Early specialization and career development [closed]

Solution 1:

Please Note: In my answer, I have used such terms as "very good", "solid", "strong" etc. in relation to "knowledge" which are subjective rather than objective. However, to be definite, I have also listed some textbooks of mathematics which, in my opinion, constitute such a knowledge; I do not claim that these textbooks are standard. Also, I have stated some of my opinions in my answer below but I do not claim that any of these opinions constitute fact.

The required breath and depth of your mathematics knowledge will largely depend on your field of specialization. In particular, I would recommend thinking roughly about which fields of mathematics you are likely to choose as a speciality. For example, the required mathematics knowledge if you choose to specialize in probability theory, for example, is different to that if you choose to specialize in algebraic geometry. You need not have any specific branches of mathematics in mind; but it is very helpful to at least be able to decide whether you would prefer to specialize in abstract algebra, analysis, topology and geometry, or in a branch which could fairly be construed as being of a different flavor to these fields.

In any case, I believe that it is at least healthy and exciting to study in some depth abstract algebra, analysis, and topology and geometry. For example, abstract algebra furnishes the basic language of much of modern mathematics (in addition to being an interesting and exciting branch of mathematics in its own right). In particular, it is very important to acquire some depth of knowledge in abstract algebra at an early stage in your mathematical career. I would say that a reasonable amount of knowledge in linear algebra is absolutely essential to modern mathematics. I also think that a good knowledge of groups, rings and fields is very important; more specifically, I think that, at a minimum, you should be familiar with the contents of Topics in Algebra by I.N. Herstein. However, if you wish to specialize in abstract algebra or in topology and geometry, then you will need to know more abstract algebra; more specifically, it is important to have a very good knowledge of homological algebra if you wish to study either algebraic topology or algebraic geometry and a very good knowledge of commutative algebra if you wish to study algebraic geometry. Also, a solid knowledge of representation theory is important for some aspects of topology and geometry.

I also believe that it is important to have a good knowledge of real and complex analysis. There are many reasons for this but the most obvious is that the real line and the complex plane are central objects of study in much of modern mathematics and will probably occur frequently in your mathematical studies. In the case of complex analysis, I feel that it is very important to be acquianted with a field of mathematics in which a range of powerful results can be derived from studying a simple definition; complex analysis is an excellent example of this. Also, complex analysis is fun because you can use complex analysis to solve problems whose statements appear to have no relation to complex analysis whatsoever; the calculus of residues is an important example of this. In summary, I think that you should be familiar with the contents of Principals of Mathematical Analysis by Walter Rudin and the first fifteen chapters of Real and Complex Analysis by Walter Rudin. I believe that this will furnish you with a reasonably strong background in analysis.

I believe that it is important to have a good knowledge of algebraic topology and differential geometry. For example, manifolds constitute some of the most familiar and important geometric objects and having a solid understanding of manifolds will certainly enrich your mathematical life. Similarly, I personally found algebraic topology to be one of the most interesting branches of mathematics; at the elementary level, one computes algebraic invariants of topological spaces and doing this for familiar spaces, such as compact two-manifolds and CW-complexes, is very exciting. Also, algebraic topology provides much of the motivation for branches of mathematics such as homological algebra and category theory. In summary, I think that it is advisable to be familiar with the first six chapters of Elements of Algebraic Topology by James Munkres and the first six chapters of An Introduction to Differentiable Manifolds and Riemannian Geometry by William Boothby.

In short, I believe that an elementary knowledge of abstract algebra, real analysis, complex analysis, algebraic topology, and differential geometry is very important in modern mathematics. Of course, in addition to this, you will need to learn more about your field of speciality (over the course of your mathematical career).

I hope this helps and best wishes in your mathematical studies!

Additional Details (January 1st, 2012)

I think that it is important to acquire a balance between breadth and depth in your mathematical education. I think, on the one hand, specializing only in algebraic geometry is not very wise and, on the other hand, learning a little amount of everything without have a specialist knowledge in any one area is also unwise. For example, algebraic geometry has strong connections to many different areas in mathematics; e.g., number theory, several complex variables, commutative algebra, and differential geometry, and becoming a research algebraic geometer usually requires having familiarity with some or all of these areas. Also, there are many correct answers to the question of which areas outside of algebraic geometry with which you should be familiar; the answer depends on which aspect of algebraic geometry you are interested. A word of warning: my answer below does not take this [the aspects of algebraic geometry with which you are interested] into account (because I do not know); I have simply given my best attempt at a general answer but algebraic geometry is too broad a subject for any general answer to be a good one.

However, time is limited and it is therefore necessary to focus on at most a few different areas. I would advise you to spend most of your time developing a solid knowledge of algebraic geometry; usually, "solid knowledge of algebraic geometry" means that you have read Hartshorne's textbook (at least chapters 1-4 of it) and can solve most or all of the exercises. I think it is important to supplement a reading of Hartshorne's textbook with commutative algebra and homological algebra; I would recommend reading Matsumura's Commutative Algebra because it (seems to) contain all the results in commutative algebra assumed or stated with proof in Hartshorne's textbook (Eisenbud's Commmutative Algebra: With a View Toward Algebraic Geometry is another great alternative). I believe to read Matsumura's textbook you will need to be familiar with basic homological algebra (the contents of appendix B of Matsumura's Commutative Ring Theory (a different textbook!) suffice).

My advice is to read Hartshorne's textbook and Matsumura's textbook concurrently. However, algebraic geometry has rich connections to areas of mathematics other than commutative algebra. Your primary aim should be to acquire a solid knowledge of these areas as well. For example, I think understanding some of the analytic theory (e.g., several complex variables) is advisable. My advice would be to read Gunning and Rossi's Analytic Functions of Several Complex Variables and also to read (at least a portion of) Principles of Algebraic Geometry by Griffiths and Harris.

I think that, if you have to choose one area of analysis to learn, then Fourier/harmonic analysis is a good choice. The techniques of Fourier analysis are very important throughout analysis and beyond and it would make sense to learn some of this theory. Since you are already familiar with the contents of Real and Complex Analysis by Walter Rudin, a good book to read on Fourier analysis would be Loukas Grafakos' Classical Fourier Analysis. The sequel to this book, Modern Fourier Analysis (by the same author), covers more advanced material (of a specialist nature) and could be read if you are interested. However, I believe that it would be a sensible investment to at least acquire a solid knowledge of the contents of Classical Fourier Analysis which include $L^p$-spaces and interpolation, maximal functions, the Fourier transform, distributions, Fourier analysis on the torus, singular integrals of convolution type, and Littlewood-Paley theory and multipliers. In addition to a background in several complex variables, this should furnish you with a good knowledge of some of the basic techniques of analysis.

Finally, I think it is good to have a knowledge of basic algebraic number theory; there are plenty of excellent textbooks on this subject, my favorite being Janusz's Algebraic Number Fields. If you have time, then it could also be a good idea to learn about the arithmetic of elliptic curves which has connections to algebraic geometry. Silverman's textbook (The Arithmetic of Elliptic Curves) is often considered standard in this area.

If you have time, then I think that reading some of the textbooks recommended above would be a very good idea. However, as I said, algebraic geometry is too broad a field for your question to have one correct answer. In practice, one would have to accumulate a knowledge of mathematics over a lifetime; the areas I mentioned above constitute important material that I think one should know most urgently. However, there are other areas which I have not even mentioned which could also be extremely important (if not essential) in some aspects of algebraic geometry; e.g., representation theory and even theoretical physics! In particular, I think it would be a good idea to ask your advisor or an algebraic geometer you know very well on guidance as to which areas of algebraic geometry you should focus on and which areas of mathematics outside of algebraic geometry are most relevant to this specialization.

PDE's is an extremely rich area of mathematics and I think it is certainly not a bad idea to learn more about them. The same is true for combinatorics. However, I think it is most wise to read small books (~ 100 pages) on these subjects just to get a flavor of the sorts of techniques in these areas. In fact, this illustrates a general strategy: if you wish to learn more about a new area of mathematics under time constraints, then it is a good idea to read a small book on the subject.

More generally, I would advise you to read as much mathematics as possible, review the mathematics you already know, think about it in new ways, and connect it to the mathematics you are currently learning. For example, if you have heard that there is a mathematical concept "X" but do not know the definition (a hypothetical situation), then you can read the Wikipedia article on it, read blogs about it, or even pick up a small textbook on it. I think, inevitably, mathematics is far too enormous to read linearly and carefully in its entirety. Therefore, you might sometimes need to make choices depending on time constraints. For example, can you learn (roughly speaking) the most basic techniques and results in the theory of "X" in a short time (a week, for example)? You might not be able to learn the theory of "X" as thoroughly as you would if you had a month of time but it is far better than knowing nothing about "X". Similarly, I think it is a good idea to pick up other branches of mathematics by reading about them as an "side activity" while learning algebraic geometry. If you get time later, then you will be able to read these branches more thoroughly and your prior experience with the subject will allow this (second) reading to be accelerated.

I hope this helps!