On Infinite Cycles in Graphs—or How to Make Homology Interesting
by Reinhard Diestel
Why is it that so many standard theorems about cycles in finite graphs, especially theorems related to their “homology,” fail for infinite graphs? This riddle has puzzled infinite graph theorists for some time, and much guesswork has been expended on its various instances. We present a surprising solution. As soon as the graph’s cycle space, or first homology group, is based not on its usual finite cycles but on its circles, the homeomorphic images of the unit circle in the space formed by the graph together with its ends, and infinite sums of such circles are permitted in generating the space, everything generalizes smoothly. The paper explains the ideas behind this approach by exploring a series of increasingly “wild” examples of “infinite cycles” that are obviously needed in order to salvage even the most basic facts about finite graph homology for infinite graphs. It then turns out that even the wildest of these are just instances of the topological notion of a circle in the graph together with its ends. The paper is extracted from a more comprehensive survey of the results obtained by this approach, as well as of the numerous open problems it suggests.
Two “Generic” Proofs of the Spectral Mapping Theorem
by Torsten Ekedahl and Dan Laksov
The “Spectral Mapping Theorem” is a simple and beautiful result from linear algebra that has many applications in algebra, geometry, and number theory. In the case of fields the result is part of the folklore of linear algebra and there exist several proofs involving only standard methods from linear algebra. The spectral mapping theorem is, however, most natural and useful in the context of commutative rings with identity. We present two easy proofs of the spectral mapping theorem. Both proofs are based upon useful classical methods, the discriminant trick and a trick for splitting generic polynomials, that are of considerable interest in themselves. The only result used in the proof is the Main Theorem of Symmetric Functions.
The Number of Halving Circles
by Federico Ardila
A set S of 2n+1 points in the plane is said to be in general position if no three points of S are collinear and no four are concyclic. A circle is called halving with respect to S if it has three points of S on its circumference, n-1 points in its interior, and n-1 in its exterior. We prove the following surprising result: any set of 2n+1 points in general position in the plane has exactly n2 halving circles!
From Hilbert’s Superposition Problem to Dynamical Systems
by V. I. Arnol’d
Problems and Solutions
Geometry and Convexity of cos √x
by Darko Veljan
A Simple Proof of Sharkovsky’s Theorem
by Bau-Sen Du
The Hairy Ball Theorem via Sperner’s Lemma
by Tyler Jarvis and James Tanton
Some Subgroups of SO3 R Isomorphic to the Free Produce (Z/2Z) * (Z/3Z)
by Kenzi Satô
Ergodic Theory of Numbers.
by Karma Dajani and Cor Kraaikamp
Reviewed by David W. Boyd
Discrete Mathematics: Elementary and Beyond.
by László Lovász, József Pelikán, and Katalin Vesztergombi
Reviewed by Arthur T. Benjamin