# To Do List

(Difference between revisions)
 Revision as of 23:45, 4 March 2012 (view source)Luis (Talk | contribs)← Older edit Revision as of 21:51, 12 March 2012 (view source)Luis (Talk | contribs) Newer edit → Line 1: Line 1: + == Things that need to be done == + We need to come up with some organization for the articles. We need to come up with some organization for the articles. Line 4: Line 6: * 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. * 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. + + * Some discussion on [[Dirichlet form|Dirichlet forms]], and maybe some models from [[nonlocal image processing]]. + + * Fractional curvatures in conformal geometry. + + * 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]]..) + + * 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. + + * Limit of $\sigma$-order operators as $\sigma \to 2^-$, maybe this should go on a page on [[Linear integro-differential operators]]. + + * 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 image processing]] + + * [[Aggregation equation]] + + == (partially) Completed tasks == * A sort of [[Starting page]] that serves as a "root" for all pages we add (ideally, any page we create should be reachable from here). (UPDATE: See discussion.) * A sort of [[Starting page]] that serves as a "root" for all pages we add (ideally, any page we create should be reachable from here). (UPDATE: See discussion.) * Definition of [[viscosity solutions]] for nonlocal equations. Also a discussion on existence using [[Perron's method]] and uniqueness through the [[comparison principle]]. * Definition of [[viscosity solutions]] for nonlocal equations. Also a discussion on existence using [[Perron's method]] and uniqueness through the [[comparison principle]]. - - * Some discussion on [[Dirichlet form|Dirichlet forms]], and maybe some models from [[nonlocal image processing]]. * Some general regularity results like [[holder estimates]], [[Harnack inequalities]], [[Alexadroff-Bakelman-Pucci estimates]], some reference to [[free boundary problems]]. * Some general regularity results like [[holder estimates]], [[Harnack inequalities]], [[Alexadroff-Bakelman-Pucci estimates]], some reference to [[free boundary problems]]. Line 21: Line 41: * A page about [[semilinear equations]] including the [[surface quasi-geostrophic equation]] and also some form of KPP. * A page about [[semilinear equations]] including the [[surface quasi-geostrophic equation]] and also some form of KPP. - * [[Nonlocal porous medium equation]] and its more regular sister, the [[Aggregation equation]]. + * [[Nonlocal porous medium equation]]. * [[drift-diffusion equations]]. * [[drift-diffusion equations]]. - * [[Nonlocal minimal surfaces]] and [[Nonlocal mean curvature flow]] (specially in connection to [[nonlocal image processing]]). + * [[Nonlocal minimal surfaces]] and [[Nonlocal mean curvature flow]]. - + - * Fractional curvatures in conformal geometry. + - + - * 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]]..) + - + - * 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. + * There is also plenty of work on [[Dislocation dynamics]] that we ought to add later on. * There is also plenty of work on [[Dislocation dynamics]] that we ought to add later on. - - * Pages about [[Homogenization]] (local and nonlocal) should appear here too. - - * Limit of $\sigma$-order operators as $\sigma \to 2^-$, maybe this should go on a page on [[Linear integro-differential operators]]. - - * 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]]. * Phase transitions involving non-local interactions, in particular, pages about [[Particle Systems]], discussing the Giacomin-Lebowitz theory and the Ohta-Kawasaki functional. * Phase transitions involving non-local interactions, in particular, pages about [[Particle Systems]], discussing the Giacomin-Lebowitz theory and the Ohta-Kawasaki functional. Line 49: Line 57: classification of nonlocal minimal cones, etc... classification of nonlocal minimal cones, etc... - *Fill up the list of [[upcoming events]] such as conferences, workshops, summer schools. + * Having a [[list of equations]] may make it easier to navigate the wiki. - * I have started a [[Literature on Nonlocal Equations]] to dump there all papers we want to reference or are already referencing on the wiki. + == Other ideas == - * Having a [[list of equations]] may make it easier to navigate the wiki. + + * Fill up the list of [[upcoming events]] such as conferences, workshops, summer schools. + + * [[User:Nestor|Nestor]] has started a [[Literature on Nonlocal Equations]] to dump there all papers we want to reference or are already referencing on the wiki.

## 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.
• 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.
• 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.

• A sort of Starting page that serves as a "root" for all pages we add (ideally, any page we create should be reachable from here). (UPDATE: See discussion.)
• Phase transitions involving non-local interactions, in particular, pages about Particle Systems, discussing the Giacomin-Lebowitz theory and the Ohta-Kawasaki functional.
• We may want to have a section on open problems, for example integral ABP for general kernels, harnack for kernels with a

less restrictive bound below, $C^{1,\alpha}$ estimate in bounded domains with a nonsmooth kernel, supercritical quasi-geostrophic, classification of nonlocal minimal cones, etc...

## Other ideas

• Fill up the list of upcoming events such as conferences, workshops, summer schools.