Reviews for Equilibrium states and the ergodic theory of Anosov diffeomorphisms (Lecture notes in mathem...

The average rating for Equilibrium states and the ergodic theory of Anosov diffeomorphisms (Lecture notes in mathem... based on 2 reviews is 3 stars.

Review # 1 was written on 2018-06-23 00:00:00 Conor ONeill This book represents a compendium of some of the major development in modern higher mathematics in areas such as number theory, topology, group theory. This book is really good at conveying the fascinating and sometime totally unexpected inter-connectedness between apparently unrelated fields in mathematics. Moreover, the author's love for mathematics, as something not just extremely useful to progress, but inherently beautiful and fun, is visible and contagious. What I like about this book is also that it goes into the math to a much more significant extent than most popular science books, yet managing to keep things at a level sufficiently easy to understand. The target audience is the competent maths amateur, interested in the relatively recent (say after WWII) developments in higher mathematics. The only issue I have is that the author in some cases gets so infuriatingly and tantalizingly close to the actual details of a major result, and then it just closes with the statement "the full result can only be understood by specialists". The resulting frustration is compounded by the fact that the bibliographic references are very limited and unsatisfactory. But this is a minor fault. Let's get into the actual contents of this book: - after an initial, introductory chapter about prime numbers and number theory, there is a pretty nice chapter (still at pretty introductory level, unfortunately) about axiomatic set theory and Cantor's famous "continuum problem". To be honest, I was expecting more detail here: for example, when talking about axiomatic set theories such as Zermelo–Fraenkel, the axiom of choice is not discussed, and the axioms of the ZF theory itself are not discussed either. Real pity, as we are dealing with the foundational basis of the whole of mathematics here. - There is then a section about complex numbers. While complex numbers are explained in a reasonable way, there is no complex analysis, and the quaternions are touched only very briefly. This is a real pity and a missed opportunity, as complex analysis is one of the most beautiful as well as useful branches of Mathematics. - This is followed by a really nice chapter titled "beauty from chaos", which explains in a really beautiful and approachable way the concepts of fractional dimensions, chaotic dynamics, and the amazingly beauty of fractals. The Menger Sponge (D=2.7268), an amazing object which has zero volume enclosed by an infinite surface area, is also presented. Fractals are not just beautiful, but also useful, and we can see examples of fractals in in the realms of engineering, electronics, chemistry, medicine, even urban planning and public policy. - We have then another very enjoyable chapter about the critically important (to science in general, not just maths) concept of symmetry (and symmetry groups). The author even manages to explain at conceptual level, and in a very approachable way, the classification theorem of finite simple groups. The "monster" (sporadic simple group consisting of complex matrices of order 196,833) is also explained. Galois groups and the solution to the general quintic equation are also explained quite well. - There is then a very nice chapter about the famous Hilbert's Tenth Problem (solution to Diophantine Equations). The degree-25 prime-generating polynomial discovered in 1977 is also explained. - This is followed by couple of very nice chapters about some fundamental concepts and results in topology, where the author manages to beautifully explain, in a clear and precise manner, fundamental concepts such as the topological invariants represented by the Euler characteristic and "orientability". A beautiful introduction to the theory of manifolds (including concepts such as associated differentiation structures) is also a real enjoyment to read. Mind-blowing facts such as the peculiar richness and complexity of the 4-dimensional space as opposed to all other dimensional situations, are also presented beautifully. By the way, the four-color problem is also nicely treated. - The final two chapters are nothing of particular brilliance, but decently good anyway; they explain Fermat's last theorem and some introductory concepts about the efficiency of algorithms. Overall, a pretty good introductory overview of some selected recent developments in modern maths, targeted at the competent amateur, with some pretty good chapters. 4-star.

Review # 2 was written on 2021-04-17 00:00:00 Bob Ferth Nice general historical survey.