Bruce Reznick, University of Illinois at Urbana-Champaign

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 and e-books 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; for Illinois, it's
IDEALS
https://www.ideals.illinois.edu/search

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.
There is some duplication, but more "fun" links at a webpage I
maintain for my students:
Fall 2019 Links.

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. ** I particularly call your attention to a collection
of 18
Reference Guides
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, housed at Wolfram. 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. 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. 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 24, 2020*