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This is one of the many examples of geometric concepts being applied to integers (in this case, the notion of derivation, although a non-linear one). Another i
by ykonstant 2y ago
This is one of the many examples of geometric concepts being applied to integers (in this case, the notion of derivation, although a non-linear one).
Another important concept is that of a curve and its ring of functions in algebraic geometry; for the integers, the curve is the prime spectrum of Z, i.e. the prime ideals generated by each prime number <p>. The ring of regular functions is precisely the ring of integers, operating as functions on prime numbers by n(p) = n modulo p.
I wonder if D has any interpretation in terms of nonlinear differential operators on Spec(Z).
- bandrami 2y agoAre you saying the derivative is a geometric concept? Tangent slope of a curve is simply one application of a derivative; it's not the derivative's identity. What the derivative is is the inverse of an inner product.
- ykonstant 2y agoCan you people stop with the inane pedantry? Yes, the derivative is a geometric concept and so is the inner product; they are at the core of what a Riemannian manifold is, they group to form the (co)tangent spaces of varieties and schemes and their derived structures produce the local geometric data of the object in question.
- lupire 2y agoThe point is that the derivative is a more general concept than just geometric, and is naturally defined with or without a geometric comtext. Of course almost anything can be modeled geometrically. You can draw a picture of almost anything. The integers themselves are obviously geometric, by drawing a kindergarten number line. https://en.m.wikipedia.org/wiki/Langlands_program https://en.m.wikipedia.org/wiki/Langlands_program
- ykonstant 2y agoThis is all semantic nonsense that devalues the original post I made about a potential connection between two arithmetic-geometric objects. But thanks for patronizing me, a research mathematician actually working on the arithmetic Langlands program, with a wikipedia link to the fucking Langlands program.
- bandrami 2y agoHuh? Inner product only needs vectors. No geometry required.
- lupire 2y agoThat's one of many definitions of derivative.
- jesuslop 2y agoAhh, nice one dares to proffer this. I can plug my own wondering about if one can formulate an aritmetic derivative that instantiates the Kähler differential concept (but I'm supposed not to ask unless I already knew the answer, so what'd'be the point).
- jesuslop 2y agorelated, https://amathew.wordpress.com/2010/02/17/kahler-differentials-and-regular-local-rings/#comment-297 https://amathew.wordpress.com/2010/02/17/kahler-differential...