I'll have a go at 4. (I don't know what is and isn't in your brain, so this may be entirely unhelpful. Even if so, it may help make it clearer what would be helpful.)

I think a better perspective on induction is that "regular induction" is just a special case of transfinite induction.

Ordinary induction over the natural numbers

When you first learn about induction, it's usually presented something like this: "To prove that P(n) for all n, prove that P(0) and that P(n) => P(n+1)." But it's not that uncommon to have induction proofs that work a slightly different way -- e.g., going from P(n) to P(2n) and P(2n+1) -- and a better way of looking at it is this: "When proving that P(n), you are allowed to assume that P(m) for all m<n".

Any proof using "n->n+1" induction is already a proof using this more powerful induction principle. (Because assuming P(n-1) is just part of what it means to assume P(everything less than n).)

Why does this more powerful principle work? Because if P(n) fails then P(m) fails for some smaller m, and then P(l) fails for some still smaller l, and so on, and you get an endless sequence of smaller and smaller numbers -- but these are all positive integers, so the sequence can't go on for ever, contradiction.

So the key fact is that there are no infinite decreasing sequences.

I find it helpful to think about this in terms of how the natural numbers are (or can be) constructed. 0 is a natural number; every natural number has a successor; and that's all of them. And in some sense this is why there are no infinite decreasing sequences: because each natural number is built by a finite increasing sequence.

Other varieties of induction

Now, the same thing works in other contexts where you have a kind of object (above, it's natural numbers) and a notion of comparison (above, it's the < relation) for which there are no infinite decreasing sequences.

(You often want to think of < as meaning "is simpler than".)

These tend to be objects with a recursive construction similar to that of the natural numbers. For instance, binary trees. The empty tree is a binary tree; if L and R are binary trees then the tree consisting of a root node whose children are L and R are binary trees; and that's all of them. So now you can do induction over binary trees: try to prove P(T) allowing yourself to assume P(T') whenever T' is a subtree of T. Or Lisp-style lists. The empty list is a list; if L is a list and x is a thing-that-isn't-a-list, then there's a list with x as head and L as tail; and that's all of them. Now you can do induction over lists: to prove P(L), allow yourself to assume that P(tail(L)).

In these cases, you can turn this "structural induction" into ordinary induction over natural numbers: use the depth of the tree, or the length of the list. But there are other cases, and we'll see one now.

Transfinite induction over sets and ordinals

Take our objects to be sets. And take < to mean "is an element of, or an element of an element of, etc.".

Why are there no infinite decreasing sequences? Because there are no infinite chains that look like $\cdots \in x\in y\in z$, going on for ever on the left. Why's that? Depending on how you look at it, it's (1) because there's an axiom (the Axiom of Foundation) that forbids it, or (2) because sets are constructed recursively, starting from the empty set, and this recursive construction guarantees it. (If you try to turn that into an actual proof, you end up back with the Axiom of Foundation, which is what guarantees that you can think of sets as being constructed recursively.)

Take our objects to be ordinals. And take < to be the usual ordinal comparison relation.

Why are there no infinite decreasing sequences? Because the ordinals are well-ordered by <. Why's that? Because (at least with the usual construction in ZF(C)) any set of ordinals is a subset of a larger ordinal, with the < relation between ordinals corresponding to the < relation within that larger ordinal.

(These examples can't be translated into ordinary integer inductions: there are far more ordinals than integers, and far more "ranks" of sets than integers. But we still have the no-infinite-descending-chains property!)

These are all the same

All of these work the same way. You have a kind of object (natural numbers, trees, lists, sets, ordinals), a notion of being smaller/simpler (comparison of integers/ordinals; subtree; tail of list; element of set), and a theorem or an axiom that guarantees that you can't keep getting smaller/simpler for ever. And, in that context, anything that's true of X whenever it's true of everything simpler than X is true of all X. And that's what induction really is.

What kind of requests would you like to get? Specific questions? Certain subjects? etc.

The more specific the better, basically.

One thing missing: what is the intuitive "geometric" interpretation of matrix transpose? Considering the transpose is used all the time, I still don't have an intuitive sense of what it does.

It's unclear to me what counts as a "geometric" interpretation here. The coordinate-free interpretation is that taking transposes is the coordinate-dependent version of taking the adjoint of a linear operator between inner product spaces. There's also an interpretation in terms of SVD which may be more explicitly geometric: taking transposes swaps the left and right singular vectors. (This is one way to think about the spectral theorem: it means a symmetric matrix is a matrix whose left and right singular vectors agree, which means they're eigenvectors. This isn't quite a proof because singular vectors aren't quite unique but it's pretty close.)

An intuitive explanation of the SVD.

I wrote a blog post about SVD; not sure whether it will give you what you want but I like it.

an explanation of the difference between transfinite induction and regular induction. I.e. IIRC, there needs to be defined a whole new kind of induction in order to do transfinite induction vs the base induction used to define things a little past N. Why is another mechanism necessary?

As gjm says, ordinary induction is a special case of transfinite induction so it's not exactly a whole new kind of induction, just a (pretty mild, all things considered) generalization.

I second this; I too would appreciate a Qiaochu’s-eye view. Also, are there hidden dependencies in the MIRI reading list? I feel like some subjects would be most profitably studied before or after others - category theory is mentioned as helping you understand the underlying structures of mathematics and generalize your understanding - so does that mean read early, or read near completion of the list?

I'd also add to 2: an intuitive explanation of eigenvectors and their properties in general. They seem to pop up everywhere in Linear Algebra and when then do, people gesture towards a set of intuitions about them that I haven't managed to pick up.

On 3: when doing a mathematics degree, you will often get a fair bit of choice of what subjects to study in depth. So any list of subjects-you-need-to-know will need to be somewhat variable -- or else end up covering more than most people would do in their actual undergraduate studies. Could you say more about what areas you're interested in?

I did indeed mean that. Nice catch!