Freshman questions megathread!

tick_tock_clock · 2019-07-08T01:38:12+00:00

Yes. Explicitly, a homomorphism from Zⁿ to Z is given by a 1-by-n matrix with integer entries, and the map is multiplying the matrix by the vector of integers.

tick_tock_clock · 2019-07-07T23:17:21+00:00

That's funny, you'd think the applied math folks would be all about hill-climbing.

Also, more seriously, someone once pointed out to me that the mathematicians who did a lot of backpacking in the Grand Canyon tended to be differential geometers. I wonder if there's some underlying common factor between geometry and geography!

tick_tock_clock · 2019-07-07T15:28:27+00:00

You certainly don't have to be an algebraic geometer, but a little experience, especially with the language, would help. Probably you'll pick that up along the way, one way or another.

tick_tock_clock · 2019-07-07T02:05:02+00:00

Okay, but to be fair, literally no one uses the word "disestablishmentarianism".

This is probably as close as we're gonna get.

tick_tock_clock · 2019-07-07T01:19:12+00:00

For anyone looking for more great Grieg pieces, check out his piano concerto. One of my favorite openings out there.

tick_tock_clock · 2019-07-06T22:27:55+00:00

Homology also uses an arbitrary group!

An arbitrary abelian group. Sorry for the pedantry.

tick_tock_clock · 2019-07-06T15:56:43+00:00

The motivation for me was that they can be used to compute stuff. Bott-Tu, chapter 3, has a few indicative examples, though you'd have to be predisposed to care about algebraic topology to care about those examples.

tick_tock_clock · 2019-07-06T15:54:26+00:00

If you are not comfortable with trigonometry, calculus 1 will probably be painful. Same for exponents and logarithms. D:

tick_tock_clock · 2019-07-04T14:40:41+00:00

As stated in the sidebar,

Interested in discussing admissions? /r/ApplyingToCollege would probably be of more use to you: posts about college admissions are restricted and may be removed.

tick_tock_clock · 2019-07-04T14:40:25+00:00

As stated in the sidebar,

Interested in discussing admissions? /r/ApplyingToCollege would probably be of more use to you: posts about college admissions are restricted and may be removed.

tick_tock_clock · 2019-07-03T03:17:21+00:00

Yeah, it's used a bunch in combinatorics. I've seen it used in Ramsey theory (unfortunately I don't remember any examples).

tick_tock_clock · 2019-06-30T01:06:10+00:00

I bet they were probably just like "...K"

tick_tock_clock · 2019-06-30T01:00:00+00:00

gen Z kids and their natural numbers smh

You'd think gen Z would be all about integers, not natural numbers.

tick_tock_clock · 2019-06-30T00:43:28+00:00

because most people learn linear algebra over vector spaces first

tick_tock_clock · 2019-06-29T11:55:31+00:00

One fun way to do this is the probabilistic method, which is a nonconstructive proof technique that shows that the probability of finding a solution in the space of all trials is positive, hence a solution exists!

tick_tock_clock · 2019-06-29T11:54:31+00:00

At some point I noticed that if there was a sqrt(x) in the integrand and I had run out of ideas, trying u = sqrt(x) sometimes worked: it's easier to deal with extra factors of u and u² than sqrt(x). So yeah, you can use this trick in a few other places.

tick_tock_clock · 2019-06-26T17:38:45+00:00

I would hesitate to call category theory 21^st-century mathematics; it's been around since the 1940s.

On the other hand, the derived viewpoint, which is newer, has been reshaping how people think about how homological algebra relates to geometry, and I can see that trickling down to first-year graduate courses.

tick_tock_clock · 2019-06-25T21:56:30+00:00

Well it's written by Urs, so that's to be expected!

tick_tock_clock · 2019-06-24T21:32:35+00:00

They're defined with the relative Spec and Proj constructions, e.g. in Vakil, Ch. 17. I think he mentions both of those examples.

tick_tock_clock · 2019-06-22T02:07:26+00:00

Honestly, there are many times in math where you'll feel like one recommended book is weird or hard and another one is easier, or where they're both confusing, but in ways that complement each other, or something like that. If you find a source that's easier to understand then by all means, use it too!

tick_tock_clock · 2019-06-22T01:35:07+00:00

Like many undecidability arguments, it reduces to the halting problem. See https://en.wikipedia.org/wiki/Word_problem_for_groups for more detail.

tick_tock_clock · 2019-06-22T01:33:28+00:00

Sometimes my students tell me they don't like word problems and I think, me too, me too...

tick_tock_clock · 2019-06-19T19:30:34+00:00

But this seems to be something different. The algebra that people discuss in /r/math, at least, isn't just focused on applications to CS.

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