#spectracled asks

[spectracled-eider]

8! Least favorite notation you’ve ever seen

Einstein Summation Convention. I maintain that if you need to add a phrase to the effect of ''where we're using ESC", it is not good notation.

Thanks for the ask!

[spectracled-eider]

*conjugates you*

My imaginary part! :(

[spectracled-eider]

It doesn't matter if it's dis or dat duck? (5)

If you're making a reference to something then I'm not aware of the reference. If you are literally asking about the new theme of my blog, yes it does matter because the url is a combination of the name of the species (spectacled eider) and spectra

[spectracled-eider]

What is your favorite proof (if you have one)

I'm a big fan of proofs that make use of what is described as the "Mayer-Vietoris Argument" in Bott-Tu. The general framework is we want to show that a particular map in (co)homology is an isomorphism for manifolds, e.g. Poincaré Duality or De Rham's Theorem. The idea is to show that first the map is an isomorphism for open sets of ℝⁿ and build up to showing that it holds for all manifolds. The key step is to use the Mayer-Vietoris sequence in combination with the Five-Lemma to conclude that if a manifold has a cover of two open sets which are homeomorphic to open subsets of ℝⁿ and so is their intersection, then the map is an isomorphism on the manifold too. Then we do an induction argument on the cardinality of an open cover where each open set is homeomorphic to an open subset of ℝⁿ and so is any pairwise nonempty intersection of open sets in the cover. This proves that the isomorphism for manifolds with such a cover that is finite, but a lot of manifolds do, e.g. all compact manifolds do. In fact all (smooth) manifolds admit "good" covers, they just aren't necessarily finite. You then have to do more to make sure we get all manifolds but this is the important part.

I'd say the easiest example to find of this is the proof of de Rham's theorem in Intro to Smooth Manifolds by Lee though I'm not sure it's the most accessible in terms of prerequisites. Maybe one day I'll give a proof so one of these theorems that use this.

Thanks for the ask!

[spectracled-eider]

For the math ask game: 8, 23, 47

8. Least favorite notation you’ve ever seen?

As answered here it's Einstein Summation Notation

23. Will P=NP? Why or why not?

From everything I understand about the problem, I don't think so. I also hope it isn't (though even if it is, that doesn't imply that finding the algorithms which solve NP problems in polynomial time is in anyway easy)

47. Just how big is a big number?

As my friend would say: at least 3. More seriously, it'd be one that can be defined using a sequence that can't be computed in finite time. (There's a fancy word that I've forgotten now)

Thanks for the ask!

[spectracled-eider]

Hey, just thought you'd be happy to hear that your posts have inspired me to pick maths back up! I got an MMath a couple years ago but focused on work for a while since I can't justify the cost of a PhD atm over other things. I've just bought one of the main reference books for my dissertation (on Symmetry in Hamiltonian Systems) with the intent of reading all the bits I didn't read for my Masters :) Thank you for the inspiration

!!!! I'm also happy when I am the reason people start/get back into doing maths! :)))

From the looks of the contents page, you might be interested in the posts I will eventually make about my 4th year project!

[spectracled-eider]

what bit of undergrad-level math would you most recommend teaching to receptive children circa thirteen years of age?

if, purely hypothetically, one were to get the chance to do so?

Probably something like modular arithmetic. It's quite easy to demonstrate and isn't too far fetched from things they'll probably already know but it is just weird and different enough to peak their interest. Plus you can do the good old trick of comparing it to how a clock works

[spectracled-eider]

math ask game: 49, 62

49. What’s your favorite number system? Integers? Reals? Rationals? Hyper-reals? Surreals? Complex? Natural numbers?

Complex numbers easily. Complex analysis has some wonderful results!

62. Are there any non-interesting numbers?

I mean, if we declared a number to be non-interesting because it didn't have any/many common nice/interesting properties surely it would then be interesting because why doesn't it exhibit interesting properties? Surely the absence of anything interesting would in and of itself be something interesting?

Thanks for the ask!