By Kenji Ueno, Koji Shiga, Shigeyuki Morita

ISBN-10: 0821832824

ISBN-13: 9780821832820

This booklet will deliver the wonder and enjoyable of arithmetic to the school room. It bargains severe arithmetic in a full of life, reader-friendly sort. incorporated are workouts and plenty of figures illustrating the most ideas.

The first bankruptcy offers the geometry and topology of surfaces. between different subject matters, the authors talk about the Poincaré-Hopf theorem on severe issues of vector fields on surfaces and the Gauss-Bonnet theorem at the relation among curvature and topology (the Euler characteristic). the second one bankruptcy addresses a number of facets of the idea that of measurement, together with the Peano curve and the Poincaré procedure. additionally addressed is the constitution of 3-dimensional manifolds. specifically, it truly is proved that the third-dimensional sphere is the union of 2 doughnuts.

This is the 1st of 3 volumes originating from a chain of lectures given by means of the authors at Kyoto collage (Japan).

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**Extra info for A mathematical gift, 1, interplay between topology, functions, geometry, and algebra**

**Sample text**

78 (1988) 263. J. Hoste, A. Ocneanu, K. Millet, P. Freyd, W. Lickorish, D. Yetter, Bull. Am. Math. Soc. 12 (1985) 239. V. Turaev, Invent. Math. 92 (1988) 527. Y. Wu, J. of Knot Theory and its Ramifications 1 (1992) 47. Y. Akutsu, M. Wadati, Commun. Math. Phys. 117 (1988) 243. Y. Akutsu, T. Degutchi, M. Wadati, Phys. Reports 180 (1989) 247. H. Kauffman, J. of Knot Theory and its Ramifications 1 (1992) 59. M. Ge, L. Wang, K. Xue, Y. Wu, Advanced Series in Math. Physics, Volume 9. H. Wenzl, Invent.

Lett. A 4 (1990) 1091. [26] P. Cotta-Ramusino, E. Guadagnini, M. Martellini, M. Mintchev, Nuc!. Phys. B330 (1990) 557. 29 ON COMBINATORIAL THREE-MANIFOLD INVARIANTS G. Felder and O. Grandjean Mathematik, ETH-Zentrum, 8092 Zurich, Switzerland 1 INTRODUCTION In a recent paperl, Turaev and Viro gave a set of axioms for certain "initial data", and a procedure to construct combinatorial invariants for three-dimensional manifolds out of these data and a triangulation of the manifold. Then they showed that examples of initial data satisfying these axioms are provided by 6j-symbols of the quantum group Uq (sI2)' when q is a root of unity.

13), and 6j-symbols can be defined: let {VA}AEJ a complete set of inequivalent irreducible representations. A(VE, VA ® VF) I. A(VE, (VA ® VB) ® Vo). A(VA, VB ® Vo). In our example, irreducible representations are one-dimensional, labeled by the elements of the group: VA = c, PA(f) = f(A), and we have the canonical isomorphism VA ® VB ~ VAB. Thus Hom(VAB, VA ® VB) = C, with canonical orthonormal basis 1, and the 6j-symbol is just a evaluated on the three independent group elements. Choosing different orthonormal bases amounts to replacing a by a cocycle in the same cohomology class.

### A mathematical gift, 1, interplay between topology, functions, geometry, and algebra by Kenji Ueno, Koji Shiga, Shigeyuki Morita

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