Home Calculus III - Lecture 5

Calculus III - Lecture 5

Lecture highlights:

  1. Euclidean structure
  2. Categories and functors


He methed a lot this lecture!

In the Greek times, they believed that the Earth is the centre of the universe. They thought that the universe is three dimensional, with a centre. The Greek philosopher Aristotle had a concept of space and time in the form of $(\mathbb{E}^3,O)\times \mathbb{E}^1$.

Newton believed that the universe has no centre, so his concept is $\mathbb{E}^3 \times \mathbb{E}^1$. Before Newton, there was Galileo. He had the Galileo spacetime, which is $\mathbb{A}^4_\mathbb{R} \to \mathbb{E}$, where the map is a surjective affine map. The inverse image of the time $t$ in $E^1$ is equivalent to $\mathbb{E}^3$. So Galileo had relativity, but Newton does not. So Newton really went backwards.

The modern relativity is Einstein spacetime. But this actually should be called Minkowski spacetime. It is \(\mathbb{A}^{4}_{\mathbb{R}}\) with the Minkowski structure, \(\left<\cdot, \cdot\right> : T_x\mathbb{A}_\mathbb{R}^4\times T_x\mathbb{A}_\mathbb{R}^4\to \mathbb{R}\). This is called the Lorentz inner product. It is translation invariant, defined as \((\overrightarrow{xy}, \overrightarrow{xz}) := (\vec{u}, \vec{v}) \mapsto \vec{u}\cdot_{\text{Lorentz}}\vec{v} = u_0v_0 - u_1v_1 -u_2v_2-u_3v_3\), where we denote \(\overrightarrow{xy}\) as $\vec{u}$, and $x$ is a point (4-dim) in the tangent space. With a translation invariant assignment of the Lorentz product to tangent space, we have the Minkowski space $M^4$. (Again I have no idea of the logical relationship here from my scattered notes).

In modern relativity, time is not absolute, and space is not absolute. We cannot talk about two events as simultaneous absolutely. You have to choose an inertial frame. Whether two events are at the same space point is not absolute in Galileo, but is absolute in Newton.

Classical physics is a kind of geometry, so mathematicians, who are familiar with classical differential geometry, can learn classical mechanics very quickly.

Lecture Notes

$ \mathbb{R}$ is either linear space of real column space with $n$ entries, or the corresponding Euclidean vectors with inner product being the dot product. It consists of vectors.

$ \mathbb{A}^n_\mathbb{R}$ is the affine space of n-tuples of real numbers. It consists of points.

$\mathbb{E}^n$ is the Euclidean space $(\mathbb{A}^n_\mathbb{R}, \text{the standard Euclidean structure})$

Tangent space

\[\left< \cdot, \cdot \right>: T_x \mathbb{A}_\mathbb{R}^n \times T_x \mathbb{A}_\mathbb{R}^n\to \mathbb{R}\]

$(\overrightarrow{xy},\overrightarrow{xz}) \mapsto (y-x) \cdot (z-x)$. This is clearly translation invariant.

Defines $x-y$ again.

Up to linear equivalences, there exists a unique $n$-dimensional real linear space, i.e. if there is $V$ satisfies this property, then $V\cong \mathbb{R}^n$. This essentially means that every such linear space has a trivialization.

Up to linear isometric equivalance, there exists a unique $n$-dimensional real Euclidean vector space. This implies that any $\mathbb{E}^n$ has an orthonormal basis. M proceeds to spend five minutes on the Gram-Schmidt process for some reason. So we have $(V, \left<\cdot,\cdot\right>) \xrightarrow[\cong]{T} (\mathbb{R}^n, \cdot)$ such that $\left<\underline{u}, \underline{v}\right> = T\underline{u}\cdot T\underline{v}$.

From now on we denote an affine space over $\mathbb{R}$ to be $\mathbb{A}$. So we drop the subscript.

$T_p \mathbb{A}$ is the tangent space of $\mathbb{A}$ at $p$.

We also have a notion of the total tangent space $T\mathbb{A} := \bigcup_{p\in \mathbb{A}} T_p \mathbb{A}$. It is the collection of all tangent vectors of $\mathbb{A}$.

${\overrightarrow{pq}:p,q \in \mathbb{A}} = \mathbb{A}^2$ is an affine space with doubled domain.

For affine spaces $\mathbb{A}, \mathbb{B}$, we have

An affine map $\overrightarrow{pq}\mapsto \overrightarrow{\phi(p)\cdot\phi(q)}$ preserves straight line. More explicitly, we have $tp+(1-t)q \mapsto t\phi(p) + (1-t)\phi(q)$.

Now we discuss categories. There is a set of analogies between monoids and categories, which are basically generalised monoids. (What about the class of objects? idk). They are generalised as they only have partial composition. You cannot have arbitrary compositions of two functions, because you have to note the compatibility of their domains and codomains. Also, it is generalised as they are not necessarily based on sets. A category might be very very large, so we call it a class. For categories which are sets, we call them small categories. For example, a monoid is a small category.

monoid homomorphism-functor

There are many categories. Once you define a mathematical object, you should make sure that there are plenty of them so its more interesting. Below is a list of cats:

  1. monoid, group
  2. $\mathscr{S}$ = cat of set maps - huge, bigger than class of sets since each hv id partial bin operation is composition
  3. $\mathsf{Vect}$ = cat of linear maps under composition
  4. the cat of matrices under multiplication
  5. cat of affine maps under composition

Bourbaki redefined things. As civilization moves forward, we have more generalisation.

We have some examples of functors. We have the tangent functor from the category of affine maps to itself. This type of functors is called an endofunctor, similar to an endomorphism. This is what we are actually talking about if we talk about differentiation.

We also have a functor, the Borel functor, which is from the category of continuous (topological) maps, to the category of measurable maps.

There are only a few great ideas in mathematics. Once you notice them, you can learn very quickly, and it is one of the ways how you discover new things. Here, the idea is completion, in the sense that the set of complex numbers is a completion of the reals. Another example is span in linear algebra. For this Borel functor, we notice that the collection of open sets is not complete in the sigma-algebra sense. You cannot have countable intersection in top, but only finite.

How do we do a Borel completion on continuous maps? To do completion, it is enough to check the generators only. Pull back generator in measurable maps to measurable continuous map. (I have no idea what this sentence means I just wrote it down).

Historically, people didn’t like these abstract things. Borel was depressed. His advisor Weierstrass said that he didn’t know what is set theory, but it’s not mathematics. Poincare held similar opinions. But this was eventually saved by Kolmogorov, who was politically strong and pushed set theory to secondary school education.

Category theory was introduced in 1940s for algebraic topology, otherwise statement of theorems would become very messy. Still nowadays, many mathematicians reject it. Only people in algebraic topology, algebraic geometry and other algebra accept it. We should have a more open mind. This allows us to learn faster.

The next few lecture will be on this functor $T$.


mic drop

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