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Topology of Numbers
- strmpnk 8y agoIt’s a shame they only mention Farey when introducing the binary tree construction. It’s called a Stern-Brocot tree (after two independent discoveries). Farey sequences end up being a specific enumeration over this tree. It’s interesting to see the range of applications this structure has. While it helped me understand a few ideas like how a Cauchy sequence might work, practical applications included things like finding approximate ratios for floating points with various limits on the scale of the denominator and single update list sorting systems that don’t rely on midpoints (they run out of precision and require relabeling rather early). The history of the concept is also worth looking at. A good overview is available in Graham, Knuth, and Patashnik‘s book “Concrete Mathematics.”
- heinrichf 8y agoHatcher is the author of famous freely available and beautifully typeset topology books http://pi.math.cornell.edu/~hatcher/#anchor1772800 http://pi.math.cornell.edu/~hatcher/#anchor1772800 (most notably "Algebraic topology"). "By special arrangement with the publisher, an online version will continue to be available for free download here, subject to the terms in the copyright notice." Some details about how he typesets his books here: http://pi.math.cornell.edu/~hatcher/AT/typography.html http://pi.math.cornell.edu/~hatcher/AT/typography.html
- a-nikolaev 8y agoIndeed, his work is spectacular. I also think that his geometric approach is quite unique and instructive (although I did not get into his books that far).
- dbranes 8y agoWhat a confusing name in an era where homotopy-theoretic methods in arithmetic geometry are flourishing. Looks Hatcher covers non of that and use "topology" to mean "geometrically motivated".
- gnulinux 8y agoI agree, I can't stop scratching my head. Maybe "Geometry of Numbers" is a better name.
- yesenadam 8y agoGeography maybe?
- gowld 8y agohttp://pi.math.cornell.edu/~hatcher/TN/TNpage.html http://pi.math.cornell.edu/~hatcher/TN/TNpage.html > we are using the word "Topology" in the general sense of "geometrical arrangement" rather than its usual mathematical meaning > perhaps the title could have been "Topography of Numbers" instead. https://www.google.com/search?q=define+topology https://www.google.com/search?q=define+topology > 2. the way in which constituent parts are interrelated or arranged. > "the topology of a computer network"
- NotAnEconomist 8y agoThis is a dumb question, but what's the difference between geometry on discrete sets and homotopy theory on discrete sets, eg, in the spirit of digital topology? Every set is open, so every set is closed. So something like a closed interval, or the product of closed intervals, is just going to look like the network that defines a line segment, square, etc... right? In that sense, it seems like the geometry of numbers and the topology of numbers are basically the same thing. But I'm going to be honest -- I know basically nothing about arithmetic geometry.
- dbranes 8y agoAs you observe one can't really recover any non-trivial number theoretic things by looking at integers with the discrete topology. The theory of discrete topological spaces is just the theory of sets. Instead you can look at things like prime ideals of integers localized at some prime, and consider algebro-geometric topologies on that
- bbeonx 8y agoOh cool, this is from Allen Hatcher. I've worked through some of his algebraic topology and it was very well written. I'm excited to look through this
- jhinra 8y agoI'm impressed how many people are here to say, "Oh, Allen Hatcher!" I did the same. It certainly says a lot about Hatcher as a textbook writer.
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- james_s_tayler 8y ago"elementary number theory from a geometric point of view" It's stunning how intuitive geometric explanations can feel. I like the geometric approach.
- Cobord 8y agoThere is also the Zariski topology of Spec Z that is the topology of numbers in the sense of algebraic geometry. Or of Z[x]
- ilyagr 8y agoThe Farey sequence that is central to this book also appears in the functional pearl on "Enumerating the Rationals" that recently appeared on HN. https://news.ycombinator.com/item?id=18515413 https://news.ycombinator.com/item?id=18515413
- markhollis 8y agoI can imagine this text will be popular among Project Euler problem solvers, given the shared themes of continued fraction, Farey Sequences etc.