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Prikvačeni tweet
Powerful: Hermann Weyl's speech at Emmy Noether's funeral. https://www-history.mcs.st-andrews.ac.uk/Extras/Weyl_Noether.html …
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Another good slide: student feedback at the end of a course.
#MTBoS https://twitter.com/HigherGeometer/status/1223921733817659394 …pic.twitter.com/NyYUiFJmcp
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Source: Della Dumbaugh, "Prospering Through Mathematics" talk at JMM 2020 https://www.youtube.com/watch?v=wxtR1rQ9xPA … at 16:20
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Emil Artin's teaching schedule was in no way special. He solved Hilbert's 17th problem, reshaped Galois theory, proved the general reciprocity law in alg. number theory—and taught a 1st year trig. class Those current Professors who manage to never teach first year students...
pic.twitter.com/v6wUBH4vPK
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I asked a question on MathOverflowhttps://mathoverflow.net/q/351751/4177
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theHigherGeometer proslijedio/la je Tweet
A new prize, the Ladyzhenskaya medal in mathematical physics, has been announced by the National Committee of Mathematicians of Russia, St Petersburg State University, and, for the inaugural prize, the Organizing Committee of the ICM. /1
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theHigherGeometer proslijedio/la je Tweet
EGA website is now live at https://ega.fppf.site (still very much in testing). Built thanks to
@stacks_project tools (https://gerby-project.github.io/ )!Prikaži ovu nitHvala. Twitter će to iskoristiti za poboljšanje vaše vremenske crte. PoništiPoništi -
Ben's paper is published! This will form part of his Masters thesis (the other part is about quartic Gauss sums) at
@MathsUOA Ben Moore, A proof of the Landsberg–Schaar relation by finite methods, Ramanujan Journal (2020) https://doi.org/10.1007/s11139-019-00195-4 … https://arxiv.org/abs/1810.06172 pic.twitter.com/Ki7FDPaRBD
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theHigherGeometer proslijedio/la je Tweet
Domenico Fiorenza Knight of the nLab at "M-Theory and Mathematics 2020" http://ncatlab.org/nlab/show/M-Theory+and+Mathematics …pic.twitter.com/LotOBRkXz7
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Can anyone think of any other properties (or variations thereof) we are accustomed to having, but that break for exponentiation x^y? Certainly right distributivity fails in general: (x+y)^z ≠ x^z+y^z. And left distributivity too: x^(y+z)≠x^y+x^z Any more? 9/8
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So I guess when students first learn about the exponentiation operation, it basically breaks everything they knew about algebraic operations of addition and multiplication (and their inverses -,÷). Exponentiation is weird! 8/8
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Interestingly, there are axiomatic theories of arithmetic (such as bounded quantifier arithmetic, IΔ_0) that cannot prove that exponentiation is a total function! People thus sometimes study the axiom system IΔ_0+EXP, which explicitly adds the exponential function as an axiom 7/
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So it's not even *power associative*, with 1 and 2 being the only numbers x with (x^x)^x = x^(x^x)—this property holds for all the nasty Cayley–Dickson algebras beyond the octonions! 6/
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Also, exponentiation is not associative, but sometimes is: (1^1)^1 = 1^(1^1) (2^2)^2 = 2^(2^2) ...but (3^3)^3 ≠ 3^(3^3) ...though (0^0)^0 = 1^0 = 1 but 0^(0^0) = 0^1 = 0 hmmm... 5/
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So maybe we want to include 0 in the domain (and define 0^0 arbitrarily, why not? I like 0^0 = 1, for now) ...but what about fixing y and trying to solve x^y = 1? ...ok, so then x = 1, and so 1 is a left inverse for every number ...except for 0, whose left inverse is 0 4/
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OK, so it doesn't have a two-sided identity, but we can ask for inverses, even for the right identity element 1, no? ...but this is trying to solve x^y = 1 ...so for all x, we have x^0=1, so 0 is a right inverse for every number ... but we didn't include 0 in the domain 3/
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it has a right identity, x^1 = x for all x.... ...but no left identity! there is no c such that c^x = x for all x ...but for each x there might be a d with d^x = x, namely d=x^{1/x} ...if 'numbers' = real numbers, this is all x, but if numbers=integers, then only for x=1 2/
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As a binary operation, exponentiation is kinda weird (let's take positive numbers as the domain). It fails all kinds of familiar axioms: 1/
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Spoke with a grad student today that I hadn't seen for a little while, whose thesis is due soon. It came up that she is trying to figure out what to do with the PhD position she has been offered. Me: I thought you were doing a PhD. She: so did I.
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