## ChatGPT and Mathematics (III)

GPT-4 was just released. Here’s a preview of what it’s capable of. I tried throwing some mathematics problems at it to check out its capability. Problem 1: Cheryl’s Birthday Problem There’s an infamous logic problem from Singapore’s primary school Olympiad … Continue reading

## ChatGPT and Mathematics (II)

This post is only tangentially related to mathematics. As an experiment on AI-assisted learning, I tried to write a web application in Javascript + WebGL. At the start of this experiment, I had some experience with Javascript but absolutely no … Continue reading

## ChatGPT and Mathematics (I)

The Main Point ChatGPT is an AI language model that has been making the news recently. There are multiple articles on the internet on what the model is capable of so this will not be the focus of our post. … Continue reading

## Commutative Algebra 64

Segre Embedding Throughout this article, k is a fixed algebraically closed field. We wish to construct the product in the category of quasi-projective varieties. For our first example, let be the projective variety defined by the homogeneous equation . We define … Continue reading

## Commutative Algebra 63

Serre’s Criterion for Normality Throughout this article, fix an algebraically closed field k. In this section, A denotes a noetherian domain. We will describe Serre’s criterion, which is a necessary and sufficient condtion for A to be normal. In the … Continue reading

## Commutative Algebra 62

Irreducible Subsets of Projective Space Throughout this article, k is an algebraically closed field. We wish to consider irreducible closed subsets of . For that we need the following preliminary result. Lemma 1. Let be a graded ring; a proper homogeneous … Continue reading

## Commutative Algebra 61

In this article, we will consider algebraic geometry in the projective space. Throughout this article, k denotes an algebraically closed field. Projective Space Definition. Let . On the set , we consider the equivalence relation: The projective n-space is the set … Continue reading

## Commutative Algebra 60

Primary Decomposition of Ideals Definition. Let be a proper ideal. A primary decomposition of is its primary decomposition as an A-submodule of A: where each is -primary for some prime , i.e. . Here is a quick way to determine if … Continue reading

## Commutative Algebra 59

Prime Composition Series Throughout this article, A is a noetherian ring and all A-modules are finitely generated. Recall (proposition 1 here) that if M is a noetherian and artinian module, we can find a sequence of submodules whose consecutive factors … Continue reading

## Commutative Algebra 58

We have already seen two forms of unique factorization. In a UFD, every non-zero element is a unique product of irreducible (also prime) elements. In a Dedekind domain, every non-zero ideal is a unique product of maximal ideals. Here, we … Continue reading