By Sergei K. Lando, Alexander K. Zvonkin

Graphs drawn on two-dimensional surfaces have regularly attracted researchers through their attractiveness and via the range of inauspicious inquiries to which they offer upward push. the idea of such embedded graphs, which lengthy appeared particularly remoted, has witnessed the looks of fullyyt unforeseen new purposes in contemporary a long time, starting from Galois idea to quantum gravity types, and has develop into a type of a spotlight of an enormous box of analysis. The publication presents an obtainable creation to this new area, together with such subject matters as coverings of Riemann surfaces, the Galois crew motion on embedded graphs (Grothendieck's thought of "dessins d'enfants"), the matrix critical technique, moduli areas of curves, the topology of meromorphic services, and combinatorial points of Vassiliev's knot invariants and, in an appendix via Don Zagier, using finite staff illustration conception. The presentation is concrete all through, with various figures, examples (including computing device calculations) and routines, and may entice either graduate scholars and researchers.

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**Extra resources for Low-Dimensional Topology II: Graphs on Surfaces and Their Applications (Encyclopaedia of Mathematical Sciences, vol.141)**

**Sample text**

To make one But if it of his hypotheses contradict be a fact that the variable does reach its limit, and if this fact be assumed as true, then why not state it in the definition of a limit ? The reason is This, also, would have plain. been very inconvenient, since the author would have found it very tion difficult to verify the correctness of his defini- by producing any variables belonging to the in- finitesimal analysis that actually reach their limits. , p. 6. f Ibid. THE PHILOSOPHY OF MATHEMATICS.

Now this difference in the definition of a limit may, at first view, appear very trifling, yet in If, at the outset reality it is one of vast importance. of such inquiries, we diverge but ever so little from the of truth, we may ultimately find ourselves strict line involved in darkness and confusion. , p. 6. follow. Is the definition of a limit, all-imthen, of the one a mere matportant idea of the infinitesimal calculus, ter of convenience, or should it be conformed to the nature of things ?

4. TEE PEILOSOPS7 OF MATHEMATICS. The mathematicians. student who 41 follows the guid- ance of the one sees everything about him, and is at every step refreshed and invigorated by the pleasing On the contrary, prospects presented to his mind. the student who pursues the analysis of the other resembles, for the most part, the condition of a man who feels his way in the dark, or consents to be led blindfold by a string in the hand of The very his guide. point of divergence in these two very different modes of development is to be found in the first definition of the definition of In the all-important term limit.