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, theyre both bowls Essentially, they have the same shape that we would see visually Geometry doesnt 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 whats 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