/sci/ - Science & Math

Is the definition of [math] \otimes_R [/math] via

[math] \operatorname{Hom}_{\Z} (M \otimes_R N, G) \simeq \operatorname{Hom}_R(M, \operatorname{Hom}_{\Z} (N, G)) [/math]

good or are others more natural to gain an intuition and think of canonical examples of tensor product of modules?

>>10832772
I mean for the OP, if he doesn't realize.

>>10832769
I mean that's a philosophical question that always ends up with a question of meaning of life in general.
What is useful research? Useful to whom? Science? Civilization? Is scientific progress necessarily good for humanity? Etc.
I do R&D at work already and I like history of mathematics I guess. Also would give me a trajectory to work out personal favorites to formalizations of mathematical frameworks.

The exponential in it's forms covers a lot of different fields and has some clean and completely worked out properties that are at the same time of interest to many practical questions, so I collected some ideas on how to write a book around it.
My favorite math history book is
https://www.amazon.com/Modern-Algebra-Rise-Mathematical-Structures/dp/3764370025
which captures how the notion of the field of algebra has evolved from Galois to Grothendieck. The discussions around Nothers work and contemporary of Hilbert are a bit tiresome (because nobody knows of all the math that didn't work out or found interest covered there) but I'd be willing to dig up some biographical work and tie it together. It's also a form of research, really, albeit more of a histographical one.

How do I get my uni to pay me for my work, providing me a research assistant position or something like that? I'll greentext so it's shorter
>2 year undergrad
>gets into professor lab just by asking him
>see and eventually assist experiments
>we get along well, I show up multiple times weekly and often stay up late, even when doing paperwork, just cause I genuinely enjoy it (not required work, he didn't ask me to come and no one is paying me or something)
>near the end of 2year
>working on what will be my thesis (his research obv) when suddenly gets an idea
>tell him, he says it's interesting and says he will help me carrying out my research, so new thesis

Now, I don't want to sound arrogant, but I believe in the few ideas I have, and I think it will give results and could be published easily.
And that's for my thesis. This study could possibly have some interesting developments (new antitumoral proteins) which hopefully I'll investigate after the first paper (thesis=isolation and characterization of those new proteins). But the idea of doing all this work, all those hours for free really bums me out. After I prove them (and myself) I can do reaserch with the first paper and tell them I have other ideas, will they consider me? Maybe after I graduate? Should I or should I not publish like that, for the sake of boating my cv for future occasions? I'm lost, I need your help guys

>>10763216
>>10763232
Thanks guys, I feel a bit more motivated.
I guess networking is one of the few things I'm doing well, weirdly enough almost no one else in my course is doing it, it seems everyone wants to be a nutritionist lol.

Is it worth to get a master degree in a general subject like molecular bio in order to stay close to one professor you are working with and could provide you with connections and opportunities? If it was for me I would go somewhere else and get a more specific master degree but this professor will let me keep publishing with him if I stay in that uni, do you think it's worth it?