In the game, this is a theorem called the "first sub-boss of Multiplication World." You will prove it by mobilizing all the theorems built up so far: mul_zero, zero_mul, succ_mul, and mul_succ. The ...

I will explain examples of linear transformations of Lie groups. Regarding left-invariant vectors, I will show the equivalence between the expression viewed from the mapping of tangent vectors and the ...

Before conducting the experiment, participants attended a training session. In this session, the details of the experiment were explained for the participants, and several items were presented to them ...

This semester I’ve been teaching an undergraduate course on Galois theory. It was all online, which meant a lot of work, but it was also a lot of fun: the students were great, and I got to know them ...

Have you ever sat in a math classroom and wondered, “When will I ever use this?” You might have asked yourself this question when you first encountered “imaginary” numbers, and with good reason: What ...

June Huh thought he had no talent for math until a chance meeting with a legendary mind. A decade later, his unorthodox approach to mathematical thinking has led to major breakthroughs. On a warm ...

In recent years, quantum probability theory has been used to explain a range of seemingly irrational human decision-making behaviors. The quantum models generally outperform traditional models in ...

The character variety of a surface is a scheme with a rich but discrete set of automorphisms given by the mapping class group of the surface. The action of this group on the real points of the ...

In order to get used to the calculus of exact squares, the first thing we have to change is to start thinking more in terms of Kan extensions rather than limits and colimits. For instance, it’s usual ...