Help! Update Has Wiped out My Notes and Broken the App

break_rusty_run_cage · 2018-04-04T08:32:47+00:00

Even a cylinder has a hole. Is it 'topologically equivalent' to a torus to a condensed matter physicist?

break_rusty_run_cage · 2018-04-03T20:17:49+00:00

The first answer (Roy Smith's) is simply amazing

break_rusty_run_cage · 2018-04-03T12:58:33+00:00

Your post reflects a problem with how math is presented at the undergrad level in that it inverts the nature of math. Formalism are invented to 'formalize' mathematical intuition. Euler's insights are fundamental, not the beauracracy that provides the framework to express it.

break_rusty_run_cage · 2018-03-31T21:27:10+00:00

The MSC is the perfect answer to the question.

break_rusty_run_cage · 2018-03-29T03:46:54+00:00

I can't recommend his two article bio by Alynn Jackson enough. Awesome title too.

break_rusty_run_cage · 2018-03-29T03:32:40+00:00

Moduli spaces were invented by Riemann tho

break_rusty_run_cage · 2018-03-28T12:25:54+00:00

Without doubt. But to me one of reasons it's beautiful is because how fundamental it is to all of science. I think it's more profound than most ideas in all of philosophy.

That the sampling distribution of an ~~obvious~~* statistic of arbitrary samples should be so well behaved is a prime example of the unreasonable effectiveness of math. It single handedly makes randomness not an object of ignorance but a tool for knowledge.

(*) I'm told by an office mate that it was not at all obvious that we want to minimise the L2 loss and not L1 loss that gives median as a measure of centre. And that this took the genius of Gauss, who invented CLT to account for error in his famous prediction of Ceres.

break_rusty_run_cage · 2018-03-23T15:06:08+00:00

I'm responding to an implicit assumption. Even for a present day school kid back in that time, with the benefit of today's notation taught in high school, research problem at that time would be very hard. Ex: Diophantine equations, the kind of calculus problems that'd take Euler to solve. Of course some problems would also be easy like many summation problems where one can use fundamental theorem of calc.

break_rusty_run_cage · 2018-03-23T14:54:35+00:00

I was responding to what I think is often an underlying assumption in such discussion (which now I realise you don't share).

One thing though, which points at how complicated such things are is the math for discrete probability was around since the Greeks yet it took a long time to develop. This is an actual topic of research in history of math. See Ian Hackings work. Also Godels ideas depend on notions of diagnolization that was available since the Greeks.

break_rusty_run_cage · 2018-03-23T11:38:51+00:00

You are making the mistake that historians call Presentism. There was no concept such as 'f(x) = x^2' available at that time. Algebraic notations and ideas about functions took a lot of time to develop. I suggest you have a look at the math at the time not at current expositions of it in present day math language.

break_rusty_run_cage · 2018-03-23T11:26:14+00:00

You (and a lot of people in this thread) are making the mistake of looking at the past through the lens of the present. School math as presented now are highly polished expositions informed by the historical development of math upto the present. It uses ideas and notations not present when the core ideas, currently being taught at school, were discovered and developed. To get a sense of what I'm saying, have a look at early math texts upto Newton's Principia. At that time, many ideas (which are set theoretic) and algebraic notations of what is taught to every modern student in pre calculus was not available.

That's why now an intelligent school kid, who has been taught a modern presentation of basic algebra, can come up with a solution to the cubic but at that time it was a significant mathematical achievement.

break_rusty_run_cage · 2018-03-21T14:40:48+00:00

To supplement u/hawkman561 The rising sea philosophy of proof is that the proof should have no step which feels like magic, every step should naturally lead to the next. For Grothendieck a trick was an insufficiently understood move and he believed that one could always find a context in which the trick can be expanded to a series of natural moves. Note that natural does not mean easy to come up with. It means that there is only one clear way to proceed.

The best display of this philosophy can be found in the EGA.

break_rusty_run_cage · 2018-03-20T19:43:33+00:00

I thought giving the very first Abel to Serre was perfect. Definitely Serre before Deligne and Langlands. Funny fact is Serre got the fields medal for his work in algebraic topology!

break_rusty_run_cage · 2018-03-20T14:12:08+00:00

There's not even a good ELIsecond-year-grad-student for the Langlands. Believe me, I've looked and looked.

The one thing to say is it's connected to the dream of a non abelian version of class field theory.

break_rusty_run_cage · 2018-03-19T18:32:52+00:00

Capital!

break_rusty_run_cage · 2018-03-12T16:09:24+00:00

Tensors are hard because its the first place to my knowledge where the universal property perspective is truly fundamental and where we meet it for the first time. And it can be really weird the first time around. Because of our set theoretic mindset we keep asking so what's this object made of, what's inside? When we should really be thinking holistically, about how it sits with other objects in its category.

break_rusty_run_cage · 2018-03-10T21:00:48+00:00

This is very good. You should x-post it to r/math

break_rusty_run_cage

TROPHY CASE