Posted by Jason Polak on 25. March 2015 · Write a comment · Categories: algebraic-topology, books

Doug Ravenel has made his book Nilpotence and periodicity in stable homotopy theory available for free download along with a list of errata, also available at the same page as the book.

Here is the official description from Princeton University Press:

Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic.

Ravenel’s first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.

At the official page of the book, you can also buy a paperback copy.

Posted by guest on 05. February 2013 · 1 comment · Categories: books · Tags: , ,
A Guest Post by Emily Shier

From Here to Infinity: A Guide to Today’s Mathematics
By Ian Stewart
1996 edition

“From Here to Infinity” is an enchanting read, which inspires both budding mathematicians, and curious outsiders alike. For mathematicians are mysterious beings to the general population; enshrouded in a cloak of cryptic symbols, they slip into another world, with an aura the ignorant classify as having a residue of chalky smoke, and mundane arithmetic.

Stewart bridges the gap between the uninformed individual and the world of mathematics with friendly, open approach. Several comprehensive chapters discuss intriguing topics, including chaos theory, knots, computer technology, algorithms, fractals, Fermat’s last theorem, and how to increase one’s odds of winning the lottery.

Never speaking down to the reader, Stewart provides many examples to illustrate a concept, which are punctuated with the occasional joke. For the reader with little exposure, the examples are fascinating, and show another side of thinking all together. However, as the examples develop, the level of math increases steeply. But, the initial feeling of frustration with a challenging idea gives way to a feeling of satisfied accomplishment with the completion of each chapter.
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Posted by Jason Polak on 30. December 2012 · Write a comment · Categories: books, number-theory · Tags: , ,

Posting has slowed a little bit this month because of holidays, but in the last couple weeks during my visit home I decided to refresh some basic knowledge of valuation theory by going through thoroughly the book “Introduction to p-adic Numbers and Valuation Theory” by George Bachman. Naturally, I wrote this quick review.

Bachman’s book is designed to be a leisurely introduction to valuation theory and p-adic numbers. It has only 152 pages and naturally cannot be comprehensive. It is, rather, an enjoyable read that does not require much advanced knowledge, though some experience with metric spaces is certainly required to fully appreciate the later chapters on the extension of valuations.
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Posted by Jason Polak on 18. November 2012 · Write a comment · Categories: books · Tags: , ,

Paul R. Halmos, who worked on fields from ergodic theory to algebraic logic and who authored the textbooks “Naive Set Theory” and “Measure Theory”, details much of his academic life in his autobiography, “I Want to be a Mathematician: An Automathography”
. In this post I shall give a few of my impressions of this book.

Halmos writes almost exclusively on his professional life as a mathematician and provides commentary and opinions on research, supervising, teaching, administrative work, and it even includes a math problem or two. In contrast, he rarely writes about his personal life or his other interests besides mathematics. As a mathematics student however, I never thought this as a deficiency and Halmos’s lively and engaging writing kept me steadily reading until the end.
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Let us recall some classic words:

Our subject starts with homology, homomorphisms, and tensors.

Saunders Mac Lane, in “Homology”

And while Mac Lane’s “Homology” and its friend by Cartan and Eilenberg are certainly fairly comprehensive sources of homological algebra, viewpoint shifts in the subject have made more recent approaches desirable. Weibel’s ‘An Introduction to Homological Algebra’ (author website, Amazon), or IHA, is just that: a modern textbook on homological algebra. Aside from a few busy semesters, during the last two years I have been slowly reading it, as I was determined to read this book cover-to-cover. Now that I have finished this, it is my pleasure to write a short review of this book.
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As in many mathematics departments, graduate students in McGill’s Department of Mathematics and Statistics have to take a comprehensive examination comprising of two parts: a written part (Part A) and an oral part (Part B). The Part B exam is based on two topics related to the student’s field of research. One of my Part B topics is algebraic groups.

The algebraic groups section will consist of some classical material found in usual sources, for which I am mainly using Humphreys’ book, and Brian Conrad’s notes on reductive group schemes, which can be found on his website.
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Posted by Jason Polak on 28. January 2012 · Write a comment · Categories: books, modules · Tags: , ,

As it happens every so often, I browse the mathematical library pseudorandomly, and look out for interesting titles; usually a prerequisite for interesting is that they have something to do with the realm of algebra. This is exactly how I found Faith’s book, with its captivating title urging me to borrow it.

Now, inevitably in mathematical research, one has to efficiently skim through papers and books to find specific ideas and facts. The unfortunate thing is that sometimes it is easy to neglect the stimulation of the idle curiosity that probably brought most mathematicians into their fields in the first place, and so I try to combat this neglect by my idle browsing and blogging.

I try not to spend too much time on this so that I progress with my degree, but I try to nurture my curiosity through reading anything that looks interesting. Returning to books, I do believe there are few worse literary follies than a graduate algebra textbook that lacks imagination in its examples and theorems and passion in its explication. I only fear that such books will tend to promote in the learning of higher algebra what most institutions have done with calculus, and that is to make it a tiresome mechanical effort, washing away the once vibrant and fanciful colours from the gentle tendrils of the mind.

But fear not! Should the mental dessication start to occur in a young algebraist’s mind; should the flames of passion dim for the wonders of the injective module, she can always turn to the entire object of this post, videlicet Faith’s “Rings and Things and a Fine Array of Twentieth Century Associative Algebra”
. I refer to the second edition, incidentally, which corrects many errors from the 1st edition.

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