I encourage Mathematics students to remember that, fundamentally, we are problem solvers. When I started college in 1991, one of the books that was recommended to me along these lines was:
- “Problem Solving Through Problems” by Loren C. Larson
This book is particularly good for students preparing to take the Putnam Mathematics Competition. I think research mathematics is different from contest mathematics, but this is often debated.
One way to understand the discipline of mathematics is to study its history. To this end, take a look at:
- “Mathematical Thought from Ancient to Modern Times” by Morris Kline
A more recent book that is much broader in scope than the history of mathematics is:
- “The Princeton Companion to Mathematics,“ edited by Timothy Gowers
The above few books are really excellent for students at the college level. I must emphasize that here I am assuming mastery of the K-12 curriculum. In Wisconsin, this curriculum is defined by the Common Core State Standards for Mathematics.
Regarding high school mathematics, I recommend a solid background in precalculus, trigonometry, basic probability and statistics before taking on calculus. Many students, in my opinion, are misled into thinking that calculus should be taken as early as possible.
Regarding Calculus, I learned it by first by reading Douglas Downing’s Calculus the Easy Way. Some of the other books in this series are also good.
With that said, students interested in majoring in mathematics should plan on taking the standard two year calculus sequence. At UWM, one popular suggestion is: Math 231, 232, 233, 234. While you are in this sequence, take Math 341 (aka 241) as early as possible.
Beyond the K-12 curriculum and standard calculus sequence, here are some excellent next steps:
The above links are to books that I recommend to advanced undergraduate and beginning graduate students of mathematics.
For students who are interested in working on research projects with me, please note that I take a combinatorial point of view, which stems from Richard Stanley’s Enumerative Combinatorics, volumes one and two. While I was in graduate school at UC-San Diego during the late 90’s, a great deal of work was being done on the theory symmetric functions, so I’m heavily influenced by I. G. Macdonald’s Symmetric functions and Hall Polynomials, and W. Fulton’s Young Tableaux. Note that the latter work very clearly identifies the relationship that combinatorics has with both geometry and representation theory.
One goal of research mathematics is to develop new tools to solve problems by exploiting symmetry. For present day examples of what I’m talking about click here. Historically, many examples fit under the heading of invariant theory. As a graduate student, I was taught to use methods described best in Roe Goodman and Nolan Wallach’s book:
The history is interesting. One might view the above work as an exposition of Hermann Weyl‘s perspective, which is influential in modern day mathematics and physics. However, I learned early on that the name that is attached to this subject is Sophus Lie. His motivation was, in part, to solve differential equations using groups of symmetry in much the same was that Galois theory is used to solve polynomial equations. Many aspects of the subject now include this Norwegian mathematician’s name. It goes without saying that the word Lie creates confusion — our goal is to tell the truth! Please help straighten this point out by telling folks that “Lie” is the name of a person. By the way, “Lie” is pronounced as “Lee.”
In any case, the history tells a big picture story. My recommendation is to read the essay by Roger Howe, A Century of Lie Theory. However, this might be complicated if you are new to Lie theory. An easier alternative is Lectures on Lie algebras and Lie groups by Carter, MacDonald, and Segal.
When I studied from the Goodman and Wallach book, I found it helpful to trace some of the ideas through other topics.
Here is a short list of these areas with links to books that I found helpful:
|Actions of algebraic groups||Lie theory||Symmetric spaces|
|Algebraic varieties||Homological algebra||Smooth manifolds|
|Commutative rings||Linear algebra||Differential Equations|
At this point, I welcome further discussion and references. Please keep in mind that the above emphasizes a very specific point of view toward three (related?) big picture research programs:
- Quantization of classical physics (including Einstein’s theory of general relativity)
- The Langlands program (both arithmetic and geometric)
- Non-standard models of computation