@MarianoSuárez-Alvarez I am reading this proposition in qing liu saying that $\text{Proj} ( B \otimes_A C)= \text{Proj} B \times_{Spec A} Spec C$ where $B$ is a graded $A$ - algebra
It seems that isomorphism involving fiber products says exactly that the projective plane is the affine line times the projective line @MarianoSuárez-Alvarez
I am currently studying the chapter entitled "Symmetry" from Michael Artin's book "Algebra" and am having some difficulties understanding the material. It is dealing with isometries, dihedral groups, ...
Would anybody recommend any other book for the lesson and somewhere were I can find problems...
I think the set of all binary operations $(a,b) (c,d) \mapsto (p(a,b,c,d), q(a,b,c,d))$, that preserve non-coprimeness, forms an ideal of $(\Bbb{Z}[x_1, \dots, x_4])^2$.
Hi. There are two water containers A & B. A has radius r and B has radius r/2. They are both put out in rain. Assuming both are getting equal amount of rain, what would be the ratio of amount of water in both containers?
From Michael Artin's Algebra: " The char polynomial of an $n \times n$ matrix $A$ has the form $p(t) = t^n - (trace A) t^{n-1} + \text{ (intermediate terms) } + (-1)^n(det A)$. There is a formula for the cubic term coefficient. You have to find that out.
@Kevin, What horrifies me is the idea that someone might use noncommutative field theory to resolve the naked singularity in the 2d dilaton gravity black holes. I can feel it is coming.
@Irina My advisor and I have alreayd generated an accurate numerical solution to this problem, but I have spent the last 4 months trying to develop an approximate analytical solution with basically no progress. My advisor seems totally happy with a numerical solution, but it annoys me. I have 1 or 2 ideas left to try and if that doesn't work I'll be moving on ot something else
@Irina It can be. My advisor has a very good intuitive grasp of mathematics so he can kind of 'cheat' and get the right answer. I'm not so experienced so I have to do things the rigorous(ish) way. Which means I spend a lot more time thinging about some technical math questions
@Kevin, I heard a nice review of attempts to find theories where the entire black hole entropy can be understood in terms of entanglement of matter fields in the background (related to "induced gravity"), and of the disasters that befell information in 2d dilaton gravity.
I am studying things, but idea comes in time, I am not much experienced with strings, but trying my best Kevin :)
@Irina It seems that it'd be nice if the entropy can be explined in terms of entanglement, but there do seem to be some serious problems with that idea. To salvage it, it seems that we'll either have to totally redefine what we think happens at the event horizon, or admit that the equivalence principle fails, or admit that there is information loss in the universe. None of which seems particularly nice.
I think people are silent because no one knows what's going on (as Polchinski admitted during his talk on the issue)
UNless you mean in the chat. In that case, its just cuz its the middle of the day and they've been overrun by physics students!
Particularly, the talks by CM theorists about topological entanglement entropy, classification of topological phases, and open questions in their field more generally were inspiring me.
@Irina Yes. It is something of a blessing for people working on this, becuase even as we exclude more and more of the supersymmetric theories at the LHC, a whole new field has been opened up that uses holography. I just heard a talk about this the other week by Brian Swingle (who is at Harvard)
He has been at least somewhat trying to incorporate tensor networks and holography (Juan Maldacena is doing something similar I think)
Haha, I see. At some point in the next few years I'll sit down and really hammer out this groups stuff. BUt its gonna be hard and I dont have any use for it yet
@Kevin, history has shown that down the path of wormholes lies nothing but madness, despair, and bad science fiction movies. People are resisting bravely, for now... :-)
@Irina Yes, the only thing I see people suggesting now is that there may be non-travesable wormholes. You can tgo through them or sue them to send information, but perhaps htey entangle the inside of the BH to some other region of space (perhaps the boundary even)
