Brave New Number Theory
Solution 1:
This is probably not the answer you are looking for but, since there has been no other answers, it may be of some use. Recently, I have enjoyed finding out how wonderful and mysterious the relationship between (algebraic) number theory and low-dimensional topology is through the links between Galois groups and homotopy/fundamental groups. In particular the framing of properties of prime numbers in class field theory and arithmetic geometry having duality correlations with knots, links and three-dimensional manifolds! This subject of Arithmetic Topology was hinted time ago by several people, like D. Mumford and B. Mazur, which has very recently got its first MARVELOUS book:
- Morishita, M. - Knots and Primes: An Introduction to Arithmetic Topology, Springer 2011.
Some of the first explorations on this were:
- Mazur, B. - Remarks on the Alexander Polynomial,
- Mazur, B. - Notes on Étale Cohomology of Number Fields,
- Morishita, M. - Analogies between Knots and Primes, 3-Manifolds and Number Rings,
- Kohno T.; Morishita M. (eds.) - Primes and Knots, AMS 2006.
The whole point is the study of the fundamental groups of topology but from a number-theoretic perspective, that is, working with the (étale) algebraic fundamental group of varieties and schemes as originally thought by A. Grothendieck in his anabelian geometry. For this and as a prerequisite to Morishita's book, along with standard arithmetic geometry, one should get the specific background on the relationship of Galois gropus and fundamental groups of schemes, provided by:
- Szamuely, T. - Galois Groups and Fundamental Groups, CUP 2009.
- Schneps, L. (ed.) - Galois Groups and Fundamental Groups, CUP 2011.
These new dualities between apparently so different worlds may open the door to a deep understanding of the mysterious arithmetic-topologic relationship.