In the various forums there have been questions that require for a clear formulation of both the question and the answer use of relevant mathematics. It is also clear that the mathematical background, and more importantly that elusive quality sometimes called "mathematical maturity" is varied among participants. The purpose of this post is to outline the landscape of mathematical topics (certainly not in an encyclopedic fashion) and to suggest some useful textbooks and monographs for further study.

There is a general misperception that mathematics is deductive, while other related disciplines such as physics are inductive. In fact mathematics research relies heavily on intuition and inductive reasoning. Mathematicians, good mathematicians, are skilled at guessing the correct solution to a problem and then proving that the guess is correct. Think about how theoretical mathematics is presented. One understands several examples, states a theorem (a good guess) then proves that the theorem is correct. Outstanding guesses are called conjectures, and the resolution of a long-standing conjecture is a major accomplishment. For example Perleman received accolades for finally proving the Poincare Conjecture and Wiles for proving Fermat’s Last Theorem, which was really a conjecture. Graduate studies in mathematics serve to hone one’s intuition (the ability to guess correctly) and ability to construct logical proofs of that which one guesses.

As one studies any of the disciplines discussed below, it is wise to recognize that what should be trying to develop is a basic, and intuitive, understanding of the structure of the subject being studied, and a facility with methods of proof that are useful in the area. The former is more important than the latter, for one can usually find some way, perhaps clumsy, to construct a proof of a true theorem, but no technique will let one prove that which is wrong.

Mathematics is roughly divided into algebra, analysis, and topology. Geometry is sometimes included with analysis, sometimes with topology, and sometimes as a fourth broad category. These categories are neither mutually exclusive nor all-inclusive. Combinatorics, for instance, can be listed under algebra or in a separate category for "finite mathematics". Distinctions will blur further as more advanced mathematics is encountered. So, be aware that the categories discussed are a bit loose, to say the least.

1.Basic undergraduate mathematics

Algebra and trigonometry

Elementary lgebra and trigonometry are really high school classes, and when taught at the university level are essentially, if not formally, remedial. Any good book with a title like "College Algebra" or "College Algebra and Trigonometry" will do nicely. A class at a junior college or university is highly recommended if you are less that very proficient in these subjects.

Calculus and Analytic Geometry

Calculus is fundamental to physics. In fact calculus was invented by Newton for the purpose of formulating laws of mechanics. It is not too much to advocate that the mathematician Newton was the first to formulate quantitative physics, and he invented calculus in order to do it. Basically Newton took as his goal to explain the laws of planetary motion as described by Kepler based on curve fits to empirical data compiled by Tycho Brahe. To do that he needed differential equations, which of course required calculus.

Good calculus texts are hard to find. I have taught more than a handful of calculus classes, and generally am dissatisfied with the over-emphasis on intricate calculations, clever substitutions, and crank-turning at the expense of understanding what derivatives and integrals really are and why one ought to care about them. This situation has only become worse with the tendency for calculus to be taught in high school.

That said, one can learn calculus, at least at the level of performing the usual operations of differentiation and integration of common functions from almost any calculus text. I personally like Mike Spivak's Calculus as it does do a better job on the concepts than most other texts, but many students find it too difficult and it is not wide used (though most professional mathematicians like it). Other good texts, more widely used, are those by Lipman Bers, Thomas and Finney, and Olmstead.

By the time one has completed a class or a self-study program from these texts one should have been exposed to derivatives, the Riemann integral, infinite series, partial derivatives and multiple integrals, and some vector notation and analytic geometry of surfaces in 3-space. One has not seen a rigorous treatment of the subject or been required to understand proofs in detail or to produce proofs by oneself.

Linear Algebra and Ordinary Differential Equations

Linear algebra is the study of vectors, linear transformations, sets of simultaneous linear equations. In a first class in linear algebra one encounters the notion of a vector space, linearity of functions, representation of linear functions by matrices, matrix multiplication, a bit on the determinant, etc. This material is critical for applications in engineering and physics. It is the foundation for multi-variable calculus and the study of dynamical systems (systems of differential equations).

Linear algebra may be studied for the first time in conjunction with differential equations, and there are some good textbooks that combine the two subjects very effectively. I personally recommend this approach, but others might prefer to study the subjects separately.

Ordinary differential equations find application throughout physics and engineering. Quite commonly the approach to solving physics problems is to first express the problem as a differential equation and then solve it. Solving differential equations, particularly in the non-linear case, often involves a trick or two combined with a clever substitution. Thus one aspect of a class on the subject is learning to recognize some common types of differential equations and the techniques that allow one to solve them.

Recommended introductory linear algebra and differential equations texts:

Linear Mathematics; an Introduction to Linear Algebra and Linear Differential Equations-- Fred Brauer

Elementary Differential Equations-- Boyce and DiPrima

Differential Equations and Their Applications-- Braun

There are many, many other texts on these subjects, and most of them will do nicely for a first encounter. Pick one (or two).

Once you are acquainted with calculus, linear algebra and differential equations, you are ready to undertake a more serious study of mathematics and physics. These are the basic tools, though many more will be needed and encountered as one studies more advanced topics.

At this stage one is equipped to do the common calculations that are encountered in physics and engineering. However, one has not yet been exposed to the deeper reasons these subjects -- the theory that explains why these calculations work.

1.Rigorous Mathematics -- first steps

Real Analysis and Topology of Euclidean Space

Real Analysis is "calculus done right". In a first class in Real Analysis one will learn about the real number system, learn what "completeness" means, what makes the real numbers complete, and understand how the existence of a limit is related to completeness. From that foundation one develops the basic theory of differential and integral calculus in a rigorous fashion, with complete proofs. In doing so one will encounter the topology of the line and of Euclidean space, begin to understand the critical concept of a compact set, and see the Heine-Borel theorem proved (which is the fact that a subset of Euclidean space is compact if and only if it is closed and bounded). In brief, one learns why calculus works.

Recommended textbooks:

Elements of Real Analysis-- Robert Bartle

Principles of Mathematical Analysis-- Walter Rudin

Foundations of Analysis-- Joseph Taylor (excellent for self-study as it concentrates on the essentials)

Calculus on Manifolds– Michael Spivak (excellent treatment of advanced calculus and the beginnings of differential geometry)

Algebra

Algebra is the study of algebraic structures and operations -- groups, rings, fields, modules, vector spaces, etc. It includes linear algebra. A basic understanding of abstract algebra is necessary for further study of mathematics, because mathematics is an interconnected subject and algebra and algebraic structures are found throughout it. The study of functional analysis, which is important for much of modern physics (see below) combines algebra, topology and analysis and is critical to the theory of partial differential equations and quantum mechanics (Hilbert spaces are just one small piece of functional analysis).

Recommended Texts:

Algebra-- Michael Artin

There are, of course, other texts, but at the advanced undergraduate level it is hard to beat Artin.

For linear algebra, Artin's book is sufficient, but there are some others that specialize in that branch of algebra and are excellent:

Finite Dimensional Vector Spaces-- Paul Halmos (excellent from the perspective of Physics and analysis)

Linear Algebra-- Hoffman and Kunze

Complex Analysis

The calculus of complex-valued functions of one complex variable is an old and beautiful subject. It is (perhaps) surprisingly different from the theory of real-valued functions. For instance, you will learn that if a complex function is differentiable even once, then it is infinitely differentiable, and in fact is locally representable as a power series. This is different from the case of real-valued functions in which any function that is simply continuous is the derivative of something and in which derivatives need not be continuous at all. In an introduction class you will earn this, and be exposed to the calculus of residues, which can be used to solve some integrals that were too difficult in your first calculus class (ever wonder how some of those terrible integral formulas in the tables were found ?). Much of advanced mathematical analysis relies on properties of the complex numbers and complex calculus and hence this subject is foundational for further study. Quantum mechanics, for instance, is heavily dependent on the theory of Hilbert spaces over the complex numbers.

Recommended texts:

Complex Variables and Applications-- Churchill (an old and standard undergraduate text)

Introductory Complex Analysis & Applications-- Derrick (one of many plain vanilla texts)

Complex Variables-- Joseph Taylor (highly recommended. An undergraduate text that is both clear and goes a bit farther than most)

Foundations for Theoretical Mathematics

This is advanced undergraduate to first or second year graduate material (maybe a bit more in places) which provides the basis for the study of modern mathematics. There is a great deal of interplay among the topics in analysis, algebra, and topology noted below and it is beneficial to study at least one area in each area concurrently.

Analysis

Major topics in the area of analaysis are the general theory of measure and integration (particularly the Lebesgue-Steltjes integral), elements of functional analysis, and complex analysis of one variable. Meaasure and integration covers the abstract theory of the integral, which includes measure theory (modern probability theory is one small part of measure theory) and the modern integral, which makes rigorous many of the operations of interchange of limits and integrals or limits and infinite sums that one sees in a “hand-waving” fashion in undergraduate engineering and physics classes. Functional analysis combines subjects previously learned in analysis with algebra and topology to develop such subjects as topological vector spaces (e.g. Banach and Hilbert spaces), the theory of the Fourier integral and Fourier series, the theory of Schwartz distributions (making rigorous not only the Dirac delta, but in in fact allowing one to take the derivative of the delta function (think about that one for a bit)) which is important in the theory of partial differential equations, Hilbert space operators and spectral decomposition, etc. Complex analysis of one variable expands greatly the subject matter that may have been learned in an undergraduate course, with much less emphasis on calculations and much more on the theoretical results that are unique to the subject.

Recommended texts:

Measure Theory– Halmos (the classic on the subject)

Measure and Integration, A Concise Introduction to Real Analysis – Richardson (a brief and excellent text based on the authors notes from a class taught by Shizuo Kakatani of the Riesz-Markov-Kakutani theorem which is a high point of the subject)

Real and Complex Analysis – Walter Rudin (a superb text that covers measure and integration, basic real analysis, basic complex analysis, and some topics in functional analysis particularly those related to Fourier analysis)

Real Analysis, Modern Techniques and Their Applications– Folland (a widely used text)

Functional Analysis– Walter Rudin (a classic and very widely used text. The best of the bunch.)

Functional Analysis– Yosida (good reference but perhaps too much for a first class)\

Complex Analysis– Ahlfors (the standard text on the subject)

Functions of One Complex Variable– Conway (good overall text on the subject)

Complex Function Theory– Heins (excellent text, but some find it too deep. I like it a lot.)

Analytic Function Theory, vols I, II– Hille (a classic text by a master of the subject)

Topology

Topology is broadly split into point set topology (aka general topology) and algebraic topology. Point set topology is foundational for all of modern mathematics, and is a mature subject with relatively little ongoing research. Algebraic topology uses topological methods to construct algebraic structures which are produce topological invariants which aid in the classification of topological spaces. Algebraic topology is an active and vibrant research area.

Topology is the study of continuous functions, often characterized as “rubber sheet geometry” in a setting sufficiently abstract to lay bare the fundamentals. A topologist is sometimes described as “someone who can’t tell the difference between a donut and a coffee cup” – a donut and a coffee cup have the same topology.

Recommended texts:

General Topology– Kelley

Topology– Dugundji

Topology, An Outline for a First Course – Ward (this is actually a text for a Moore method course. Theorems and examples are stated, but the student is expected to construct the proofs for himself and work out the examples. This is an excellent way to learn topology, but expect to work hard.)

Topology– Hocking and Young (covers both point set topology and elements of algebraic topology)

Algebraic Topology: An Introduction– Massey (a gentle introduction)

Algebraic Topology– Maunder (a common introductory text and overview)

Algebraic Topology– Spanier (a not-so –gentle treatment but an excellent book)

Homotopy Theory– Hu (homotopy is one important aspect of algebraic topology and this is a good text on that topic)

Homology Theory– Hilton and Willey (homology s a second important topic in algebraic topology)

Algebra

Algebraic structures permeate mathematics. Therefore, it is necessary to be familiar with abstract algebra in order to pursue studies in almost any area of modern mathematics. Galois theory is often used as an example of what a complete mathematical theory should be. Research in algebra has now large been absorbed into the discipline of algebraic geometry, which is an advanced topic of considerable difficulty (Wiles proof of the Fermat theorem lies within the domain of algebraic geometry for instance).

The basic algebraic structures are groups, rings, fields, modules, and vector spaces. One studies these structures and functions that preserve the operations attendant to them. All of this finds application in analysis, topology, geometry and more algebra.

Basic algebra should be mastered before going on to more advanced topics such as commutative algebra.

Recommended texts:

Algebra– Serge Lang ( an excellent graduate text. Be alert for mistakes and be ready to correct them.)

Algebra– van der Waerden (a classic based on notes from a class by Emmy Noether)

Lectures in Abstract Algebra, vols I, II, III – Jacobson (another classic)

Commutative Algebra with a View Toward Algebraic Geometry– Eisenbud

Commutative Algebra, vols I, II – Zariski and Samuel

Introduction to Commutative Algebra– Atiyah and McDonald

Differential Geometry

Differential geometry is the study of calculus on topological spaces that are locally Euclidean, but may be much more complex when viewed in the large. The surface of a sphere, for instance, is a 2-dimensional manifold. Differential geometry began with the study of surfaces by Gauss and in higher dimensions was formulated and studied by Riemann. It was the work of Riemann that provided the mathematical foundation on which Einstein built the general theory of relativity.

Differential geometry combines analysis, topology, and algebra into mathematical machine of great power. Modern differential geometry is very abstract, a great deal of the hard work being incorporated into the definitions. Theorems can flow fairly easily from the definitions and constructions, but it takes quite a bit of intellectual work to understand how anyone ever came up with the definitions.

The mathematics of differential geometry (specifically pseudo-Riemannian or semi-Riemannian geometry) is absolutely essential to any serious study of general relativity. Spacetime is an intrinsic manifold (not embedded in anything) and this concept requires the theory of differentiable manifolds in order to make any sense at all.

There are several ways to develop the basic theory of differentiable manifolds. They look rather different at the outset but all end up in the same place. One will find several of those approaches represented in the texts listed below.

Recommended texts:

Foundations of Differentiable Manifolds and Lie Groups– Warner (a thin but dense text that is an excellent introduction for those willing to work hard)

Differential Geometry(vols I.II, III, IV,V) – Michael Spivak (Spivak’s attempt to explain how people came up with the definitions and to give a comprehensive treatment of much of the subject. Highly recommended but heavy.)

Foundations of Differential Geometry– Kobayashi and Nomizu (the classic reference in the subject. Excellent but not for the faint of heart)

Differential Geometry and Symmetric Spaces– Helgason (an excellent treatment and a personal favorite)

Differential Geometry, Lie Groups and Symmetric Spaces– Helgason (more recent revision of Helgason’s earlier book)

Lectures on Differential Geometry– Sternberg

Differential Forms in Algebraic Topology– Bott and Tu (this is as much a text on algebraic topology as on differential geometry, is more abstract than the other books listed, but Bott is do damn good that I could not pass up listing this book. Bott actually started out in studying electrical networks and electrical engineers may know his name from Bott-Duffin synthesis in network theory. )

Riemannian Geometry– Petersen

Riemannian Geometry– do Carmo

Semi-Riemannian Geometry with Applications to Relativity– O’Neill

Manifolds, Tensor Analysis and Applications– Abraham, Marsden and Ratiu

Finale

Mathematics is a huge field. Even if you master all of the material above, you will have just scratched the surface. But if you have obtained a basic understanding of the above areas, then you are equipped to delve deeper into those specialized area that may interest you.