2009 October 6th - questions

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1 Sune

Is it possible to explicitly define (without AC) a total ordering on a set with cardinality greater than \mathbb{R}?


2 Yuhao

Let K be the function field of an algebraic curve C. Then by the birational nature of the genus, it determines the genus g of the curve. How can we see g directly from the field?

(One way is to PICK a function in K and we get a ramified morphism C
\rightarrow P^1, then use Riemann-Hurwitz to compute the genus; However, I think there should be a more "intrinsic" way to see that, i.e. without picking a function in an arbitrary manner. The genus is a well-defined invariant for any extension of the ground field of tr.deg. 1, right?)


3 Anon

Determine explicitly a partition of the plane into two sets A and B such that neither A nor B contains (the image of) a non-constant continuous curve.


4 An H

Let K denote the cyclotomic field obtained by adjoining pth roots of unity to Q, where p is prime. Let q be a prime element in the ring of integers of K which is coprime to p. Let L denote the Kummer extension of K by adjoining a pth root of q to K.

Question: is it always true that p divides the exponent of the prime ideal (1-\zeta_p) in the relative discriminant D_{L/K}? (as usual, \zeta_p denotes a primitive pth root of unity)


5 Critch

In the categories Set and AbGrp (and I believe any topos), finte limits (e.g. kernels, products, equalizers...) commute with directed colimits (aka direct limits). Give an example category where this fails.


6 Anton

Can a coequalizer in the category of Schemes fail to be surjective? (Note: it must hit all closed points in the target, because otherwise the closed point could be removed to make the coequalizer smaller.)

See the question: [1]


7 Darsh

Let f: \mathbb{C}\cup\infty \to \mathbb{C}\cup\infty be rational.

In what generalized notions of convergence does the sequence x,f(x),f^2(x),\ldots converge?


8 Harold

In the category of smooth manifolds, when does the fibre product of two maps exist?

A) Necessary conditions

B) Sufficient conditions (e.g. that each map is a fibration).


9 James T

Which bounded linear maps on a Hilbert space are the exponentials of other maps? I.e., what is the image of the map exp: B(H)\to B(H)?


10 Pablo

Give a simple example of a (necessarily infinte dimensional) Lie algebra that is not the lie algebra of any (necessarily infinite dimensional) lie group.


11 Theo



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