How this blog works

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This ‘web log’ (etymology of ‘blog,’ as far as I know of) is intended to be a batch of notes that I make upon academic studies. Among the notes that I feel worth sharing, to be precise.

Some of the notes will be written in (math) English, while some else will be written in Korean (my mother tongue). It is mostly due to the fact that I am bit uncomfortable to use English to represent convoluted ideas which is not math.

Every post will accompany the ‘update log’ below, to announce that some sort of revision is made after the first post.

Update Log

  • 210916: Created

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An Abstract Nonsense for Gödel Incompleteness

Although Gödel’s proof of his incompleteness is renowned for the formal construction of the sentence “I cannot be proven,” there are ways to illuminate this within the framework of abstract nonsense. From the nLab point of view, this is a consequence of Lawvere’s fixed point theorem, as shown in Lawvere 1969 or Yanofsky 2003. However, since I am not entirely sure if I understood the work correctly, I will present my own re-invention of the wheel below.

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Playing with the Ax–Grothendieck Theorem

We discuss a construction question that naturally arises from the Ax–Grothendieck theorem. In particular, this comes from a field automorphism between the complex numbers and the ultraproduct of prime algebraically closed fields of positive characteristics.

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Elliptic curves of bidegree (2,2)

One way to see an elliptic curve is to view it as a smooth bidegree (2,2) curve in $\mathbb{P}^1\times\mathbb{P}^1$. This fact itself comes from the adjunction formula, but we suggest a way to derive a bidegree (2,2) formula from the Weierstrass equation. Based on that, we see how this connects with the tropicalized actions of Vieta involutions on elliptic curves.

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