Lev Landau, a great theoretical physicist, wrote a series of books which he called the 'Theoretical Minimum' which contained everything he considered elementary for an aspiring theoretical physics researcher. Leonard Susskind, is apparently interested in continuing this tradition. In fact, there's an interesting discussion of Lev Landau's text on the physics stack-exchange.

My question is whether there have ever been significant attempts to do a similar thing in mathematics. If so, why don't these texts have the same importance in the mathematical community?

Here's the list of the texts that had to be mastered:

  1. Mathematics I. Integration, ordinary differential equations, vector algebra and tensor analysis.
  2. Mechanics. Mechanics, Vol. 1, except §§ 27, 29, 30, 37, 51 (1988 russian edition)
  3. Field theory The Classical Theory of Fields, Vol. 2, except §§ 50, 54-57, 59-61, 68, 70, 74, 77, 97, 98, 102, 106, 108, 109, 115-119 (1973 russian edition)
  4. Mathematics II. The theory of functions of a complex variable, residues, solving equations by means of contour integrals (Laplace's method), the computation of the asymptotics of integrals, special functions (Legendre, Bessel, elliptic, hypergeometric, gamma function)
  5. Quantum Mechanics. Quantum Mechanics: Non-Relativistic Theory, Vol. 3, except §§ 29, 49, 51, 57, 77, 80, 84, 85, 87, 88, 90, 101, 104, 105, 106-110, 114, 138, 152 (1989 russian edition)
  6. Quantum electrodynamics. Relativistic Quantum Theory, Vol. 4, except §§ 9, 14-16, 31, 35, 38-41, 46-48, 51, 52, 55, 57, 66-70, 82, 84, 85, 87, 89 - 91, 95-97, 100, 101, 106-109, 112, 115-144 (1980 russian edition)
  7. Statistical Physics I. Statistical Physics, Vol. 5, except §§ 22, 30, 50, 60, 68, 70, 72, 79, 80, 84, 95, 99, 100, 125-127, 134-141, 150-153 , 155-160 (1976 russian edition)
  8. Mechanics of continua. Fluid Mechanics, Vol. 6, except §§ 11, 13, 14, 21, 23, 25-28, 30-32, 34-48, 53-59, 63, 67-78, 80, 83, 86-88, 90 , 91, 94-141 (1986 russian edition); Theory of Elasticity, Vol. 7, except §§ 8, 9, 11-21, 25, 27-30, 32-47 (1987 russian edition)
  9. Electrodynamics of Continuous Media. Electrodynamics of Continuous Media, Vol. 8, except §§ 1-5, 9, 15, 16, 18, ​​25, 28, 34, 35, 42-44, 56, 57, 61-64, 69, 74, 79-81 , 84, 91-112, 123, 126 (1982 russian edition)
  10. Statistical Physics II. Statistical Physics, Part 2. Vol. 9, only §§ 1-5, 7-18, 22-27, 29, 36-40, 43-48, 50, 55-61, 63-65, 69 (1978 russian edition)
  11. Physical Kinetics. Physical Kinetics. Vol. 10, only §§ 1-8, 11, 12, 14, 21, 22, 24, 27-30, 32-34, 41-44, 66-69, 75, 78-82, 86, 101.

Note: Landau's texts were necessary for an exam which aspiring researchers had to pass in order to do research in theoretical physics. However, only 43 people passed the exam between 1933 (when Landau first administered it) and 1961(Landau suffered a car accident).

  • $\begingroup$ Feel free to roll back my edit to the title if it isn't what you meant. I also wondered about "specified" or even "come up with" instead of "developed". $\endgroup$ – J W Jan 2 '17 at 12:02
  • $\begingroup$ @JW I think 'developed' is fine because I don't think Theoretical Physics has less breadth/depth than mathematics. It's more a matter of personality/boldness. I think Landau was unique in that way. $\endgroup$ – user93511 Jan 2 '17 at 13:08
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    $\begingroup$ @JW Maybe the Bourbaki group was similar to Landau in terms of ambition but I think their texts are for people already doing research, not people who are about to get into mathematical research. $\endgroup$ – user93511 Jan 2 '17 at 13:09
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    $\begingroup$ Could you explain what you mean by "a theoretical minimum"? I really don't know what you're talking about and I shouldn't have to go read a bunch of other pages to find out even that. $\endgroup$ – David Richerby Jan 2 '17 at 18:07
  • $\begingroup$ @DavidRicherby That's a very good question. I just added a short description. $\endgroup$ – user93511 Jan 2 '17 at 23:27

At the same time that Lev Landau (and Yevgeny Lifshitz) were publishing the volumes of the course of theoretical physics, the Russian mathematician Vladimir Smirnov published the six volumes of his Course in Higher Mathematics. I think that, for those years, these volumes can be considered a good starting knowledge for a mathematician, and that even today are readable and interesting (if you can find the volumes, here an old French edition).

This edition is in four parts and six volumes. The first part is essentially about differential and integral calculus in one and two variables. The second part is more varied; it contains chapters on ordinary differential equations, multiple integrals, vector analysis and differential geometry, Fourier series and an introduction to partial differential equations. The third part (two volumes) is divided into two volumes: the first about linear algebra and group representations, the second about complex analysis and some special functions. The fourth part has a first volume about integral equations and the calculus of variations, and a second volume on partial differential equations.

Clearly it is an old setting of the matter, with an exposition strongly oriented to applications, expecially to physics. And certainly it is the result of a complex cultural phenomenon, related to the Cold War and the technological and scientific confrontation with the Capitalist World.

You can consider that, in the same years, in Europe, there was also the Bourbaki's ''school'' that published a series of volumes on the fundamentals of mathematics, with a completely different and more abstract approach. The ten volumes of the ''Éléments de mathématique'' published by this group of mathematicians was certainly very important and influenced many modern branches of mathematics, but the volumes are not didactical and really difficult to read.

  • $\begingroup$ This appears to be the closest thing to Landau's work so far. Is it a coincidence that he's also Russian? It might have been a cultural phenomenon. $\endgroup$ – user93511 Jan 5 '17 at 17:14
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    $\begingroup$ Would you mind listing the courses he covered? $\endgroup$ – user93511 Jan 5 '17 at 17:18
  • $\begingroup$ The first part is essentially about differential and integral calculus in one and two variables. the second part is more varied; it contains chapters on the ordinary differential equations, multiples integrals, vectorial analysis and differential geometry, Fourier series and an introduction to partial derivative equations. $\endgroup$ – Emilio Novati Jan 5 '17 at 20:35
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    $\begingroup$ Clearly it is an old setting of the matter. And certainly it is the result a complex cultural phenomenon. You can consider that, in the same years, in Europe, there was also the Bourbaki's ''school'' that pubblished a series of volumes on the foundamentals of mathematics, with a completly different (and more abstract) approach. $\endgroup$ – Emilio Novati Jan 5 '17 at 20:41
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    $\begingroup$ @JW: I've added to my answer, but I apologize for my bad english :) $\endgroup$ – Emilio Novati Jan 6 '17 at 9:19

In 1991, Vladimir I. Arnold compiled a list of 100 mathematical problems in a paper titled "A mathematical trivium". (You can view all 100 problems in that pdf.)

In that paper he says:

The compilation of model problems is a laborious job, but I think it must be done. As an attempt I give below a list of one hundred problems forming a mathematical minimum for a physics student. Model problems (unlike syllabuses) are not uniquely defined, and many will probably not agree with me. Nonetheless I assume that it is necessary to begin to determine mathematical standards by means of written examinations and model problems. It is to be hoped that in the future students will receive model problems for each course at the beginning of each semester, and oral examinations for which the students cram by heart will become a thing of the past.

Although the problems are aimed at physics students, most of the problems are purely mathematical.

I don't know how famous these problems are, but there is an entire subforum on the Art of Problem Solving (AoPS) Fourms dedicated to these problems.

  • $\begingroup$ I just looked through some of the problems. They're pretty cool. I'll try to work through them during this Christmas break. :) $\endgroup$ – user93511 Jan 2 '17 at 7:58
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    $\begingroup$ theoretical minimum in math :D is what you learn in your phd degree of math or physics $\endgroup$ – Jose Garcia Jan 2 '17 at 11:36

There is Garrity's All the Mathematics You Missed: But Need to Know for Graduate School, published in 2001. The author states in the preface:

The goal of this book is to give people at least a rough idea of the many topics that beginning graduate students at the best graduate schools are assumed to know.

Topics covered (rapidly) in the book include linear algebra, real analysis basics, vector calculus, point-set topology, Stokes' Theorem (classical and differential forms), curvature, geometry, complex analysis, countability, the Axiom of Choice, abstract algebra, Lebesgue integration, Fourier analysis, differential equations (ODEs and PDEs), combinatorics and probability, and algorithms.


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