# To Do List

(Difference between revisions)
 Revision as of 17:15, 17 November 2012 (view source)Nestor (Talk | contribs)← Older edit Revision as of 17:16, 17 November 2012 (view source)Nestor (Talk | contribs) Newer edit → Line 11: Line 11: * Fractional curvatures in conformal geometry. * Fractional curvatures in conformal geometry. - * Create a page on [[Non-local Lagrangian | Non-local Lagrangians]] and mention relevant examples. + * Create a page on [[Non-local Lagrangian | Non-local Lagrangians]] and mention relevant examples, note Non-local Lagrangian is a term used by particle physicists and it is not necessarily the same as a [[Dirichlet form]]. * We need  to explain further the  [[Extension technique]] and its connection with fractional powers of the Laplacian and Conformal geometry. Required background: [[Geometric Scattering Theory]], [[Ambient Metric Construction]], [[GJMS Operators]] and [[Singular Yamabe Problem]]..) * We need  to explain further the  [[Extension technique]] and its connection with fractional powers of the Laplacian and Conformal geometry. Required background: [[Geometric Scattering Theory]], [[Ambient Metric Construction]], [[GJMS Operators]] and [[Singular Yamabe Problem]]..)

## Things that need to be done

We need to come up with some organization for the articles.

The list below can be a starting point to click on links and edit each page. The following are some of the topics that should appear in this wiki.

• We better start thinking hard about writing the Introduction to nonlocal equations. We gotta start somewhere, so any random idea or small thing you want to write should go in there. This is high priority, since if someone reads one page of this wiki, it will likely be this one.
• Fractional curvatures in conformal geometry.
• Create a page on Non-local Lagrangians and mention relevant examples, note Non-local Lagrangian is a term used by particle physicists and it is not necessarily the same as a Dirichlet form.
• It would be wise (once the wiki is more mature) to add pages about the Boltzmann equation, since it is one of the more "classical" and better known integro-differential equations.
• Pages about Homogenization (local and nonlocal) should appear here too. (CITATIONS still needed)
• Given the recent works of Osher/Gilboa and Bertozzi/Flenner on Ginzburg-Landau on graphs we should have an article on the natural similarities between non-local operators and elliptic operators on graphs.
• Nonlocal Phase Field Equations this will include many types of evolution equations such as nonlocal reaction-diffusion, models arising from nonlocal ising models, nonlocal cahn-hilliard, nonlocal biological aggregation, sharp interface limits (giacomin-lebowitz, bates-et-al, de massi-pressutti, etc...)
• Boundary Harnack inequality it would be good to possibly include this in the description of what is different/same regarding local vs. nonlocal equations.
• Describe, with references, the connection between holder estimates of viscosity solutions of stationary and or parabolic equations and the existence and uniqueness of solutions to the relevant Martingale Problem
• Discussion of Jean Picard's paper "ON THE EXISTENCE OF SMOOTH DENSITIES FOR JUMP PROCESSES" (regularity for pure jump processes with non-degenerate, translation invariant generators)
• Add Courrège's Structure theorem for operators and other structure theorems (generators of Feller semigroups, Hunt processes... etc)