New Algorithm for Characteristic Classes

[mjungmath]

The current algorithm for characteristic forms is comparably slow. 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 (cf. #31691).

The idea is that characteristic (cohomology) classes are generated by Chern/Pontryagin/Euler classes. The generators can be computed by an Faddeev-LeVerrier-like algorithm (cf. #30681).

Depends on #32270
Depends on #32272
Depends on #32396

CC: @slel @egourgoulhon @tscrim @mkoeppe @tobiasdiez

Component: manifolds

Keywords: characteristic_classes

Author: Michael Jung

Branch/Commit: ac5cbbf

Reviewer: Travis Scrimshaw, Eric Gourgoulhon

Issue created by migration from https://trac.sagemath.org/ticket/29581

[mjungmath]

Branch: u/gh-mjungmath/new_algorithm

[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

[mkoeppe]

Commit: 1a803eb

[slel]

comment:4

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.

[sagetrac-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

[sagetrac-git]

Changed commit from 1a803eb to cb442eb

[sagetrac-git]

Changed commit from cb442eb to 15c7341

[sagetrac-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

[mjungmath]

Author: Michael Jung

[mjungmath]

Changed keywords from none to characteristic_classes

[fchapoton]

comment:8

Correct syntax for crosslinks is

.. SEEALSO::

[fchapoton]

comment:9

• 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.

[mjungmath]

comment:10

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.

[mjungmath]

comment:11

Thanks for your advice. :)

[sagetrac-git]

Changed commit from 15c7341 to d372293

[sagetrac-git]

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

d372293

Trac #29581: Strange typos reverted

[mjungmath]

comment:15

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! :)

[sagetrac-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

[sagetrac-git]

Changed commit from d372293 to 19815c4

[egourgoulhon]

comment:112

#18036 is probably relevant here.

[tscrim]

comment:113

I think you want to follow the change from #32386 since you are wanting to do symbolic ring computations. Otherwise you end up with

sage: I.parent()
Number Field in I with defining polynomial x^2 + 1 with I = 1*I

[egourgoulhon]

comment:114

Replying to @tscrim:

I think you want to follow the change from #32386 since you are wanting to do symbolic ring computations. Otherwise you end up with

sage: I.parent()
Number Field in I with defining polynomial x^2 + 1 with I = 1*I

In src/sage/symbolic/all.py there is a deprecation warning against from sage.symbolic.expression import I:

lazy_import("sage.symbolic.expression", "I", deprecation=(18036,
        "import I from sage.symbolic.constants for the imaginary unit viewed as an element of SR, or from sage.rings.imaginary_unit for the element of ZZ[i]"))

However, this warning does not seem effective...

[egourgoulhon]

comment:115

In a fresh Sage session, we have (since #18036 I guess)

sage: I.parent()
Number Field in I with defining polynomial x^2 + 1 with I = 1*I
sage: from sage.symbolic.constants import I as I_symb
sage: I_symb.parent()
Symbolic Ring
sage: SR(I) is I_symb
False

I would have expected True for the last output (i.e. to have I unique as a symbolic expression).

[tscrim]

comment:116

It's not so unnatural to have copies being made. It would also be a lot of work for SR to check that some potentially very complicated expression is equal to I.

Okay, then follow the deprecation and import from sage.symbolic.constants. The main thing is to not use the one from from sage.rings.imaginary_unit import I.

[sagetrac-git]

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

44f8cfa

Trac #29581: import I from symbolic.constants

[sagetrac-git]

Changed commit from 4e2e8b8 to 44f8cfa

[mjungmath]

comment:118

Okay, there we go. Thank you for this feedback!

[mjungmath]

Changed reviewer from Travis Scrimshaw to Travis Scrimshaw, Eric Gourgoulhon

[tscrim]

comment:120

Green bot. LGTM.

[vbraun]

comment:121

[docpdf] ! Package inputenc Error: Unicode character ϑ (U+03D1)
[docpdf] (inputenc)                not set up for use with LaTeX.
[docpdf] 
[docpdf] See the inputenc package documentation for explanation.
[docpdf] Type  H <return>  for immediate help.
[docpdf]  ...                                              
[docpdf]                                                   
[docpdf] l.73283 ...1}{\PYGZsh{} use spherical coordinates}
[docpdf]                                                   
[docpdf] ? 
[docpdf] ! Emergency stop.

[egourgoulhon]

comment:122

Indeed, any new Unicode character has to be declared in

src/sage/docs/conf.py

for the pdf doc to be generated successfully.

[sagetrac-git]

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

ac5cbbf

Trac #29581: remove unicode character

[sagetrac-git]

Changed commit from 44f8cfa to ac5cbbf

[mjungmath]

comment:124

That should solve it. Hopefully.

[tscrim]

comment:125

That is one way to solve it.

[vbraun]

Changed branch from u/gh-mjungmath/new_algorithm to ac5cbbf