Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeNov 20th 2011
   • (edited Nov 20th 2011)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI saw a question on MO that intrigued me. If one has a polynomial of one variable and looks at the symmetries you get the Galois group of the corresponding extension. What happens if you take a polynomial in two variables? The inclusion of the field k into k[x,y]/ f(x,y) induces a map of schemes and my instinct is to look for the object of symmetries of that, but it should be higher-dimensional, hence it might be more like a 2-group than a group. I may be very wrong on this, but wondered if anyone who was in on the discussions on 2-geometry has seen anything like that. It seems to be to be like a fibration with some curves as fibres. Any thought would be very welcome. :-)

   I saw a question on MO that intrigued me. If one has a polynomial of one variable and looks at the symmetries you get the Galois group of the corresponding extension. What happens if you take a polynomial in two variables? The inclusion of the field k into k[x,y]/ f(x,y) induces a map of schemes and my instinct is to look for the object of symmetries of that, but it should be higher-dimensional, hence it might be more like a 2-group than a group. I may be very wrong on this, but wondered if anyone who was in on the discussions on 2-geometry has seen anything like that. It seems to be to be like a fibration with some curves as fibres. Any thought would be very welcome. :-)

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 20th 2011
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexMinhyong Kim has some ideas on a Galois theory for two variables. Follow the link to the talk from [here](http://golem.ph.utexas.edu/category/2009/12/galois_theory_in_two_variables.html).

   Minhyong Kim has some ideas on a Galois theory for two variables. Follow the link to the talk from here.

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeNov 20th 2011
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexWill do, Ta.

   Will do, Ta.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 20th 2011
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI see one can't drag the cursor to the end of the talk. Now I'm wondering how much Minhyong says about two variable Galois theory. Perhaps it's only just briefly mentioned. (I was there, but it was a while ago.)

   I see one can’t drag the cursor to the end of the talk. Now I’m wondering how much Minhyong says about two variable Galois theory. Perhaps it’s only just briefly mentioned. (I was there, but it was a while ago.)

5. • CommentRowNumber5.
   • CommentAuthorTim_Porter
   • CommentTimeNov 20th 2011
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThe cursor drag works well for me????? The two variable Galois theory is lurking behind all this I feel. I not he uses conjugation action several times and that always hides a homotopy , but in the particular cases that I undestood, I could not see the crossed module behind the construction. but it may be that it is a groupoid on which the crossed module is acting or something like that. That begins to feel 2-torsorish. ;-)

   The cursor drag works well for me????? The two variable Galois theory is lurking behind all this I feel. I not he uses conjugation action several times and that always hides a homotopy , but in the particular cases that I undestood, I could not see the crossed module behind the construction. but it may be that it is a groupoid on which the crossed module is acting or something like that. That begins to feel 2-torsorish. ;-)