Yourself, librarians, classmates, current professors, former professors of yours, other mathematics 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 math blogs, 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 Amazon.com or Barnesandnoble.com for books in print; AddALL.com compares prices from various sites. Also, 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. Most recent research papers are available online at the home page of at least one of their authors. These are often also posted at the Mathematics ArXiv before they are published. Many electronic journals and some paper journals have part or all of their text available online. The possible sources of research are these papers, but also 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 and, increasingly, online from the granting institutions.
c. Research material -- secondary sources. Review papers (see the Bulletin and Notices of the AMS), advanced textbooks, videotapes of lectures downloadable from MSRI and, amazingly enough, YouTube if you use the correct search terms. You should know about the expository journals: Amer. Math. Monthly, Math. Mag., Math. Intelligencer, College Math. J., L'Enseign. Math., Elem. der Math., SIAM Journals (many others). These contain explanations intended for non-experts! Many articles more than 5 years old are available for on-line viewing at UIUC machines through JSTOR. 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 at MathSciNet if you're at a UIUC machine. If you aren't on-line, you can look at annual MR indices by author and subject and bound collections of MR reviews by topic. Other sources include Zentralblatt, Ref. J. (Russian) and the Science Citation Index (in the Chem Library). For older material (1868--1942), see Jahrbuch Project.
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 many guides to mathematics, recent and ancient, in print and electronic, and in English and many other languages.
a. Our Departmental Home Pages -- UIUC Department of Mathematics and UIUC Mathematics Library. You owe it to yourself to explore these magnificent resources, which have links to dozens of useful websites, many of which appear separately here. An excellent summary of general references available from UIUC machines is at the Main Library: Online reference collection.
b. Other websites A constantly updated on-line mathematics encyclopedia can be found at Eric Weisstein's Mathworld. This is very useful for faculty, grad students and undergraduates. The publisher Springer-Verlag has the Online Encyclopedia of Mathematics, which assumes more background. Most mathematicians and all academic departments have their own websites. Search through your favorite engine, or use Mathematics websites from Penn State. Professional organizations have entered the 21st century with AMS blogs and MAA social media. The MR subject classification can be worked out via Search MSC Database. 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 MacTutor, run by the University of St. Andrews in Scotland. Finally, a fun place to find out who your professors' advisors were is Mathematical genealogy.
c. 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, but see TeX Help Pages. You should also be aware of Young Mathematicians' Network, which is mainly aimed at mathematics grad students and new PhDs.
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 March 2009, Special Years in various subjects, Math Library New Journal table, meals with mathematicians, 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 gain self-confidence (or learn how 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 identifies members of many professional organizations and contains a large fraction of all active US mathematicians. Don't be shy about dropping in at new places when you're traveling, and use this list, or the various websites to see who is there. 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 and knowledgeable undergrads 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' , but never discount the "pressed fiber medium" or, (especially!) the human factor.
Thanks to many people who have made suggestions, including John Gray, Phil Griffith and John McKay.
Last modified: January 28, 2015