Hello everyone. Does anyone have some sort of intuitive idea about what a presheaf is? I'm doing the Yoneda lemma in class and I've still not really got a way to think about presheaves.
This is the context it's most natural to me in. It's simply a presheaf on the poset category of open sets of a space (with a single morphism $U \to V$ if $U \subset V$.) In this case, the way to think of it (IMO) is as local data; one has a set assigned to each open set, and an inclusion/refinement map when passing to a smaller open set...
@MikeMiller To every $U\subset B$, assign the set $\Gamma(U)$ of sections of $\pi$ over $U$, $\Gamma(U) = \{ s\colon U\to E : s\text{ continuous }, \pi\circ s = \operatorname{id}_U\}$.
@MikeMiller Well, I wouldn't think of them as sheaves usually. The other way round more, a sheaf is sort-of a covering (not quite in the topology sense), at least if well-behaved.
@Alyosha The functor here is $\Gamma$. We assign each open set $U$ to the set of sections of $\pi$ over $U$. Now, if $V \subset U$, then each section over $U$ restricts to a section over $\pi$. This is what $\Gamma$ does to maps.
I have to admit I don't have a very strong intuition for general presheaves; they're just functors, I don't really see how to get more out of it than that. In that case my understanding is heavily influenced by Yoneda.
I now have 3 wishes. (1) Solve my mental problems. (2) Go to grad school in the US. (3) Find a job in academia in the US. I hope they all come true. Pray for me.
@robjohn I would like to ask you a question. If one day you had to do something morally right to help others, yet it is illegal and you could be jailed or fined or blacklisted and monitored by the authorities, what would you do?
@JasperLoy To some degree, however, I think it is more of a spontaneous decision. If you put it to a thought out decision, it becomes more of a pragmatic decision and will usually come out to be more self-serving.
@robjohn Thank you for your response. I will ask some more people about it (telling them the details which I won't share in this chat) and then make a decision. If I disappear one day from this chat, you know I have gone to jail.
I don't mean to be spammy, but does anybody know what the notation F[G] means when F is a field and G is a set (maybe necessarily a group, I don't know)?
A problem in my algebra class: Let G denote a finite group and let n denote the number of conjugacy classes of G. Show that the center of C[G] has dimension n as a vector space over the complex numbers.
It's a little over my head right now. I need to just stare at it. It seems like staring at it always helps eventually; the intuition comes out of thin air, and then seems obvious. Ah, but that's just math.
Rings seem easier to deal with than groups, to me. Is that a common sentiment?
I'd assume so because it's less abstract, but now you're mucking with distributivity without guaranteed multiplicative commutativity, so there's weirder algebraic manipulation. Maybe it balances out.
In both cases, I think the easiest way to think about them are "Normal subgroups/ideals are the kernels of homomorphisms". It's just that the standard definition of ideal is a lot simpler than the standard definition of normal subgroup!
Not sure where to put this, but I just wanted to thank the MSE community for how much knowledge it's brought me. Without it I would not be anywhere as knowledgable at math as I am right now.
Taking the Math Subject GRE made me realize how much I learned from here.
It is a really great and supportive community with a lot of knowledgeable people. :) Hope your GRE went well. What are you hoping to go to grad school for?
@DanZimm What sources do you recommend for learning some PDE? I want to learn some, but have no background in PDE theory itself (but plenty in other areas of analysis).
@MikeMiller I think if you are solid in multivariable calc and somewhat proficient in ODEs you can pick it up and do the first couple chapters in the least
that's all you really seem to need for classical PDE theory
for more advanced it would help to know functional analysis and measure theory
and again be very proficient in calculus ;P
@usukidoll yes, you can also view it by seeing $\lvert x \rvert$ is an even function and $\sin$ is odd, so to make up $\lvert x \rvert$ you won't have any terms in the $\sin$ series. You can graph it with mathematica
I mean if you're looking at just the periodic extension you literally "copy" the graph over and over to your extended interval
Hmm, interested and intimidated. The thing is, I'm not a PhD student, and I'm not even in the math department, but the professor of the phd level matrix class I'm in right now thinks I should take it (cause I'm doing well in his class)
I like to be challenged, so I will probably take it, since my other choices for my schedule are all things I took as an undergrad. I was just curious about what to expect
@Fargle I'm hardly an expert there, I know what I know and I don't what I don't, which happens to be quite a lot. I'm an undergrad too (hopefully grad in a year)
It's neat to see the gears turning behind algebraic systems, from matrices to functions to sets to numbers, and see that they're all very similar mechanisms with different, characterizing properties.
Topology is fascinating as well. I've taken a particular interest in it.
I think I might eventually like to study knot theory or algebraic topology--both of those because of the influence of my mentor here, Dr. Thistlethwaite.
I'd like to obtain as broad and deep a basis as possible in my undergraduate years. I can take 3 math classes a semester all the way until I graduate, and I'll graduate in four years, so I can have intro grad-level knowledge of every subfield taught at my school.
It seems to me like a lot of the developments in mathematical history have been those that marry and/or blur the lines between two or more subfields of math.
What real analysis tools would you recommend me for getting the closed form of the integral below?
$$\int_0^{\infty} \frac{\log(x)}{\cosh(x) \sec(x)- \tan(x)} \ dx$$
What real analysis tools would you recommend me for getting the closed form of the integral below?
$$\int_0^{\infty} \frac{\log(x)}{\cosh(x) \sec(x)- \tan(x)} \ dx$$
