ct.category theory - What is a deformation of a category?

I have several naive and possibly stupid questions about deformations of categories. I hope that someone can at least point me to some appropriate references.

What is a deformation of a (linear, dg, A-infinity) category? Is it a "bundle of categories" over a scheme? How can you make such a notion rigorous? Maybe via stacks? Suppose we take some nice scheme $X$ and we consider $D^btext{Coh}(X)$; if we deform $X$, then do we also get a corresponding deformation of the derived category? What does this corresponding deformation "look like"? Are we deforming the morphisms? The objects? Both?

Similarly, what about in the situation where we have a category of modules over an algebra $A$? If we deform the algebra, then do we also get a corresponding deformation of the category? Again, what does it "look like"?

And finally, in either of the above cases, are there deformations of the respective categories that don't correspond to deformations of $X$ or of $A$ respectively? I expect the answer to be "yes"; then my next question is: Are there any nice examples of such deformations that can still be described in an explicit or geometric way?

I am most of all interested in concrete examples, and less interested in general theory.

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Edit 1: I probably should have mentioned this when I first posted this (almost 3 months ago now!), but somehow I forgot. Kontsevich has been at least implicitly talking about deformations of categories since at least 1994, in the original paper introducing homological mirror symmetry. The idea (or "philosophy") seems to be that the deformation theory of a category should have something to do with its Hochschild (co)homology. But I still do not understand this connection, at least in any sort of generality. Perhaps this is explained in some of the papers already listed in the answers below --- what I'd most like to see is how to relate "deformation of a (linear/dg/A-infinity) category", however one defines it, to Hochschild (co)homology.

Perhaps it's somehow obvious... but I'm pretty dense and would like to see it spelled out...

So I hope someone can explain this to me, or point me to a spot in a paper where it is explained.

I'm adding a bounty to this question just for the heck of it.

Edit 2: See my answer below.

Edge-maximizing projective transformation on polytopes

Let P and Q be simple polytopes such that P = Q ∩ H and let H be a halfspace with normal vector n. Let proj_n(e) denote the length of the projection of edge e onto vector n.

Consider the set E of edges of Q that cross through H, ie, edges with one endpoint in H and one outside H.

For any e ∈ E, does there always exist a projective transformation φ of P such that for every f ∈ E, proj_φ(n)(φ(e)) ≥ proj_φ(n)(φ(f))?

In other words, is there always a projective transformation making an arbitrary edge crossing H maximal with respect to H?

My first approach would be to choose φ to extend the endpoint of e as far as possible, but this does not appear to be sufficient to guarantee that proj_φ(n)(φ(e)) is maximized without adjusting other edges of Q.

ag.algebraic geometry - What are examples illustrating the usefulness of Krull (i.e., rank > 1) valuations?

Dear Pete,

This is a fairly general comment, rather than a precise answer:

General valuations were studied very deeply by Zariski, who used them as one of the
cornerstones of his investigations of birational geoemtry (including various forms
of his Main Theorem, resolution of singularities, and so on). His picture was that
different valuation rings (dominating the given local ring $mathcal O_P$ at a point
$P$ in a surface $X$, say) correspond to different germs of curves passing through the point $P$. The rank one valuations are just the germs of algebraic curves passing through $P$,
but the other valuations are like transcental curves that give additional information.

I believe that Krull himself didn't see that higher rank valuations would be related to geometry, but that it was Zariski (who studied Krull very carefully) who saw the applicability, and introduced them (and the closely related concept of normality) into
algebraic geometry. (I believe that Zariski writes about this somewhere, although I can't
remember where.)

This point of view persisted in algebraic geometry up until the Grothendieck revolution.
Maybe one of the last results proved using this view-point was the Nagata embedding theorem.

Apparently Grothendieck was very unhappy with valuation theory (and indeed tried to keep it out of Bourbaki's commutative algebra texts, without success), and the only vestige that survived in his view-point was the valuative criteria for separatedness and properness.

Let me close by using this soap-box to encourage people to read Zariski's papers. They are quite wonderful.

EDIT: I just remembered that Lang's book on algebraic geometry, which is a kind of an abridged version of Weil's Foundations, uses valuation theory at various points. (I remember that it is treated at the beginning of the book in the section dealing with foundational concepts in algebra, but I don't remember now exactly what he proves with it later on. But it is a fairly short book, so one could flip through it and see. The whole book doesn't go all that far, and so it's possible that it appears just because it was so endemic to the commutative algebra and algebraic geometry of that time period, rather than because he does anything particularly special with it.)

EDIT: After thinking about this question, I realized that the above statement about rank one valuations corresponding to germs of algebraic curves passing through a point is misleading. See my answer
here for a (hopefully) more correct discussion of the geometric intution behind various valuations.

cell biology - number of RNA pol II molecules in a given human nucleus?

it is highly variable even in a single cell type.

Dont know for a human cell but this is the range for ecoli: 1500-11400

check this site out. got this number from there (its quite useful for questions like these):

http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=101440&ver=3&trm=RNA%20polymerase

You can estimate protein copy numbers and there are techniques available for doing so. One of such techniques:

http://www.mcponline.org/content/11/3/O111.009613.long

You can also calculate concentration using traditional techniques like western blot and elisa and calculate approximate copy-numbers (you have to also calculate average cell volume and number of cells for protein extraction.

galois theory - When is sin(r pi) expressible in radicals for r rational?

Perhaps this question will not be considered appropriate for MO - so be it. But hear me out before you dismiss it as completely elementary.

As the question suggests, I would like to know when $sin(ppi/q)$ can be expressed in radicals (in the way that $sin(pi/4) = sqrt{2}/2$ and $sin(pi/3) = sqrt{3}/2$ can). Let $alpha = sin(x)$, and consider the field extension $mathbb{Q}[alpha]$. Using $(cos(x) + isin(x))^k = cos(kx) + i sin(kx)$ together with the binomial formula and the Pythagorean identity relating sine and cosine, we can see that $sin(kx)$ lies in a solvable extension of $mathbb{Q}[alpha]$. Thus $sin(ppi/q )$ is expressible in radicals if $sin(pi/q)$ is.

To handle $sin(pi/q)$, we start by using the same trick (which most people also learn in an elementary trig class). Write $-1 = (cos(pi/q) + isin(pi/q))^q$, use the binomial theorem to expand, compare imaginary parts, and express the right-hand-side in terms of sine using the Pythagorean identity. This gives an explicit equation for any $q$ one of whose solutions is $sin(pi/q)$. This equation is not a polynomial in $sin(pi/q)$ since it involves terms of the form $sqrt{1 - sin^2(pi/q)}$, but it is enough to prove that $sin(pi/q)$ is algebraic.

So I am curious about the number theoretic properties of this equation. What can be said about the Galois group of its "splitting field" over $mathbb{Q}$? Can we at least determine when it is solvable? Note that if the prime factors of $q$ are $p_1, ldots p_k$ and we can express each $sin(pi/p_j)$ in radicals, then the same is true for $sin(pi/q)$. So it suffices to consider the case where $q$ is prime. That's about all the progress I have made.

molecular biology - How long can I store extracted RNA?

I keep my RNA in 1mM sodium citrate pH 6.4. Citrate is a chelator, and helps trap the divalent metals many RNAses need to work. The lower pH also helps inhibit RNAse activity. EDTA could work as a chelator, but it only chelates well at pH's where RNAse activity is worse. Citrate gives both chelation and low pH.

Of course I also keep my RNA at -80.

"Fermat's last theorem" and anabelian geometry??

Do I remember a remark in "Sketch of a program" or "Letter to Faltings" correctly, that acc. to Grothendieck anabelian geometry should not only enable finiteness proofs, but a proof of FLT too? If yes, how?

Edit: In this transcript, Illusie makes a remark that Grothendieck looked for a connection between "FLT" and "higher stacks". BTW, here a note on (acc. to Illusie) Grothendieck's favored landscape.