Posts
-
The Lie bracket commutes with the differential of a Lie group homomorphism
Donaldson had written that "it follows from the definitions", but I needed to ponder on it for a while...
-
The definition of smooth structure compatibility
Say $X$ is a manifold and $p$ is a point in $M$. If $(U, \varphi_1)$ and $(V, \varphi_2)$ are charts about $p$, you probably know that we call the two charts compatible if $\varphi_2 \circ \varphi_1^{-1}: \varphi_1 (U \cap V) \to \varphi_2 (U \cap V)$ is a diffeomorphism. This definition had been mysterious to me, and as I was learning about galilean structures today, I could find it more understandable...
-
Understanding the $h$-cobordism theorem I
This is the first in a series of posts where some steps through understanding the $h$-cobordism theorem will be discussed, as I go along. As writing down everything is time-consuming, I will only be mentioning some technicalities that interest me, but will provide the main ideas of other parts as well, so that these posts maintain a form of progress through the end...
-
About Gödel's Program
The Continuum Hypothesis (denoted by $ \mathsf{CH}$ from now on) is the following claim. There exists no cardinal number between the cardinal of natural numbers and that of real numbers. Continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's famous 23 problems presented in 1900. Many mathematicians tried to settle this question, but for a while, no one couldn't find a counter-example subset of reals, either couldn't prove $ \mathsf{CH}$. The culmination of over 80 years of mathematical endeavor illuminated that, $ \mathsf{CH}$'s truth value, in the sense that would be discussed in the following paragraphs, is independent of the world of mathematics...