New Algorithm for Characteristic Classes

[gh-mjungmath]

The current algorithm for characteristic forms is comparably low. The worst case scenario showed computations times about 1h in dimension four.

With this ticket, we want to replace the current algorithm and implement characteristic cohomology classes as sub-ring of the de Rham cohomology ring.

The algorithms are based on the Faddeev-LeVerrier? algorithm. The idea is that characteristic cohomology classes are determined by an element of the polynomial ring over Chern classes, Pontryagin classes and/or Euler classes.

[mkoeppe]

New commits:

​3b36f0d	Trac #29570: alternating_form returns correct element
​85e7970	Trac #29570: Typo fixed, doctest added, returned element preferably non-zero
​1a803eb	Trac #29581: New Algorithm for Char Classes

[slelievre]

Please add a short description in the "Description" field of the ticket, and the author's full name in the "Authors" field of the ticket. Don't forget to set to needs_review when this is ready for review.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​730046c	Merge branch 'develop' into t/29570/diff_form_bug
​130ae3f	Trac #29570: NotImplementedError for non-generic ring elements
​bba21e1	Trac #29570: Strange typo fixed...
​ba3b4b9	Trac #29570: correct parent, vectorfield_module changes reverted
​841e1bf	Merge branch 't/29570/diff_form_bug' into t/29581/new_algorithm
​cb442eb	Trac #29570: new algorithm added and code cleaned

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​15c7341

Trac #29581: new algorithm added and code cleaned

[chapoton]

Correct syntax for crosslinks is

.. SEEALSO::

[chapoton]

• do not use this:

  if distinct_real == False
  

but

if not distinct_real

(at least in two places)

• The following change is wrong:

  -        - ``cmatrix`` -- curvature matrix
  -
  +            - ``cmatrix`` -- curvature matrix
           OUTPUT:
   
  -        - ``I/(2*pi)*cmatrix``
  +            - ``I/(2*pi)*cmatrix``
  

as one should not indent inside INPUT or OUTPUT blocks (because they end with only one colon). And moreover, one does want empty lines to separate these things.

This is also incorrect:

-            self._dual_exterior_powers[p] = ExtPowerDualFreeModule(self, p)
+                self._dual_exterior_powers[p] = ExtPowerDualFreeModule(self, p)

Why did you change the indentation ? it was ok.

[gh-mjungmath]

I am sorry. At this stage, this is just a draft. The indentations are typos coming from using alt+tab (at least I guess so). I will fix this soon.

[gh-mjungmath]

Thanks for your advice. :)

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​d372293

Trac #29581: Strange typos reverted

[gh-mjungmath]

An example for the application of the new algorithm would be really nice. For instance a Todd class or A-Hat class. But I have no idea for a suitable one. If anyone does, or at least knows someone who does, I would really appreciate it. Thanks! :)

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​c975bd2	Merge branch 'develop' into t/29581/new_algorithm
​19815c4	Trac #29581: Readability of one line

[mkoeppe]

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.

[mkoeppe]

Setting a new milestone for this ticket based on a cursory review.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​3c9b80a

Trac #29581: implement algorithms for chern, pontryagin and euler

[gh-mjungmath]

Do you think it is necessary to give those cohomology classes an own parent or would DeRhamCohomologyRing as parent suffice?

The point is that addition and multiplication is performed differently among those elements. Moreover, it is possible to compare them.