Question 1 is asking, if you perform a delay-coordinate embedding to reconstruct the dynamics of a system from a scalar time series

And you assume that all the conditions of the embedding theorems are true, and that you did all the measurement properly

Then are we guaranteed the reconstructed dynamics have the same geometry, or the same topology?

We are guaranteed that it has the same topology, so part b is Yes and part a is No

Recall from 8.2 that this embedding, for example, has the same topology, but clearly  from a visual inspection  does not have the same geometry as the Lorenz attractor

The theorems of delay-coordinate embedding only guarantee correctness of topology, not of geometry

For Question 2, These two shapes have similar geometry, and this is true, theyre both bowls

Essentially, they have the same shape that we would see visually

Geometry doesnt care how many holes are in an object, so the fact that this colander is full of holes is not important to geometry

But the fact that they have the same visual appearance, or the same geometry, is whats important

So this question is true

Question 3 asks if these two shapes  a bowl and a donut  have the same topology

And this is false

The donut has a hole in it, whereas the bowl does not, so the donut and the bowl have different topologies

Question 4 asks if these two shapes  the coffee mug and the donut  have the same topology

This is a very famous example in topology

Because the coffee mug can be continually deformed, without piercing any holes, to the donut, and vice versa, a coffee mug and a donut have the same topology

So this is true

For Question 5, does a bowl and a colander have the same topology?

And this is false

The colander has hundreds of holes in it, and so it has very different topology than a bowl

Although they do have the same geometry

So this question is false

Question 6 states that you need to have direct, untransformed measurements of at least one state variable for delay-coordinate embedding to work, and this is false

You only need to measure a smooth, generic transformation of at least one state variable for delay-coordinate embedding to work

Recall the examples from the lecture in 8.2, where you actually measured x * y  z in the Lorenz equation

This is definitely a transformed measurement of all state variables, and that worked just fine