[Math] What are good articles/books on the psychology of mathematical research

booksreference-requestsoft-question

I am thinking about advanced texts similar to Polya's 'How to solve it?'. Quite a few good articles of such a kind are published under Philosophy of Mathematics, but that dwells on a very different domain generally.

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Jacques Hadamard, The Psychology of Invention in the Mathematical Field.

Description (from the Library Journal):

Thoughtful and articulate study of the origin of ideas. Role of the unconscious in invention; the medium of ideas — do they come to mind in words? in pictures? in mathematical terms? Much more. "It is essential for the mathematician, and the layman will find it good reading."

There is something interesting in page 118 on the Riemann Hypothesis:

At the death of Riemann, a note was found among his papers, saying 'These properties of $\zeta(s)$ (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be.

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[Math] japanese/chinese for mathematicians

Here are a few for chinese:

Commercial Press Staff. English-Chinese Dictionary of Mathematical Terms. New York: French & European Publications, Incorporated, 1980.

De Francis, John F. Chinese-English Glossary of the Mathematical Sciences. Reprint. Ann Arbor, MI: Books on Demand.

Dictionary of Mathematics. New York: French & European Publications, Incorporated, 1974.

He Xiuhuang. A Glossary of Logical Terms. Hong Kong: Chinese University Press, 1982.

Science Press Staff. English - Chinese Mathematical Dictionary. Second Edition. New York: French & European Publications, Incorporated, 1989.

Science Press Staff. Chinese-English Mathematical Dictionary. New York: French & European Publications, Incorporated, 1990.

Science Press Staff. New Russian - Chinese Dictionary of Mathematical Terms. New York: French & European Publications, Incorporated, 1988.

Silverman, Alan S. Handbook of Chinese for Mathematicians. Berkeley, CA: University of California, Institute of East Asian Studies, 1970.

Source: here

I have never read any of these books, and I honestly doubt it that they have all the mathematical terms (especially in higher more sophisticated fields). Don't expect to be able to write "diffeomorphism between manifolds" in chinese or japanese immediately. I suggest you take a look at these references in your public library and get one that helps you the most. To be honest, I am also interested I have several chinese papers I really want to read. I would first try anything with the latest jedict/edict/cedict, and then try something else like the above references.

Learning Shimura Curves – A Good Roadmap

First of all, Kevin is being quite modest in his comment above: his paper

Buzzard, Kevin. Integral models of certain Shimura curves. Duke Math. J. 87 (1997), no. 3, 591--612.

contains many basic results on integral models of Shimura curves over totally real fields, and is widely cited by workers in the field: 22 citations on MathSciNet. The most recent is a paper of mine:

Clark, Pete L. On the Hasse principle for Shimura curves. Israel J. Math. 171 (2009), 349--365.

http://alpha.math.uga.edu/~pete/plclarkarxiv7.pdf

Section 3 of this paper spends 2-3 pages summarizing results on the structure of the canonical integral model of a Shimura curve over $\mathbb{Q}$ (with applications to the existence of local points). From the introduction to this paper:

"This result [something about local points] follows readily enough from a description of their [certain Shimura curves over Q] integral canonical models. Unfortunately I know of no unique, complete reference for this material. I have myself written first (my 2003 Harvard thesis) and second (notes from a 2005 ISM course in Montreal) approximations of such a work, and in so doing I have come to respect the difficulty of this expository problem."

I wrote that about three years ago, and I still feel that way today. Here are the documents:

http://alpha.math.uga.edu/~pete/thesis.pdf

is my thesis. "Chapter 0" is an exposition on Shimura curves: it is about 50 pages long.

For my (incomplete) lecture notes from 2005, go to

http://alpha.math.uga.edu/~pete/expositions2012.html

and scroll down to "Shimura Curves". There are 12 files there, totalling 106 pages [perhaps I should also compile them into a single file]. On the other hand, the title of the course was Shimura Varieties, and although I don't so much as attempt to give the definition of a general Shimura variety, some of the discussion includes other PEL-type Shimura varieties like Hilbert and Siegel moduli space. These notes do not entirely supercede my thesis: each contains some material that the other omits.

When I applied for an NSF grant 3 years ago, I mentioned that if I got the grant, as part of my larger impact I would write a book on Shimura curves. Three years later I have written up some new material (as yet unreleased) but am wishing that I had not said that so directly: I would need at least a full semester off to make real progress (partly, of course, to better understand much of the material).

Let me explain the scope of the problem as follows: there does not even exist a single, reasonably comprehensive reference on the arithmetic geometry of the classical modular curves (i.e., $X_0(N)$ and such). This would-be bible of modular curves ought to contain most of the material from Shimura's book (260 pages) and the book of Katz and Mazur Arithmetic Moduli of Elliptic Curves (514 pages). These two books don't mess around and have little overlap, so you get a lower bound of, say, 700 pages that way.

Conversely, I claim that there is some reasonable topology on the arithmetic geometry of modular curves whose compactification is the theory of Shimura curves. The reason is that in many cases there are several ways to establish a result about modular curves, and "the right one" generalizes to Shimura curves with little trouble. (For example, to define the rational canonical model for classical modular curves, one could use the theory of Fourier expansions at the cusps -- which won't generalize -- or the theory of moduli spaces -- which generalizes immediately. Better yet is to use Shimura's theory of special points, which nowadays you need to know anyway to study Heegner point constructions.) Most of the remainder concerns quaternion arithmetic, which, while technical, is nowadays well understood and worked out.

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[Math] japanese/chinese for mathematicians

Learning Shimura Curves – A Good Roadmap

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