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Recently I got an offer letter for a master program (with courses like algebraic topology, algebraic geometry (first chapter of Hartshorne), differential geometry, representation theory, etc... ).

Now I have three months before I go to there and start my program. My question is, should I use these three months to review the fundamental subjects in mathematics, i.e Abstract Algebra, Topology, Analysis, and Linear Algebra or should I pre-study those master programs before I go?

To be honest, I am not firm enough in those fundamental subjects as I didn't do exercises at all when I took those courses (but I managed to score well, mainly because my school only gives questions in the lecture notes). Also, I have almost forgotten some results in linear algebra (like the change of basis etc.)

At first, I would prefer to revise (by doing exercise) first before going to some advanced courses. But then a friend of mine suggested me to study further courses, where he says if I am stuck somewhere, then I go back for related topics. (For example, if I need a change of basis in the study of manifolds, then I go back to revise it.)

Which one would be better for my case, given the time constraint and the mathematical background?

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    $\begingroup$ I think having strong basics is more important. $\endgroup$ – Sahiba Arora Jun 29 '20 at 9:38
  • $\begingroup$ You can't study all of undergraduate abstract algebra, topology, analysis and linear algebra well in three months, so you'll have to make choices even if all you do is review that material. I think your goals should depend on what courses you anticipate taking in your master's program. (Are you taking all the ones you listed in the fall?) Also, if you do choose to review earlier material rather than study ahead, you should do it in books written at a graduate level of sophistication. $\endgroup$ – Anonymous Jul 1 '20 at 7:27
  • $\begingroup$ @Anonymous For your suggestion, I think I could temporarily put aside linear algebra first as my master's program's first term is not heavily dependent on linear algebra. In fact, the master program I got is ICTP mathematics diploma (ictp.it/research/math/math-diploma.aspx). I plan to revise the materials by using the study notes by Dexter Chua (dec41.user.srcf.net/notes) . I am not really sure what is the graduate level, but you may help me to clarify it by looking at the notes by him, for example, metric spaces and topological spaces for point set topology $\endgroup$ – Nothing Jul 2 '20 at 13:42
  • $\begingroup$ @LingMinHao All the things in the first term of the sample plan are basic material without too many prerequisites. You have some time before the more advanced courses you mentioned, which are in the second term. I'm not sure it's a good idea to use lecture notes from undergraduate courses at Cambridge as the basis for your revision. Students are still building mathematical maturity at that stage, especially in the first two years. A set of books that are supposed to take things from the beginning, but starting from an appropriate level of general maturity, are those... $\endgroup$ – Anonymous Jul 2 '20 at 19:51
  • $\begingroup$ by Anthony Knapp: Basic Algebra and Basic Real Analysis: math.stonybrook.edu/~aknapp/download.html I think they're intended exactly for situations like yours. An alternative for the topology/real analysis part would be Real and Functional Analysis by Serge Lang, and you can refer to his Algebra for reference (for the main text, not really the exercises). If you really intend to study algebraic geometry in the winter, I highly recommend you read through Introduction to Commutative Algebra by Atiyah and Macdonald before you start. If your time horizon were different... $\endgroup$ – Anonymous Jul 2 '20 at 19:57
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As I said in the comments, I think the important thing is that if you feel you need to go over basic material, you should do it at a higher level of sophistication than an undergraduate would.

If you had more time, you could do this in books that go into greater depth in algebra and real and complex analysis, but my recommendations here will focus on books that will allow you to move faster while also reviewing the undergraduate material you'd like to know better.

I would recommend you see if these books work for you:

  • Basic Algebra and Basic Analysis by Anthony Knapp.

They are meant to introduce you to basic graduate-level material in algebra and analysis, but start by reviewing more elementary subjects, such as linear algebra and metric spaces. (There are also sequels, Advanced Algebra and Advanced Analysis.)

I'd like to mention that because linear algebra in vector spaces is treated before modules in Basic Algebra, important facts about linear algebra (for example giving a clearer account of Jordan canonical form) are delayed until Chapter 8, and some of it is done in the exercises there. This material could have been expanded.

As supplements to those books by Knapp, I would suggest the following:

  • Real and Functional Analysis by Serge Lang. This covers topology, integration theory, functional analysis, differential calculus in Banach spaces (with the basic theorems on differential equations) and, quite briefly, manifolds. This could actually be an alternative to Basic Analysis.

  • Algebra, also by Serge Lang. I am suggesting this as a reference, not as a textbook to read. It is very well-organized for reading chapters independently of each other. Here you will find fuller explanations, for example, of the linear algebra material I mentioned above.

  • Real and Complex Analysis, by Walter Rudin. I would recommend Chapter 10 as a supplement to (or substitute for) Knapp's appendix on complex analysis. It can be read independently of the other chapters and covers very succinctly the most important material that would appear in a basic undergraduate course. (The rest of the book is also excellent, but I'm not actually recommending you read it now due to your time constraints.)

Added. Since you mention change of basis, let me explain how I remember this. If $e = (e_i)$ and $e' = (e'_i)$ are bases for the same space $V$, then the transition matrix from $e$ to $e'$ is the matrix $P$ whose columns are the vectors of $e'$ written in the basis $e$.

If $x$ is a vector, write $X$ for its coordinates in the basis $e$ and $X'$ for its coordinates in the basis $e'$. Now there is only one formula to remember: $X = PX'$. (Proof: $P$ is the matrix of $\operatorname{Id}_V \colon (V, e') \to (V, e)$.) If you remember this, you will find all the other formulas.

For example, say $T \colon V \to W$ has matrix $M$ in bases $e$ and $f$. What is the matrix $M'$ of $T$ in bases $e'$ and $f'$? Answer: We have $X = PX'$, $Y = QY'$ and $Y = MX$. What we want is $Y' = M'X'$. A simple calculation shows that we must take $M' = Q^{-1}MP$.

Likewise, say a bilinear form has matrix $M$ in basis $e$. What is its matrix $M'$ in basis $e'$? Answer: We have $X = PX'$, $Y = PY'$ and the bilinear form is given by $X^t MY$. What we want is $(X')^t M' Y' = X^t MY$. It follows that we must take $M' = P^t M P$.

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Pre-study and then that will give you the idea of what to look out for in review.

Mostly, you never really get the basics from studying the basics over and over. I recall reading that you never really understand topic X until you get to topic X+1. I doubt a calculus student will really understand limits from studying the calculus book again and again. Said student will actually understand limits better from starting to study elementary real analysis, where the student learns the concepts of supremum and infimum.

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