I want to study anabelian geometry, but unfortunately I'm having difficulties in finding some materials about it. If you could offer me some books/papers/articles I would be glad.
Anabelian Geometry – Study Materials and Resources
ag.algebraic-geometryanabelian-geometryct.category-theorygalois-theoryreference-request
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My suggestion, if you have really worked through most of Hartshorne, is to begin reading papers, referring to other books as you need them.
One place to start is Mazur's "Eisenstein Ideal" paper. The suggestion of Cornell--Silverman is also good. (This gives essentially the complete proof, due to Faltings, of the Tate conjecture for abelian varieties over number fields, and of the Mordell conjecture.) You might also want to look at Tate's original paper on the Tate conjecture for abelian varieties over finite fields, which is a masterpiece.
Another possibility is to learn etale cohomology (which you will have to learn in some form or other if you want to do research in arithemtic geometry). For this, my suggestion is to try to work through Deligne's first Weil conjectures paper (in which he proves the Riemann hypothesis), referring to textbooks on etale cohomology as you need them.
The trouble with the quasicrystals is that the literature in this area is dominated by non-mathematical or pseudo-mathematical papers and books. In particular, just extracting a mathematical definition of a quasicrystal from this literature is not so easy. This situation is well-illustrated by the wikipedia article on quasicrystals and the MO discussion of this topic at What is the relation between Quasicrystals, Riemann Hypothesis, and PV numbers?
The two papers that I found most enlightening and mathematical, address this problem head-on (more on this below):
[1]. A. Hof, "On diffraction by aperiodic structures", Commun. Math. Phys., 169 (1995), p. 25-43.
[2]. J-B. Gouere, "Quasicrystals and almost periodicity", Commun. Math. Phys., 255 (2005), p. 655-681.
Both papers prove some nontrivial mathematical theorems, on the basis of these theorems one can then form two, somewhat different, mathematical definitions of a (quasi)crystal. Senchal's 2-page long survey paper (see Mahmud's answer) or, better, her book (see Joseph's answer) is a good introduction, the trouble is that she does not prove anything in the book and that she could be sloppy with her definitions, for instance, she conflates functions, measures and distributions, which are needed for defining crystals.
If you look in Senchal's book, you first get the following physical definitions of crystals: "A crystal is any solid with essentially discrete diffraction diagram." (This includes both traditional crystals and quasicrystals.) The word "essentially" will be the difference between different mathematical definitions, which one derives from [1] and [2].
A (mathematical) quasicrystal is a tiling $T$ of ${\mathbb R}^n$ by convex polytopes satisfying certain properties:
Since general tilings are hard to work with, pretty much everybody assumes that $T$ is a Voronoi tiling of ${\mathbb R}^n$ based on a certain discrete subset $N\subset {\mathbb R}^n$, which a geometer would call a separated net.
Unfortunately, just having a separated net is not enough in order to deal with the "diffraction" issue. There is a disagreement which tilings are allowed as crystals and which are not. There is a general agreement that at least all periodic tilings (the ones where $N$ is a finite union of orbits of a discrete group of translations) and certain tilings constructed by Penrose and others via projection method, should be counted as crystals and quasicrystals respectively. I will call these two classes of tilings as "standard."
a. Here is Hof's definition of a crystal (Senchal's definition is taken from Hof's paper). Hof in [1] takes the auto-correlation function (actually, a distribution) $\gamma$ of $N$ and computes the (appropriately defined) Fourier transform $\hat\gamma$ of $\gamma$ (this is a mathematical interpretation of the "diffraction diagram"). Then $\hat\gamma$ is a measure $\mu$ which, in general, splits as a sum of two measures $\mu_d+\mu_c$: Discrete part $\mu_d$, which is supported on a certain countable subset of ${\mathbb R}^n$ and continuous part $\mu_c$. He proposes that "essentially discrete diffraction" means that $\mu_d$ is nonzero. Hof then proves that "standard" tilings indeed have nontrivial $\mu_d$ (According to [2], Hof even proves that $\mu_c=0$ in this case, but I did not check this). The trouble with this definition is that, as far as I can tell, there is no known purely geometric interpretation of the condition $\mu_d\ne 0$ in terms of the next $N$ itself (at least, none existed 6 years ago).
b. Gouere [2] (his work is an extension of Hof's approach and of a work by Lagarias) works with a slightly different definition, i.e., that $\mu_c=0$ (such sets $N$ are called Patterson sets). His main result is a purely geometric interpretation (actually, several interpretations) of this condition, see Theorem 1.1 in [1]: Patterson sets are the sets which are almost periodic with respect to Besikovitch's metric.
Remark 1. As far as I can tell from reading Freeman Dyson's paper here, definition of a quasicrystal that Dyson proposes is the one with $\mu_c=0$.
Remark 2. Gouere does not propose that a Patterson set is the right definition of a crystal, this is just my take on his paper. Condition $\mu_c=0$ is more limited, but, in view of [2], is geometric and also covers "standard" examples, while the condition $\mu_d\ne 0$ is more general, but is nongeometric.
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Best Answer
There is this very beautiful survey
Nakamura, Hiroaki; Tamagawa, Akio; Mochizuki, Shinichi
The Grothendieck conjecture on the fundamental groups of algebraic curves
http://www.math.sci.osaka-u.ac.jp/~nakamura/zoo/rhino/NTM300.pdf
You could also have a look at
Szamuely, Tamás
Heidelberg Lectures on Fundamental Groups
http://www.renyi.hu/~szamuely/heid.pdf