# 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 ?

#### 2 Yuhao

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

(One way is to PICK a function in and we get a ramified morphism , 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 denote the cyclotomic field obtained by adjoining th roots of unity to , where is prime. Let be a prime element in the ring of integers of which is coprime to . Let denote the Kummer extension of by adjoining a th root of to .

Question: is it always true that divides the exponent of the prime ideal in the relative discriminant ? (as usual, 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 mathoverflow.net question: [1]

#### 7 Darsh

Let be rational.

In what generalized notions of convergence does the sequence 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 ?

#### 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.