Yourself, librarians, classmates, current professors, former (grad and undergrad) professors of yours, other mathematics graduate students, professors and students in other areas, emeritus professors, departmental visitors, mathematicians (in and out of academia) you have never met, people hanging out in the mail room, people hanging out in the coffee room, people hanging out on newsgroups, yourself. Don't be afraid to ask "dumb" questions: a mathematician who "knows all the answers" is a mathematician too vain or scared to ask the real questions.
a. For basic courses and topics Your textbook, your classnotes, your classmates' classnotes, notes from previous incarnations of the course and from prerequisites, texts and old exams in the math library, available texts in libraries (check via on-line catalogue), books in the various MAA and AMS expository series, Schaum's and other Outlines, Dover paperbacks (cheap!), books like X for Engineers. To find paper resources, check http://www.amazon.com or http://www.barnesandnoble.com for books in print; http://www.AddALL.com/ compares prices from various sites. Also, http://www.bookfinder.com is a meta-search engine for finding used books. Many publishers offer discounts on new books to AMS conference attendees. Some people put give-away books in the Math Library, or on tables outside offices.
b. Research material -- primary sources. Research papers, personal lecture notes taken at research talks, conference proceedings, Springer Lecture Notes, etc., monographs, books, collections of important papers in a field, special issues of journals (especially the overviews of the issues), oeuvres of famous mathematicians (especially the introductions to the papers). Ph.D. theses are available from Dissertation Abstracts through the Main Library. Some people put give-away piles of preprints and reprints in various places around the Department.
c. Research material -- secondary sources. Review papers (see the Bulletin and Notices of the AMS), advanced textbooks, videotapes of lectures downloadable from http://www.msri.org and elsewhere, expository journals: Amer. Math. Monthly, Math. Mag., Math. Intelligencer, College Math. J., L'Enseign. Math., Elem. der Math., SIAM Journals (many others). Look for elementary articles by advanced authors.
d. Reviews of research papers. The principal place to find research papers reviewed is Mathematical Reviews, which is rapidly evolving from a print medium to an electronic medium, accessible via http://www.ams.org/mathscinet/search. If you aren't on-line, you can look at annual MR indices by author and subject, bound collections of MR reviews by topic and MR on CD-ROM (Silver Platter). Other sources include Zentralblatt, Ref. J. (Russian) and the Science Citation Index (in the Chem Library). For older material (1868--1942), see the Jahrbuch project http://www.emis.de/MATH/JFM/JFM.html.
e. Other sources Librarians are smart and know a lot. Histories of particular subject areas and of mathematics in general, biographies and autobiographies of mathematicians, reviews of books you want to study. For problem solving, read How to Solve It or other books by Pólya and look for problem collections; for writing mathematics, read How to write mathematics from AMS; study the style of books you cherish. There are dictionaries and encyclopedias of mathematics, recent and ancient, in print and electronic, and in English and many other languages.
a. Newsgroups and mailing lists. The UIUC computer system no longer subscribes to newsgroups, but they can be accessed via Google groups, which tells me there are 434 math groups. I don't know what's good at the moment. You should also be aware of the Young Mathematicians' Network at http://concerns.youngmath.net/, which is mainly aimed at mathematics grad students and new PhDs. Many specialties have their own mailing lists; don't be afraid to ask to join them.
b. Our Departmental Home Page -- http://www.math.uiuc.edu You owe it to yourself to explore this magnificent resource, which has links to dozens of useful places. An excellent summary of general references available from UIUC machines is at a website from the Main Library: http://gateway.library.uiuc.edu/rex/erefs/.
c. Other websites A constantly updated on-line mathematics encyclopedia can be found at Eric Weisstein's World of Mathematics website, http://mathworld.wolfram.com/. This is very useful for faculty, grad students and undergraduates. An encyclopedia which assumes more background is run by the publisher Springer-Verlag: http://eom.springer.de/ Many mathematicians have their own websites and nearly all departments have them as well. Search through your favorite engine, or use the worldwide list at http://www.math.psu.edu/MathLists/Contents.html. You can download papers and even computer programs by ftp or through a browser, and electronic journals are burgeoning. Some paper journals have part or all of their text available online. When you start writing papers, you can make them available via the ``Mathematics ArXiv" at http://front.math.ucdavis.edu/. The MR subject classification can be worked out via http://www.ams.org/mathscinet/otherTools.html. It seems that every large department and every funding agency has its own idiosyncratic way of organizing mathematical knowledge into areas. A good place to find mathematical history is at http://turnbull.mcs.st-and.ac.uk/history/Search/historysearch.html, which is the MacTutor site of the University of St. Andrews in Scotland. Finally, a fun place to find out who your professors' advisors were is http://genealogy.math.ndsu.nodak.edu/.
d. E-mail and other tools Just about every research mathematician around the world uses e-mail. It's faster and cheaper than surface mail, much cheaper than phone calls, and minimizes time-zone and language problems. Learn LaTeX, Mathematica (or some other symbolic computation language); at least learn to touch-type. I'm not an expert on this area at all.
a. Urbana Lectures in courses, seminars, named lecture series, colloquia (get used to being confused at these), mini-conferences (Illinois Number Theory, etc.), AMS regional meeting here in 1999, Special Years (e.g. number theory in 1999-2000), Math Library New Journal table, meals with fellow students and faculty and visitors, coffee room seminars, problems left on blackboards, listen to others talk in the hallways and keep notes of your "wild" ideas. Avoid mathismo: if you "talk the talk", you better be able to "chalk the chalk", but be assertive in trying to learn gain self-confidence (or learn to fake it). If you convince yourself that you are doomed to failure, you are.
b. Road trips Other good mathematics departments offer everything in a. as well. The Combined Membership List is on paper, or at http://www.ams.org/cml/, which contains a large fraction of all active US mathematicians. The paper copy gives geographical lists of members as well. Don't be shy about dropping in at strange places when you're traveling. There are dedicated Mathematics Centers all over the place, including MSRI (Berkeley), IAS (Princeton), DIMACS (New Jersey), IMA (Minnesota), and many others. Go to Joint Meetings (AMS-MAA-AWM) in January and MAA Summerfests, Joint Summer Research Conferences and Regional AMS Meetings (special sessions), disciplinary conferences, International Congresses of Mathematics and of Math Education. Almost every conference has its own website now. Graduate students are always welcome (and often subsidized) at conferences; don't let (nominally) high registration fees or travel costs discourage you, support can often be found.
c. Virtual The electronic milieu is happenin' and many collaborators will never meet. But never discount the "pressed fiber medium" or the human factor.
Last modified: February 12, 2007