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Topology of Numbers (cornell.edu)
153 points by espeed 3 months ago | hide | past | web | favorite | 20 comments



Hatcher is the author of famous freely available and beautifully typeset topology books 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


Indeed, 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).


It’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.”


What 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".


I agree, I can't stop scratching my head. Maybe "Geometry of Numbers" is a better name.


Geography maybe?


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

> 2. the way in which constituent parts are interrelated or arranged.

> "the topology of a computer network"


This 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.


As 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


Well, hold on now.

I think the language and machinery of topology, even when just reconstructing the language of sets in a discrete setting, highlights interesting facets of numbers.

eg, if you look at the inverse image of various mappings, and particularly in cases where you can iterate this via a function from a set into itself, you can start building up meaningful comments on certain classes of number theory problems.

But I am curious what you mean by "prime ideals of integers localized at some prime", since I know what (prime) ideals are, but am not sure I follow what you mean by localized


The category of sets embeds into the category of topological spaces as a full subcategory, the essential image of which are the discrete spaces. Hence the equivalence I claimed is a precise statement.

Look up ring localization.


> Hence the equivalence I claimed is a precise statement

They're obviously equivalent.

My point is that what's easily noticeable in one incarnation of the theory is different than what's easily noticeable in the other incarnation (or if you prefer, expressible), and switching our language for the same abstract structure can highlight different interesting features of it. And further, there's still utility to using topological perspectives and language to discuss the integers or naturals, even if it's equivalent to set theory.

I do appreciate the reference to ring localization -- will have to look at that further.


Genuinely tried reading the "examples" you wrote in this thread, can't make any sense out of it. Happy to discuss it if you clarify what you mean.

Just to address your original comment in this thread, perhaps it's relevant to note the following. Consider the homotopy theory of the category of nice topological spaces. The full subcategory of topological spaces supported on discrete topological spaces inherits a homotopy theory. This inherited homotopy theory is equivalent to the trivial homotopy theory on the category of sets: where weak equivalences are isomorphisms. This is the sense in which discrete spaces don't have an interesting homotopy theory, at least naively.

(This statement you can precise in your favorite model for the homotopy theory of spaces, via infinity categories, model categories etc.)


(the embedding is full & faithful)


Oh 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


I'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.


"elementary number theory from a geometric point of view"

It's stunning how intuitive geometric explanations can feel. I like the geometric approach.


The 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


I can imagine this text will be popular among Project Euler problem solvers, given the shared themes of continued fraction, Farey Sequences etc.


There is also the Zariski topology of Spec Z that is the topology of numbers in the sense of algebraic geometry. Or of Z[x]




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