## Interesting books and survey articles

It may be nice to collect in a post introductory books and survey articles in various areas in mathematics, so that all of us may take a look. This post is supposed to be edited by anyone.

Analysis
1. Complex Analysis (Ahlfors)
I am not an analyst myself so I don’t know if I am in the right position to recommend a book in analysis. Nevertheless this book is one of my favorite math books. It is a bit harder than Brown and Churchill. This book emphasizes much on the geometric side of complex analysis and at the same time doesn’t lose its rigor. However, my mathematical friends either love it or hate it very much, part of the reasons is that those who don’t usually think in pictures may feel a bit hard to follow its argument (the book uses phrases like “lies on the same side of …”, “outside a large circle” etc, instead of using the more usual “analytic” language of $z\in H$, $z\notin B_R(0)$ etc). Also the book keeps a rather informal style of introducing tools and proving results at the same time, instead of the usual Theorem-Proof style. I actually find it more natural and advantageous, but you may find it difficult to find the “proof” of a theorem, as it may be hidden somewhere in the texts before the theorem. -KKK

Geometry/Topology
1. Differential geometry of curves and surfaces (do Carmo)
Classic. I think is the best among all the elementary DG books for curves and surfaces in Euclidean space. Assume only year 1 calculus and linear algebra. Lots of pictures. It emphasizes on concepts and geometric ideas more than plain calculations (unlike Oprea). (However you should be warned that there are two sides of Differential Geometry, the “differential” side stresses on differential analysis and calculations and the “geometric” side stresses on the geometric intuition and imagination. The are equally important. Nevertheless, I think for those who are new to this subject, pictures should be the first intuitive ideas one should get, before starting any calculations. Just like calculus: before introducing the definitions of derivative and integral, one should at least have a faint idea that they correspond to the slope and area under a function. ) The exercises are also useful, some are quite challenging. -KKK

2. Riemannian Manifolds: An Introduction to Curvature (J. Lee)
A very good (and very reasonably sized) introduction to RG. It only talks about things which are absolutely necessary and doesn’t spend time on other things which should be left to a second course (e.g. Hodge theory or deRham that kinds of stuff). Quite some amount of pictures which, in my opinion, is at least as useful as the calculations themselves. Basically, it proves every theorem stated, which you may skip if you are confident that you can do it. The exercise are not extremely motivating, they are useful for practicing your calculations. -KKK

3. Comparison Theorems in Riemannian Geometry (Cheeger, Ebin)
This is certainly not an introductory book for RG, even if you have studied DG before. But it is a very classic book which talks about comparison (more global in nature) theorem in RG. It is very concise, sometimes too short. Be warned that it contains some mistakes in its calculations (or even some statements) which you should be able to spot. Still useful today even it was written about 40 years ago. Should be used for a second course in RG. -KKK

4. Morse Theory (J. Milnor)
I can say that all books written by Milnor are good. The presentation is elegant, and using just elementary calculus and analysis, it can discuss results as deep as Bott periodicity theorem (unfortunately, I haven’t read it). It even contains a chapter (though not the focus of the book) on the essence of RG in only 13 pages(!), which contains the contents of perhaps a one-semester RG course (up to geodesic and completeness)!! I think Chapter 3 is one of the most exciting parts. It applies Morse theory to study the variations of geodesics on Riemannian manifolds and draw relations betweens topology and curvatures. Using analogous methods, Morse theory can also be applied to various “infinite dimensional” spaces which gives rise to Floer theory, relating to fields as far as mirror symmetry (Louis Lau is an expert on this), etc. [The two other books of Milnor, Characteristic Classes and Topology from the Differentiable Viewpoint (haven’t read too much since I have learned that stuff from another book: from C to C, which 馬後砲ly speaking is not very well-written) are also good. But I don’t want to “promote” a single author too much, so perhaps others can write something about these too books. ]

If you want a non-typewritten copy, you can ask Yat-Ming for a version which even John Milnor haven’t seen before! -KKK

Algebra

Probability

Some comments: It is about simulation and is very important nowadays. Here is a typical problem: construct a random permutation of a standard 52-deck. Note that there are 52! configurations, and it is not practical to list them all (and then randomly choose an integer between 1 and 52!).

Now consider this method: Start with the usual ordering {1, 2, 3, …, 52}. Randomly pick $i$ and $j$ and interchange card $i$ and card $j$. Stay the same if $i = j$. Repeat this many times. “After a while” the deck should be well shuffled.

Mathematically, this corresponds to a random walk on the symmetric group $S_{52}$. Start with the identity $X_0 = e$. Given $X_n$, we pick a transposition $T_{n+1}$ at random and set $X_{n+1} = T_{n+1} X_n$. The stochastic process $\{X_n\}$ is then a Markov chain on $S_{52}$. It can be shown that the distribution of $X_n$ tends to the (equilibrium) uniform distribution on $S_{52}$ as $n \rightarrow \infty$. This justifies the above method. However, we should also know the rate of convergence. In general, this requires sophisticated mathematics, ranging from group representation to Nash-Sobolev inequalities.

Others