the course was taught.

  2. I haven't read a page of the textbook on the syllabus while the book I used is one I own and is the primary reference for the subject on my shelf.

Which book should I write in my application?

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  • People still use textbooks?
    – JeffE
    Dec 12 '17 at 15:06
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I am the Graduate Coordinator for a math department in which we ask graduate applicants to list their course textbooks.

I encourage you to follow the directions and list the textbooks that were used. By and large, this is a good way for us to figure out the approximate level and content of courses like "algebra," "analysis" and "topology" with standard names but widely varying content. A complex analysis course taught out of

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka

is likely to be quite different from one taught out of

Complex Analysis, Lars Ahlfors

If you feel that the chosen textbook was below the level of the course and you supplemented with a more advanced/rigorous textbook, you may as well list that textbook too, saying something like "I myself used..."

If the instructor has online lecture notes, definitely include a link to them: that gives a really good idea of what was covered.

Note to international readers: having standard textbooks for advanced undergraduate math classes is much more, um, standard in the United States than in many other parts of the world. We certainly know that international students may come from less of a "textbook culture." (This need not be a bad thing in itself, but it is one of several ways in which it can be harder to evaluate the profile of an international applicant than a domestic one.)

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    To complement the international part: in some countries (e.g. Italy, France, Germany) there are many (good) undergraduate textbooks, but they are usually written in the local language and are quite probably unknown to most of the US audience. Another issue is the possibly different approach to undergraduate courses.
    – Massimo Ortolano ♦
    Dec 12 '17 at 13:19
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    E.g. I didn't have calculus during the first year but a proof-based analysis course.
    – Massimo Ortolano ♦
    Dec 12 '17 at 13:21
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    @Massimo: Right, but the increased specialization of e.g. the average European university student over the average American university student usually works in favor of the European student from the perspective of a grad school admissions committee. (And for what it's worth: in my first year in an American university I took...a proof-based analysis course.)
    – Pete L. Clark
    Dec 12 '17 at 13:41
  • Thank you very much for your thorough and informed answer. I will take your advice. I think I panicked that admissions committee will read the name of a book and write off my background in an area as insufficient.
    – user38770
    Dec 12 '17 at 20:31
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How about leaving it blank or crossing it out and mentioning in the cover letter that your professors did not follow a textbook but rather designed their own lecture? I also studied math and in the whole 5 years until the end of the master, I only had one lecture that was slightly based upon a book; all others used lecture notes written by the professor. Thus, I'd say your case is not that uncommon and should be understood.

If you know of lecture notes being available to the public you might mention these of course; just in case someone wants to take a look.

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