Ideas in Mathematics The idea of continuity. We have a beautiful formula [\lvert B^A \rvert = \lvert B \rvert ^ {\lvert A \rvert}.] But what happens when at least one of $A$ and $B$ are empty? Th...

Lecture highlights: More details about the construction of our category of mappings between subsets of Euclidean spaces The velocity functor Pushforward map Definition of smooth manifold ...

Lecture highlights: We just talk about the category of smooth maps between subsets of Euclidean spaces in this lecture. We prove that these maps indeed form a category. Graphs are naturally ...

Lecture highlights: $T$ is a functor A big picture for what is to come. Meth Few meth today, mostly the cold hard truth. Well is mathematics truth? I don’t know. (For the few of you who wa...

Lecture highlights: symmetries quadrics high school mathematics multivariable calculus high school mathematics Meth Obviously he like these stuff, that’s why he mething a lot these p...

Lecture highlights: Euclidean structure Categories and functors Meth He methed a lot this lecture! In the Greek times, they believed that the Earth is the centre of the universe. They tho...

Today we talk about affine spaces. We only care about stuff over the real numbers in this course. Lecture highlights: Definition We have a ‘dictionary’ between vector space and affine space ...

Update 2024-09-09: Look at how this went. p-adic distance ultra-metric $\sigma$-finite measure: the measure space can be decomposed into a countable sequence of sets with finite measure c...

I cannot claim to have a summary of the whole lecture, that’s why I switched the wording to ‘highlights’ Lecture highlights: tensor algebra graded vector space determinants M: Do abstrac...

Calculus on Mengnifolds I have no idea what happened this lecture. Lecture highlights: Notations Review of linear algebra The set of bases of $V$ may be naturally identified with the set ...