Here are the latest problems from the 20 Questions seminar for the query-hungry :)
Is it possible to explicitly define (without AC) a total ordering on a set with cardinality greater than ?
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?)
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)
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.
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: [http://mathoverflow.net/questions/63/can-a-coequalizer-of-schemes-fail-to-be-surjective]
Let be rational.
In what generalized notions of convergence does the sequence converge?
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 ?
Give a simple example of a (necessarily infinte dimensional) Lie algebra that is not the lie algebra of any (necessarily infinite dimensional) lie group.
[http://mathoverflow.net/questions/134 See mathoverflow.net].