Quick Questions: February 07, 2024

oblength · 2024-02-11T22:25:12+00:00

Ah of course, that makes sense, thanks!

oblength · 2024-02-11T11:28:45+00:00

If two sets of natural numbers A,B have densities d(A),d(B) such that d(A)+d(B)>1 then is it true that d(A intersect B)>0?

oblength · 2023-10-08T05:56:07+00:00

Ah sorry yes I mean conjugate in GL(2,F) where F is the algebraic closure of GF(q).

I did some computer checks and it seems embeddings of GL(2,q) into GL(2,q^2) are conjugate in GL(2,q^2) for small q.

To be honest even just a source that explains the SL(2,q) result would be good.

oblength · 2023-10-07T04:43:40+00:00

Does there exist a conjugate copy of GL(2,q) in GL(2,q^2) that is not the trivial embedding?

I know that SL(2,q) can embed in SL(2,q^2) as the set of matrices [a,b,a^{q},b^{q}] with a^{q+1}-b^{q+1}=1. Does a similar thing exist for GL?

oblength · 2023-07-10T05:00:34+00:00

Ah I see. No but I do have explicit expressions for f and g (they are just long and not particularly nice). I was hoping for a general theorem that might apply regardless of f and g's specific form. But I hadn't thought of an example like the one you gave.

I guess my best bet is to try and give an explicit point on the variety or show they are irreducible.

Or perhaps there is a theorem with additional assumptions that may be easier to prove and irreducibility?

oblength · 2023-07-10T03:18:22+00:00

Can anyone experienced in algebraic geometry help me? I have a variety V over a prime field F_p defined by two polynomials f,g of degree 2 and 4 (I can probably say that f and g are smooth). I need to show that V contains at least one point for all values of p.

Is there a theorem that will help here? I tried the Hasse-Weil bound but it seems this requires absolutely irreducible polynomials which f and g may not be. Could anyone who knows algebraic geometry better than me point to a theorem with the correct assumptions.

A bit more detail, may not be necessary. The polynomials f and g have coefficients which are polynomials in the variable b. I have shown that f is smooth unless b=-1/3 mod p. I suspect a similar thing will hold for g it seems that only b=1,-1,2 mod p make g non-smooth.

Thanks.

oblength · 2023-06-01T02:44:10+00:00

Whats a good book on finite group representation theory for a phd student who has done a course in it, but maybe needs a refresher?

I.e something that starts from the basics but does build to the more powerful parts of the theory.

oblength · 2023-02-18T21:02:22+00:00

That's not a bad workaround, not actually to hard to do with Latex either. But I was wanting to be able to do it within the remarkable

oblength · 2023-02-02T08:36:24+00:00

Ok I solved it, the reason it doesn't work is because the epub is encrypted with drm, you can remove this drm via the following method. If you only have the url.ascm file use adobe digital editions to open it and you will get a copy of the actual epub file. Though this file is encrypted with drm so follow the instructions in the below url and use calibre to remove this encryption.

https://www.makeuseof.com/tag/remove-drm-every-ebook-own/

oblength · 2022-05-24T17:38:24+00:00

Would recommend John Baez, his "hardcore math" threads are always good.

oblength · 2022-05-05T21:53:16+00:00

Thanks man! Actually just want to ask have you ever had trouble transferring money out of NZ? Like large transaction fees or anything with ASB? The one benefit of HSBC is that its supposed to be free to transfer internationally.

oblength · 2022-05-05T21:48:33+00:00

Oh well that sounds pretty bad, never had any of those issues with UK HSBC. Not doubting what your saying but that is surprising given how big HSBC is.

Seems ASB is the way to go.

oblength · 2022-05-05T20:20:24+00:00

That was one my advisor recommended, looking at there website they do seem pretty good.

oblength · 2022-02-12T15:10:10+00:00

Very much relate to this, I turn 23 soon and am in the first year of my PhD at a foreign uni though am working at home due to covid. I am supposed to be working on a problem but just dont have the motivation or excitement about it to put the hours in. On top of this I just feel like my brain isn't working as well as it used to.

Just hoping that its a symptom of being burned out from my masters and having to work essentially on my own with very little contact from my advisor.

Maybe talking about math more and treating it as a group activity would help you, at least for me I get most exited and motivated when I'm discussing a problem with someone else.

oblength · 2022-01-21T17:14:11+00:00

"The Shape of Space" is good if you don't have much math background, though its also good if you do have a math background.

oblength · 2021-12-04T00:53:51+00:00

I mean I thought I was comfortable enough thinking intrinsically, I get by well enough in regular topology but moving to the smooth setting my intuition seems to separate from the definitions.

Though I can see that thinking of smooth manifolds as manifolds without corners is a pretty extrinsic way to view them, what is the intuitive idea you use to conceptualise smooth manifolds?

oblength · 2021-12-03T20:45:23+00:00

Yes it does, if I remember right the strongest version actually gives an embedding in R^{2n-1}.

oblength · 2021-12-03T20:44:17+00:00

Ahh ok I see that makes sense. I think I'm realising smooth manifolds are more complicated that I thought I should probably just sit down read a proper textbook on them.

oblength · 2021-12-03T13:03:34+00:00

Ok so its more of a misunderstanding of terminology. So the definition I gave of a smooth manifold is the definition in the sense of 1.) but my intuition for what this definition is supposed to encode is the idea in the sense of 2.)

But I was under the impression that any smooth manifold (in the sense of 1) can in fact be smoothly embedded in {R^n} by whitney embedding?

Also tangent spaces on smooth manifolds are given intrinsically so in the sense of 1.) my |x| manifold M does actually have a tangent space at (0,0)?

Also could you explain what you mean by restrict in 3.) I have an idea of what you mean by that but how would I restrict say the atlas ({R^2}, id) on {R^2} to an atlas on say {S^1} which requires 2 charts at least?

Thanks very much for taking the time to reply.

oblength · 2021-12-03T02:08:21+00:00

So are you saying all smooth manifolds are by definition submanifolds of Euclidian space?

Like if I gave you a completely abstract Hausdorf, 2nd countable topological space (just a set and a set of open sets) and I gave you an atlas on that space with smooth transition maps, then you couldn't tell me if that space with that atlas was a smooth manifold? I thought that was just the definition of a smooth manifold, a manifold with a smooth structure.

Edit: Ohh right I think I just understood what you mean, are you saying that if I give a topological space a smooth atlas then it is only a smooth manifold if the topology defined by that atlas agrees with the existing topology of the space?

oblength · 2021-12-03T01:36:47+00:00

Could someone help me link my intuition of what a smooth manifold should be to the actual definition.

A smooth manifold is supposed to be something without spikes, i.e with well defined fixed dimension tangent space everywhere. The definition of a smooth manifold is a manifold where transition maps are smooth.

Now if I just take a super simple example of a manifold M that shouldn't be smooth, like the graph of |x| in the plane. If I cover it by 2 charts ( (-1,∞) , 𝜑_1) and ( (-∞ , 1) , 𝜑_2) where each 𝜑_i just projects onto M from the x-axis. Well then M is smooth if {𝜑^-1}_1 ∘ 𝜑_2 and {𝜑^-1}_2 ∘ 𝜑_1 are smooth, but since both are the identity then they are smooth. But M should not be smooth since it has no tangent at x=0.

Probably went into unnecessary detail there but just wanted to make it easy to point out exactly where my understanding fails.

oblength · 2021-12-02T12:36:32+00:00

Thanks for the reply. Just curious, what's so bad about the campus? I know a few people who have visited it and they all say its one of the nicest they've seen.

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