theHigherGeometer

@HigherGeometer

Mathematician in Adelaide. Tweets own views, retweets not. I am boycotting Elsevier journals.

Vrijeme pridruživanja: ožujak 2019.

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  1. Prikvačeni tweet
    18. srp 2019.

    Powerful: Hermann Weyl's speech at Emmy Noether's funeral.

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  3. 13.a) Name one critical reasoning skill you learned this fall in Number Theory and describe how to plan to leverage it in the future.


A proof always relates back to definitions.
A lot of things break down to the basics.
So mastering the basics will help me to understand more complex things
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  4. 2. velj

    Source: Della Dumbaugh, "Prospering Through Mathematics" talk at JMM 2020 at 16:20

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  5. 2. velj

    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...🙄

    Artin's Teaching Schedule (Spring 1945)

Math 103a, Trigonometry
math 210b, Advanced Calculus
Math 213, Differential Equations
Math 322, Graduate Seminar

source https://www.youtube.com/watch?v=wxtR1rQ9xPA at 16:20
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  6. 2. velj

    I asked a question on MathOverflow

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  7. 1. velj
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  8. proslijedio/la je Tweet
    31. sij

    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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  9. proslijedio/la je Tweet
    26. sij

    EGA website is now live at (still very much in testing). Built thanks to tools ()!

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  10. 30. sij

    Ben's paper is published! This will form part of his Masters thesis (the other part is about quartic Gauss sums) at Ben Moore, A proof of the Landsberg–Schaar relation by finite methods, Ramanujan Journal (2020)

    Screenshot of abstract of article "A proof of the Landsberg–Schaar relation by finite methods", available at https://doi.org/10.1007/s11139-019-00195-4
    Abstract of Ben Moore's talk from AustMS 2019, entitled "Quartic Gauss sums and twisted reciprocity"
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  11. proslijedio/la je Tweet
    30. sij

    Domenico Fiorenza Knight of the nLab at "M-Theory and Mathematics 2020"

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  12. 29. sij

    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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  13. 29. sij

    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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  14. 29. sij

    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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  15. 29. sij

    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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  16. 29. sij

    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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  17. 29. sij

    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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  18. 29. sij

    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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  19. 29. sij

    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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  20. 29. sij

    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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  21. 29. sij

    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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