Yes, you need graduate real/complex analysis plus algebra
tronter said:
I am looking to study the most "useful" math. By useful, I mean math that is important in the real world. Probability I know is very important. But is Real Analysis and Abstract Algebra really necessary to study the more "useful" applied math?
Is Boas book good to get most of the useful math you need?
You mean the textbook on mathematical methods by Mary L. Boas? If so, I happen to think that is a very good book which should provide a solid background for junior year undergraduate mathematics.
To answer your question, we might need more information about what your goals are.
Naturally, I won't let lack of information stop me from trying to answer your question! Let's consider the prequisites for a handful of the most useful areas of "applicable mathematics" (applicable to physics, chemistry, biology, engineering, economics, you name it).
Nothing in mathematics has proven more useful than
information theory (see
https://www.physicsforums.com/showthread.php?t=183900), and Shannon's information theory rests firmly upon
ergodic theory, which rests upon
probability theory, which rests upon
measure theory, one of the most important topics in a good senior year or first year graduate analysis course. (Note too the prominent role played by
group actions in my survey--- this is a topic in a good senior year or first year graduate course in algebra.) Very few things in mathematics have proven more useful than
representation theory (see my post # 4 in
https://www.physicsforums.com/showthread.php?t=185965), and this rests upon
Lie theory, which rests upon graduate level analysis, algebra, and the theory of
manifolds. And nothing in mathematics has proven more useful than the theory of
differential equations, which --- you probably saw this coming!--- really requires first year graduate analysis (e.g. to understand integral transforms, operators on function spaces, harmonic analysis--- which is closely related to representation theory and even information theory, by the way). Most people who try to use DEs in model building encounter the problem of trying to solve nonlinear PDEs; here, the only really general tool is the
symmetry analysis of the equation, which requires Lie theory.
One huge area which is barely hinted at in Boas is
combinatorics and
graph theory. Indeed, the most important theorem in mathematics, the
Szemeredi lemma, involves ergodic theory and graph theory. See
http://www.arxiv.org/abs/math/0702396 (note the application of Szemeredi extends even to
number theory) and then one of the best books published to date, Bollobas,
Modern Graph Theory (which has almost no prerequisites).
Ditto the others about statistics. See my post # 5 in
https://www.physicsforums.com/showthread.php?p=1416394 and David Salsburg,
The Lady Tasting Tea. I should point out that mathematical statistics is closely related to Shannon's information theory, for example via the
Principle of Maximal Entropy. See
http://www.math.uni-hamburg.de/home/gunesch/Entropy/stat.html And--- again, you probably saw this coming!--- statistics rests upon probability theory, and proper understanding of topics such as
moments requires analysis, while proper understanding of
factor analysis (a method for "lying with statistics") requires
finite dimensional euclidean geometry and finite dimensional linear algebra, so you might as well go the limit and study
Hilbert spaces for preparation.
Pattern recognition can be viewed as part of information theory, but a relatively minor thread in the grand tapestry of applied mathematics.
Kleinian geometry or
exterior calculus, or
computational algebraic geometry, or any items from a long list of further topics, would be far more important in the grand scheme of things.
Summing up: I have discussed prerequisites for five of the most useful areas of "applicable mathematics": information theory, representation theory, differential equations, combinatorics/graph theory, and statistics. Mastering the most important techniques and results in any of these areas requires graduate level analysis (real and complex) and graduate level algebra. So plan on taking these!
I don't think you need to study philosophy, unless you can find a good course on the philosophy of statistics which gets well into the titantic struggle between "frequentists" and "Bayesians". IMO, philosophy is useful, even essential, wherever mathematics meets the real world, but traditional undergraduate courses on the philosophy of mathematics are utterly useless to prepare you for any real-world philosophical conundrums you would be likely to encounter in the 21st century. IMO, philosophers are doing society and their own discipline a great disservice by failing (for the most part) to engage contempary mathematics. The reason is probably that philosophical analysis of modern mathematics requires a Ph.D. in mathematics as well as background in philosophy, so IMO almost no credible philosophy of modern mathematics or statistics yet exists. I have nominated this as perhaps the important challenge to scholars in the new century.
