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I'm interested to know what mathematical and physics topics Einstein learnt during his graduation.Did he know about advanced mathematical topics such as Hamiltonian mechanics ,Lagrangian mechanics, Manifold and Differential topology .

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  • $\begingroup$ "During his graduation" doesn't make much sense. Do you mean "During his schooling?" Also, which institution are you referring to? $\endgroup$
    – HDE 226868
    Jan 17, 2015 at 14:32
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    $\begingroup$ Quick comment: Einstein certainly did not know about differential topology or other advanced topics in mathematics. In fact, the development of general relativity was delayed by several years because Einstein was not aware of differential geometry (Riemannian geometry, in particular). $\endgroup$
    – Danu
    Jan 17, 2015 at 20:12

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It is very difficult to access what he really knew from reading or from doing the things himself, but this paper address the simpler questions, what classes he took and what were his scientific contacts in the student years:

http://arxiv.org/ftp/arxiv/papers/1205/1205.4335.pdf

Let me also add that "manifolds" and "differential topology" were not formalized at that time, and these words were not used. On the other hand, some physicists and mathematicians has quite good understanding of these things. For example, the preliminary chapter of Maxwell's Treatease on electricity and magnetism shows that he had quite a good understanding of differential forms, homology and cohomology, Morse theory and Hodge theory (all in dimension 3, of course), the things which were formally invented or discovered at a much later time. There is little doubt that Einstein knew about this book, and it was easily available to him, but how much did he actually read or understand of it, probably cannot be determined.

EDIT. I actually did not read the paper referred above when I wrote this answer. Now I read it carefully, and it is indeed amazing:-) Einstein had Adolf Hurwiz as his math professor, one of the greatest mathematicians of that time. Another professor was Hermann Minkowski. But Einstein's classmates say he skipped their seminars. Instead, as a friend recalls:

Einstein felt that "the most fascinating subject at the time that I was a student was Maxwell's theory"

Somehow it happened that mathematicians of 19 and early 20 century did not read Maxwell's book (and of course they do not read it nowadays. Who cares about 19-th century books!) There is a very interesting paper:

Dyson, Freeman J. Missed opportunities. Bull. Amer. Math. Soc. 78 (1972), 635–652. http://www.ams.org/journals/bull/1972-78-05/S0002-9904-1972-12971-9/S0002-9904-1972-12971-9.pdf Especially pp. 638-9. My own experience confirms everything Dyson says there.

He discusses exactly this subject: what would mathematics gain if mathematicians read Maxwell. Maxwell actually anticipated much of the most important mathematics of the 20-th century.

It is amazing how mature Einstein was in his student years: he knew what to study:-) Instead of attending lectures, he studied himself "the works of Bolzmann, Kirchhoff, Helmholtz, Hertz etc.", according to his own recollection, and perhaps of Maxwell as well.

EDIT. I recently found a Russian translation of Einstein's paper "Electrodynamics of moving bodies", 1905 and read it. It is frequently mentioned that this paper has no reference list, and Einstein gives no credit to anyone. But Maxwell is mentioned frequently, (without exact references).

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    $\begingroup$ I find it hard to accept the statement that Maxwell had a quite good understanding of differential forms, Morse theory and Hodge theory if they weren't invented at his time. Moreover, it took quite sometime for it become known. Vector formalism wasn't even created when Maxwell wrote his Treatise. Wouldn't it be better to say that he possibly had a good intuition about these objects and situations? $\endgroup$ Jan 17, 2015 at 18:24
  • $\begingroup$ @Mark Fantini: If you yourself are familiar with these theories, I recommend you to read carefully the introductory part of the Maxwell book, and then we can discuss the subject again. $\endgroup$ Jan 17, 2015 at 18:54
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    $\begingroup$ It is in volume I and is called PRELINIMARY. p. 1-31. $\endgroup$ Jan 17, 2015 at 19:41
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    $\begingroup$ @MarkFantini: In the context of topography, Maxwell's paper "On Hills and Dales" contains the essential idea of Morse theory, that of relating the topology of a space to the number and type of critical points of a function on the space. He deduces statements such as that a function on the sphere (the earth) with p maxima (peaks) and q minima (bottoms) must have p+q - 2 critical points that are not extrema (passes). He refers to what we call the Euler characteristic as "Listing's rule". $\endgroup$
    – Dan Fox
    Apr 30, 2018 at 14:06
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    $\begingroup$ @BenCrowell: The "Preliminary" section of Maxwell's treatise on Electricity and Magnetism referenced by Eremenko contains a clear defintion of vectors, indications that things like stress and strain are not represented by vectors (they are tensors), definitions of line and surface integrals, discussion of orientations, definition of the gradient of a potential (p. 27, using i, j, k vector notation!), statements of what are called Stokes's and Gauss's theorems, etc. No one who doesn't understand differential forms in $\mathbb{R}^{3}$ can write all that. $\endgroup$
    – Dan Fox
    Apr 30, 2018 at 14:13
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Arthur I. Miller's Albert Einstein’s Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911) has on p. 81 a section "On Einstein's Knowledge of Electrodynamics in 1905".

I don't have access to it at the moment, but if I remember correctly, that section lists the books and monographs by Lorentz, Hertz, Föppl, Boltzmann, Helmholtz, Kirchhoff and others that Einstein had "definitely", "very probably", or "maybe" read by that time.


Edit 1: Having found the book, here goes (it's mostly electrodynamics, except Kirchhoff's book on mechanics):

A. "Definitely read, and evidence":
   1. Lorentz (1892, 1895).
   2. Hertz (Collected Works, 1893).
   3. Föppl (1894).
   4. Boltzmann (1891, 1893), von Helmholtz (1897) and Kirchhoff (1876).

B. "Very probably":
   1. Drude (1900).
   2. Abraham-Föppl (1904).

C. "Maybe":
   1. Lorentz (1904).
   2. Lorentz Festschrift (Arch. Néerl. (2) 5 (1900)).


Edit 2: Einstein’s Collected Papers reproduce his ETH Record and Grade Transcript which shows all the classes he took for credit. An Appendix has details of the course syllabi. (To the OP’s specific question, see Geiser’s courses on Infinitesimalgeometrie, and Minkowski’s on Analytische Mechanik + Anwendungen.)


Edit 3: There is also an entire book on the subject.

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The following screenshots (linked in a prior answer) are taken directly from Volume 1 of "The Collected Papers of Albert Einstein" and are all the courses he was required to take during his undergraduate years at the Polytechnic of Zurich. The grades are also shown. As we can see, Einstein did not know about differential geometry, or to be more precise, about absolute differential calculus which was the theory of tensors developed by Ricci Curbastro, Levi Civita and others in the decades before. He will learn these topics in a second time, while working on General Relativity. He will have many correspondences with Tullio Levi-Civita, who helped him with the math as well as his good friend mathematician Grossmann.

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