# A Mathematical Gift, 1: The Interplay Between Topology, by Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada

By Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada

This e-book will carry the wonder and enjoyable of arithmetic to the study room. It deals severe arithmetic in a full of life, reader-friendly type. integrated are routines and lots of figures illustrating the most thoughts.

The first bankruptcy provides the geometry and topology of surfaces. between different issues, 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 variety of points of the concept that of measurement, together with the Peano curve and the Poincaré technique. additionally addressed is the constitution of three-d manifolds. particularly, it really is proved that the three-d sphere is the union of 2 doughnuts.

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

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**Sample text**

Of course, the ergodic hypothesis has raised an ergodic problem, and thus it has stimulated an enormous eﬀort to ﬁnd rigorous 5 As we shall see at the end of the present chapter, digital computers have made possible a major leap forward in the study of the microscopic-dynamical origin of macroscopic properties of physical systems. Besides a deepening of our theoretical understanding of the subject, this has also led to the development of molecular dynamics, a precious practical tool for ab initio computations.

51) which is an equation valid for any substance: one says that this equation expresses the law of corresponding states. We ﬁnd here a property of universality that is well veriﬁed experimentally [32]. Experimentally, the transition between the gas phase and the liquid phase occurs at constant temperature and pressure. If we start from a state in which only liquid is present, the addition of heat results in the conversion of some liquid into gas until all the liquid is converted into gas. During the phase transition both P and T remain constant.

1) can be insuﬃcient to describe the great richness of the physical properties of matter at macroscopic and mesoscopic levels, properties that probably can be explained only if more-complicated internal structures of molecules, as well as other kinds of a forces of quantum nature or the interaction with a self-consistent electromagnetic ﬁeld, are considered. However, the domain of application of classical statistical mechanics can be extended very far from that of its original formulation. To give an important example, let us mention that a mathematical relationship can be established between classical statistical mechanics and quantum ﬁeld theory, a seminal relationship that has been widely exploited in recent years and that witnesses that the theoretical relevance of classical statistical mechanics, in particular for what concerns phase transitions, is not limited to classical physics.