Zulip Chat Archive

Stream: maths

Topic: Normed spaces


view this post on Zulip Heather Macbeth (May 16 2020 at 01:07):

This continues a discussion from the new members stream. I would like to discuss some possible changes to normed_space.basic. Here is the branch (not yet PR'd).

view this post on Zulip Heather Macbeth (May 16 2020 at 01:07):

  1. Implement normed_algebra.to_normed_space, prior discussion

view this post on Zulip Heather Macbeth (May 16 2020 at 01:07):

  1. Implement a class normed_module (axiom norm_smul_le, ∥c • x∥ ≤ ∥c∥ ∥x∥), and prove that a normed module over a field is a normed space (axiom norm_smul, ∥c • x∥ = ∥c∥ ∥x∥). By the way, I am not sure that a "normed module" is a particularly standard object, but it seemed to me like a useful way to abstract out the norm-respecting property of the C(a, k)-module C(a, E) (for E a normed k-space).

view this post on Zulip Heather Macbeth (May 16 2020 at 01:08):

Changes that I considered but did not currently implement:

view this post on Zulip Heather Macbeth (May 16 2020 at 01:08):

  1. Add [@simp] attribute to norm_smul (∥c • x∥ = ∥c∥ ∥x∥) and related lemmas, prior discussion. I think this is possibly a good idea in principle, but it breaks a few existing proofs in normed_space.operator_norm, normed_space.banach etc.

view this post on Zulip Heather Macbeth (May 16 2020 at 01:08):

  1. Introduce consistency among naming in
    instance normed_top_monoid : topological_add_monoid α := ...
    instance normed_top_group : topological_add_group α := ...
    instance normed_top_ring : topological_ring α :=
    instance normed_space.topological_vector_space : topological_vector_space α E := ...

view this post on Zulip Heather Macbeth (May 16 2020 at 01:09):

Comments on any of these would be welcome. Regarding 2, it was suggested in the previous thread that once normed_module is added, normed_space should cease to exist except as an abbreviation for a normed module over a field.

view this post on Zulip Heather Macbeth (May 16 2020 at 01:09):

Sebastien Gouezel said:

I think vector_space is just an abbreviation for module. We could have normed_module, and then normed_space as an abbreviation of normed_module.

Johan Commelin said:

Sure. And norm_smul would be a lemma with the assumption [normed_field k]

view this post on Zulip Heather Macbeth (May 16 2020 at 01:09):

I did not implement this yet, because, if I understand correctly, this would require going through the library to all existing instances of normed spaces, and modifying the existing construction (using the axiom norm_smul) to instead check the new axiom (norm_smul_le). Correct? Thoughts?

view this post on Zulip Bryan Gin-ge Chen (May 16 2020 at 01:23):

I haven't been following this, but @Yury G. Kudryashov recently opened #2693 which seems quite relevant.

view this post on Zulip Heather Macbeth (May 16 2020 at 01:28):

Oops, looks like @Yury G. Kudryashov and I both took the same lesson from our discussion yesterday ...

view this post on Zulip Heather Macbeth (May 16 2020 at 01:29):

Sorry Yury, do you want to incorporate my other changes into your refactor? Or vice versa?

view this post on Zulip Yury G. Kudryashov (May 16 2020 at 02:33):

Feel free to take over this

view this post on Zulip Yury G. Kudryashov (May 16 2020 at 02:33):

I'm sorry for starting this without announcing here.

view this post on Zulip Yury G. Kudryashov (May 16 2020 at 02:36):

Note that I simplified several proofs in bounded_continuous_function

view this post on Zulip Heather Macbeth (May 16 2020 at 02:37):

Yes, I think this is likely to be one advantage of replacing norm_smul with norm_smul_le.

view this post on Zulip Heather Macbeth (May 16 2020 at 02:38):

Off the top of your head, are there other instances of normed spaces whose construction will be simplified by this?

view this post on Zulip Yury G. Kudryashov (May 16 2020 at 02:38):

operator_norm?

view this post on Zulip Heather Macbeth (May 16 2020 at 02:39):

Also, do you agree with @Johan Commelin / @Sebastien Gouezel 's proposal to make normed_space an abbreviation for normed_module over a field?

view this post on Zulip Heather Macbeth (May 16 2020 at 02:39):

normed_module having the norm_smul_le axiom

view this post on Zulip Yury G. Kudryashov (May 16 2020 at 02:39):

Yes.

view this post on Zulip Heather Macbeth (May 16 2020 at 02:40):

And do you have any opinion on the other possible changes? https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/Normed.20spaces/near/197768469

view this post on Zulip Yury G. Kudryashov (May 16 2020 at 02:41):

1, 2, 3: sure
4: I don't care about instance names

view this post on Zulip Heather Macbeth (May 16 2020 at 02:43):

Ok great.

view this post on Zulip Heather Macbeth (May 16 2020 at 02:50):

Your PR has nicer code than mine. If you like I can finish chasing down the things that your PR breaks (because of the norm_smul to norm_smul_le change.)

view this post on Zulip Heather Macbeth (May 16 2020 at 02:51):

Then when that is merged, make the space to module change as a second PR.

view this post on Zulip Yury G. Kudryashov (May 16 2020 at 02:52):

Whatever is more convenient for you.

view this post on Zulip Yury G. Kudryashov (May 16 2020 at 02:53):

Your PR has nicer code than mine.

I have a bit more (~1 year) experience with Lean and mathlib.

view this post on Zulip Heather Macbeth (May 16 2020 at 05:42):

Johan Commelin said:

Yup, that's right. So all their constructions should be prepended with normed_space.mk' or some other name that provided a second constructor.

@Yury G. Kudryashov , what do you think of incorporating this suggestion (a second constructor) into your PR, so that there are not a bunch of le_of_eq's added to the library from normed space constructions for which the equality is more natural than the inequality? (I can make the change, just wanted to check your opinion.)

view this post on Zulip Sebastien Gouezel (May 16 2020 at 08:29):

@Heather Macbeth , @Yury G. Kudryashov Do you mind if I work a little on Yury's branch, incorporating your definition of a normed module and registering a normed space as an abbreviation of a normed module?

view this post on Zulip Yury G. Kudryashov (May 16 2020 at 08:45):

I don't mind.

view this post on Zulip Yury G. Kudryashov (May 16 2020 at 08:46):

Note that @Heather Macbeth has this definition in her branch.

view this post on Zulip Heather Macbeth (May 16 2020 at 08:48):

Sure, go ahead!

view this post on Zulip Heather Macbeth (May 16 2020 at 08:53):

@Yury G. Kudryashov my branch did it the old way, with a normed space as a separate class with to and of methods; no point keeping it.

view this post on Zulip Sebastien Gouezel (May 16 2020 at 09:17):

I have started playing with it, and I have broken everything :) More precisely, I have stumbled on an issue with typeclass inference that reminds me very strongly of the issue we had with pi_instance some time ago. Here is a MWE:

import analysis.normed_space.basic

section normed_module
set_option default_priority 100 -- see Note [default priority]
class normed_module (α : Type*) (β : Type*) [normed_ring α] [normed_group β]
  extends module α β :=
(norm_smul_le :  (a:α) (b:β), a  b  a * b)
end normed_module

variables {𝕜 : Type}  {n : } {E : fin n  Type}
[A : normed_field 𝕜] [N : i, normed_group (E i)] [i, normed_module 𝕜 (E i)]

-- Two different ways to say: Π (i : fin n), module 𝕜 (E i)

lemma all_right :
Π (i : fin n),
    @module.{0} 𝕜 (E i)
      (@normed_ring.to_ring.{0} 𝕜
         (@normed_field.to_normed_ring.{0} 𝕜 A))
      (@normed_group.to_add_comm_group.{0} (E i) (N i)) :=
by apply_instance


lemma fails :
Π (i : fin n),
    @module.{0} 𝕜 (E i)
      (@comm_ring.to_ring.{0} 𝕜
         (@nonzero_comm_ring.to_comm_ring.{0} 𝕜
            (@local_ring.to_nonzero_comm_ring.{0} 𝕜
               (@field.local_ring.{0} 𝕜
                  (@normed_field.to_field.{0} 𝕜 A)))))
      (@normed_group.to_add_comm_group.{0} (E i) (N i)) :=
by apply_instance -- fails
-- all_right  -- works

I defined normed modules, and then I express in two ways that if you know that you have normed module structures on every E i with respect to some normed field, then you have module structures on the E i with respect to this normed field. The difference is that the instance paths from normed field to ring are different in the two lemmas. In one of them, by apply_instance works, in the other one it fails. But the two lemmas are defeq.

view this post on Zulip Yury G. Kudryashov (May 16 2020 at 09:50):

And I guess Lean picks up the wrong path. What do you think about leaving this change to another PR?

view this post on Zulip Sebastien Gouezel (May 16 2020 at 09:56):

You mean the introduction of normed modules? Yes, sure, it can wait for another PR. Still, I think the problem above is important, and should be fixed instead of worked around: this is probably not specific to normed spaces, and could bite us in many places. @Mario Carneiro , @Gabriel Ebner , what do you think?

view this post on Zulip Heather Macbeth (May 16 2020 at 16:23):

@Yury G. Kudryashov do you mind if I also incorporate the change 1 I mentioned above? (normed_algebra.to_normed_space). It's unrelated, but only a few lines:

instance to_normed_space {𝕜 : Type*} (𝕜' : Type*) [normed_field 𝕜] [normed_ring 𝕜']
  [h : normed_algebra 𝕜 𝕜'] : normed_space 𝕜 𝕜' :=
{ norm_smul_le := begin
    intros s x,
    calc s  x = ((algebra_map 𝕜 𝕜') s) * x : by {rw h.smul_def', exact rfl}
    ...  algebra_map 𝕜 𝕜' s * x : normed_ring.norm_mul _ _
    ... = s * x : by rw norm_algebra_map_eq,
  end,
  ..h }

view this post on Zulip Yury G. Kudryashov (May 16 2020 at 18:14):

Go ahead.

view this post on Zulip Sebastien Gouezel (May 16 2020 at 20:18):

@Mario Carneiro , since you're here, did you have time to have a look at https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/Normed.20spaces/near/197786186?

view this post on Zulip Sebastien Gouezel (May 16 2020 at 20:22):

It's really a weird behavior. I have investigated it a bit further, and it behaves exactly like the example we had with pi_instance some time ago. There are two terms that are definitionally equal (even with the attribute tactic.transparency.reducible), but apply_instance fails to spot this. The outer terms are proved equal with the attribute tactic.transparency.reducible, but not inner terms. More precisely,

lemma outer_works :
  (Π (i : fin n),
    @module.{0} 𝕜 (E i)
      (@normed_ring.to_ring.{0} 𝕜
         (@normed_field.to_normed_ring.{0} 𝕜 A))
      (@normed_group.to_add_comm_group.{0} (E i) (N i)))
   =
  (Π (i : fin n),
    @module.{0} 𝕜 (E i)
      (@comm_ring.to_ring.{0} 𝕜
         (@nonzero_comm_ring.to_comm_ring.{0} 𝕜
            (@local_ring.to_nonzero_comm_ring.{0} 𝕜
               (@field.local_ring.{0} 𝕜
                  (@normed_field.to_field.{0} 𝕜 A)))))
      (@normed_group.to_add_comm_group.{0} (E i) (N i))) :=
by tactic.reflexivity tactic.transparency.reducible  -- works

lemma inner_fails :
  (@normed_ring.to_ring.{0} 𝕜
         (@normed_field.to_normed_ring.{0} 𝕜 A))
  =
  (@comm_ring.to_ring.{0} 𝕜
         (@nonzero_comm_ring.to_comm_ring.{0} 𝕜
            (@local_ring.to_nonzero_comm_ring.{0} 𝕜
               (@field.local_ring.{0} 𝕜
                  (@normed_field.to_field.{0} 𝕜 A))))) :=
by tactic.reflexivity tactic.transparency.reducible  -- fails

view this post on Zulip Sebastien Gouezel (May 16 2020 at 20:23):

With tactic.transparency.instances, everything works.

view this post on Zulip Mario Carneiro (May 16 2020 at 20:50):

This sounds like "working as intended" though

view this post on Zulip Mario Carneiro (May 16 2020 at 20:50):

reflexivity reducible is supposed to unfold only reducible definitions, not instances

view this post on Zulip Mario Carneiro (May 16 2020 at 20:53):

doesn't simp have a canonize_instances option that is supposed to fix issues like this?

view this post on Zulip Sebastien Gouezel (May 16 2020 at 20:56):

I don't think it works as intended. You have

lemma all_right :
Π (i : fin n),
    @module.{0} 𝕜 (E i)
      (@normed_ring.to_ring.{0} 𝕜
         (@normed_field.to_normed_ring.{0} 𝕜 A))
      (@normed_group.to_add_comm_group.{0} (E i) (N i)) :=
by apply_instance

lemma fails :
Π (i : fin n),
    @module.{0} 𝕜 (E i)
      (@comm_ring.to_ring.{0} 𝕜
         (@nonzero_comm_ring.to_comm_ring.{0} 𝕜
            (@local_ring.to_nonzero_comm_ring.{0} 𝕜
               (@field.local_ring.{0} 𝕜
                  (@normed_field.to_field.{0} 𝕜 A)))))
      (@normed_group.to_add_comm_group.{0} (E i) (N i)) :=
by apply_instance -- fails
-- all_right  -- works

lemma outer_works :
  (Π (i : fin n),
    @module.{0} 𝕜 (E i)
      (@normed_ring.to_ring.{0} 𝕜
         (@normed_field.to_normed_ring.{0} 𝕜 A))
      (@normed_group.to_add_comm_group.{0} (E i) (N i)))
   =
  (Π (i : fin n),
    @module.{0} 𝕜 (E i)
      (@comm_ring.to_ring.{0} 𝕜
         (@nonzero_comm_ring.to_comm_ring.{0} 𝕜
            (@local_ring.to_nonzero_comm_ring.{0} 𝕜
               (@field.local_ring.{0} 𝕜
                  (@normed_field.to_field.{0} 𝕜 A)))))
      (@normed_group.to_add_comm_group.{0} (E i) (N i))) :=
by tactic.reflexivity tactic.transparency.instances

In the first lemma, the instance is found. In the second lemma, the instance is not found. But from the point of view of instance resolution, these two things should be the same as the third lemma shows that they are equal for tactic.reflexivity tactic.transparency.instances.

view this post on Zulip Mario Carneiro (May 16 2020 at 20:58):

This is apparently an issue with typeclass inference, and I'm not qualified to fix it in the C++

view this post on Zulip Mario Carneiro (May 16 2020 at 20:58):

it is not using instance reducibility for some subproblems

view this post on Zulip Mario Carneiro (May 16 2020 at 20:59):

does simp fix it?

view this post on Zulip Sebastien Gouezel (May 16 2020 at 21:00):

In my use case, the instance has to be inferred automatically even to let me state my lemma, so simp is not really an option...

view this post on Zulip Sebastien Gouezel (May 16 2020 at 21:01):

So, your guess if that apply_instance is passing the flag tactic.transparency.reducibility instead of tactic.transparency.instances somewhere. Yes, that would perfectly explain the behavior I am seeing.

view this post on Zulip Mario Carneiro (May 16 2020 at 21:02):

I guess you could locally hide a bad instance?

view this post on Zulip Mario Carneiro (May 16 2020 at 21:02):

like I doubt you need local rings for this

view this post on Zulip Sebastien Gouezel (May 16 2020 at 21:05):

Yes, I could certainly do that. But we have already been bitten by this with the pi_instance issue (where it was solved by cleaning up the tactic to make sure that it generates diamonds that are solved with tactic.transparency.reducibility), so I am afraid it will come up again later. If you think this is a C++ problem, all I can do is invoke @Gabriel Ebner (but not now, it's the week-end) and pray :)

view this post on Zulip Patrick Massot (May 16 2020 at 21:12):

We need the Zulip developers to implement deferred pings.

view this post on Zulip Sebastien Gouezel (May 16 2020 at 21:18):

I have opened an issue with all the details, and I'll do the manual pinging later.

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 00:37):

BTW, @Heather Macbeth what is your goal with the normed_space/normed_algebra PRs? I mean, do you want to formalize something specific or just filling in a hole in the library?

view this post on Zulip Heather Macbeth (May 17 2020 at 02:52):

When I noticed these things, I was trying to prove that under appropriate hypotheses C(X, 𝕜) is a Banach algebra.

view this post on Zulip Heather Macbeth (May 17 2020 at 02:52):

I am not sure what I will try next, maybe Holder spaces.

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 03:01):

Good idea!

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 03:02):

I pushed normed_ring instance, then realized that now we do two loosely related changes in one PR.

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 03:03):

So possibly this part should be reverted and moved to another PR. Do you have a normed_algebra instance?

view this post on Zulip Heather Macbeth (May 17 2020 at 03:04):

Do you mean, that C(X, 𝕜) is a normed ring?

view this post on Zulip Heather Macbeth (May 17 2020 at 03:04):

Or do you mean an internal change to normed_space regarding normed rings?

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 03:07):

I mean this instance

view this post on Zulip Heather Macbeth (May 17 2020 at 03:08):

Ah great. Yes, I had some draft code doing approximately that :)

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 03:08):

The next instance should be something like instance [nonempty α] {R : Type*} [normed_algebra 𝕜 R] : algebra 𝕜 (α →ᵇ R).

view this post on Zulip Heather Macbeth (May 17 2020 at 03:08):

But keep it!

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 03:08):

then normed_algebra

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 03:08):

Do you have these instances, or I should write&push mine?

view this post on Zulip Heather Macbeth (May 17 2020 at 03:12):

Just checked .. I have algebra but not yet normed_algebra

view this post on Zulip Heather Macbeth (May 17 2020 at 03:13):

That was why I caught the missing normed_algebra.to_normed_spaceinstance

view this post on Zulip Heather Macbeth (May 17 2020 at 03:13):

If you like I can finish tomorrow and push as part of the same PR

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 03:17):

I've just created a new branch bcf-normed-algebra. I'm going to revert some changes in the norm-smul branch. It would be nice if you'll push the two missing instances to the new branch tomorrow, then open a new PR.

view this post on Zulip Heather Macbeth (May 17 2020 at 03:17):

Great, will do.

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 03:18):

I can fix merge conflicts once norm_smul will be merged.

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 03:26):

reverted This means that we shouldn't merge norm-smul into bcf-normed-algebra.

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 03:45):

I'd love to have the following fact about Hölder continuous functions in mathlib: if f : X → Y is Hölder continuous with exponent α, then dimHY1αdimHX\operatorname{dim}_H Y \le \frac{1}{α}\operatorname{dim}_H X. This is an easy exercise but we have neither Hölder continuous functions nor Hausdorff dimension.

view this post on Zulip Heather Macbeth (May 17 2020 at 03:47):

Yes, I noticed that Hausdorff dimension is missing. Also a good project.

view this post on Zulip Heather Macbeth (May 17 2020 at 03:49):

I would like to try to get C2,αC^{2,\alpha} estimates for the Newtonian potential

view this post on Zulip Heather Macbeth (May 17 2020 at 03:50):

Probably there are a lot of things missing before that can be done, but it would be a useful test to show what the library is missing.

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 03:51):

Then you'll probably need FTC. We have an abandoned PR with it. I hope to take over this PR soon. Unfortunately, I'm getting distracted by other tasks.

view this post on Zulip Heather Macbeth (May 17 2020 at 03:51):

I saw that you are doing the implicit function theorem, that would be really great.

view this post on Zulip Heather Macbeth (May 17 2020 at 03:52):

People could make some examples of manifolds once you do that!

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 03:53):

It's mostly done but I need complemented subspaces to write a good API.

view this post on Zulip Heather Macbeth (May 17 2020 at 03:56):

There also needs to be a theory of strict k-times-differentiable functions, right?

view this post on Zulip Heather Macbeth (May 17 2020 at 03:56):

To get inverse and implicit function theorems in arbitrary differentiability

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 04:05):

Do we need f to be strictly differentiable k times?

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 04:05):

Once we have the inverse function, its differentiability follows from differentiability of the original function.

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 04:06):

See this lemma

view this post on Zulip Heather Macbeth (May 17 2020 at 04:15):

Maybe that's right. Then one ends up with manifolds whose structure groupoid (if I have learned the library's language correctly yet :slight_smile: ) consists of functions which are 1-time strictly differentiable, k-times differentiable?

view this post on Zulip Sebastien Gouezel (May 17 2020 at 07:07):

The plan is rather to work on reals or complexes, where C^k functions are automatically strictly differentiable.

view this post on Zulip Kevin Buzzard (May 17 2020 at 08:26):

People could make some examples of manifolds once you do that!

There are no examples of anything in mathlib. We don't even have dihedral groups. For some reason examples don't form part of the philosophy. I would love to see examples. When we made schemes the computer scientists said we should make examples, and then we proved that the spectrum of a ring was a scheme and somehow this was enough because we can in theory make examples of rings

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 08:46):

Some examples of manifolds are useful for general theory. E.g., there are theorems about SnS^n and RPn\mathbb RP^n.

view this post on Zulip Kevin Buzzard (May 17 2020 at 08:47):

I think Kenny made the scheme P^1 at some point but the gluing was hard work

view this post on Zulip Kevin Buzzard (May 17 2020 at 08:48):

I'm sure S^n would be straightforward once you have the basic theorems. How would you do projective space though?

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 08:49):

It would be nice to have

  • PE{\mathbb P}E for a vector space EE;
  • embedding of Pn\mathbb P^n into an affine space of larger dimension (don't remember if there is a general formula for this).

view this post on Zulip Kevin Buzzard (May 17 2020 at 08:51):

Do you have a plan for quotients? What I'm saying is that in alg geom they were a challenge

view this post on Zulip Kenny Lau (May 17 2020 at 08:51):

I thought about PE earlier (like a year ago maybe), and I decided that PE requires Grassmannian which requires linear algebra

view this post on Zulip Kevin Buzzard (May 17 2020 at 09:15):

Putting a manifold structure on the quotient of a manifold by a lie group action might be a challenge. Will there be underlying type theory problems which mathematicians don't usually notice?

view this post on Zulip Kevin Buzzard (May 17 2020 at 09:17):

I'm thinking G/HG/H with for example G=GL2(R)G=GL_2(\mathbb{R}) and HH the upper triangular matrices.

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 09:18):

Specifically for the projective space we can come up with an explicit atlas

view this post on Zulip Kevin Buzzard (May 17 2020 at 09:42):

I would have said the same thing about the alg geom projective space but it was much easier said than done in practice

view this post on Zulip Patrick Massot (May 17 2020 at 09:55):

We'll need affine charts for projective spaces at some-point (but first we'll need affine spaces...). But this is not the morally correct way to put a manifold structure on them. We need quotient manifolds.

view this post on Zulip Heather Macbeth (May 17 2020 at 13:56):

Yury G. Kudryashov said:

It would be nice to have

  • PE{\mathbb P}E for a vector space EE;
  • embedding of Pn\mathbb P^n into an affine space of larger dimension (don't remember if there is a general formula for this).

I believe Pn\mathbb P^n embeds (non-holomorphically) into the space of Hermitian (n+1)×(n+1)(n+1)\times(n+1) matrices.

view this post on Zulip Heather Macbeth (May 17 2020 at 13:57):

But a non-holomorphic embedding only allows one to construct it (via the implicit function theorem) as a real manifold, not as a complex manifold.

view this post on Zulip Heather Macbeth (May 17 2020 at 13:59):

A bounded holomorphic function is constant, so there is no holomorphic embedding.

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:00):

I think constructing projective n-space as a complex manifold might be quite challenging, given what we have.

view this post on Zulip Heather Macbeth (May 17 2020 at 14:02):

Kevin Buzzard said:

Putting a manifold structure on the quotient of a manifold by a lie group action might be a challenge. Will there be underlying type theory problems which mathematicians don't usually notice?

The right theorem is, the quotient by a free proper action is a manifold.

view this post on Zulip Heather Macbeth (May 17 2020 at 14:02):

I think, for this, one needs existence & uniqueness for ODEs

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:03):

In Lean this will be a definition, not a theorem, and it might be challenging.

view this post on Zulip Reid Barton (May 17 2020 at 14:03):

But you also can't construct CP^n as a complex manifold this way, can you?

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:03):

Not the C×\mathbb{C}^\times action on a vector space of one dimension higher, minus the origin?

view this post on Zulip Heather Macbeth (May 17 2020 at 14:03):

Yes, I think that's fine.

view this post on Zulip Reid Barton (May 17 2020 at 14:03):

Isn't it not proper?

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:03):

Oh maybe I don't know what proper means

view this post on Zulip Reid Barton (May 17 2020 at 14:04):

Maybe I don't know!

view this post on Zulip Heather Macbeth (May 17 2020 at 14:04):

Proper in this case means that the map G×MM×MG\times M \to M \times M is proper.

view this post on Zulip Heather Macbeth (May 17 2020 at 14:05):

I think that's fine here.

view this post on Zulip Heather Macbeth (May 17 2020 at 14:05):

Anyway, even if not, one could take the quotient of S2n+1S^{2n+1} by S1S^1.

view this post on Zulip Reid Barton (May 17 2020 at 14:05):

Yes, I was just dimly recalling this definition. That's fine then.

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:05):

Then Reid's point is that this doesn't give the complex structure

view this post on Zulip Reid Barton (May 17 2020 at 14:07):

The (Cn+1{0})/C×(\mathbb{C}^{n+1} - \{0\})/\mathbb{C}^\times definition is fine though, I was just using the wrong definition of proper quotient.

view this post on Zulip Heather Macbeth (May 17 2020 at 14:09):

Heather Macbeth said:

The right theorem is, the quotient by a free proper action is a manifold.

Just skimming the proof .

view this post on Zulip Heather Macbeth (May 17 2020 at 14:10):

First, implicit function theorem to show that the G-orbits are properly embedded submanifolds

view this post on Zulip Heather Macbeth (May 17 2020 at 14:11):

Then, existence & uniqueness for ODEs, with smooth dependence on initial conditions, to put charts on the G-orbits via the exponential map, uniformly across a slice.

view this post on Zulip Heather Macbeth (May 17 2020 at 14:17):

Kevin Buzzard said:

I think constructing projective n-space as a complex manifold might be quite challenging, given what we have.

But why can projective n-space not currently be constructed from n+1 affine charts?

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:17):

Ask @Kenny Lau

view this post on Zulip Heather Macbeth (May 17 2020 at 14:17):

Is it just that it's still a pain to differentiate actual functions?

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:18):

The problem, I believe, is that quotients are very intuitive to mathematicians but my limited experience with using them in Lean to make complex objects has been extremely painful.

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:19):

You have two copies of affine 1-space, and then you take the disjiont union, make the quotient, and now we have the underlying type

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:19):

but now you need to actually make a structure which has been formalised but which is truly complicated

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:19):

and building that structure will involve actually checking all the axioms

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:20):

and Patrick's experience checking that various definitions of manifolds were equivalent was a lot more difficult than he expected.

view this post on Zulip Heather Macbeth (May 17 2020 at 14:20):

I see. So when you construct it from n+1 charts, you are not actually avoiding a quotient construction. You are just introducing a messier quotient construction.

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:20):

Now we've gone with one. Maybe it was the best for making an API or something.

view this post on Zulip Heather Macbeth (May 17 2020 at 14:20):

disjoint union of n+1 copies of Cn\mathbb{C}^n via identification of points

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:20):

But now let's see whether it's any good for making an example.

view this post on Zulip Heather Macbeth (May 17 2020 at 14:21):

rather than C\mathbb{C}^* quotient of Cn+1{0}\mathbb{C}^{n+1}\setminus\{0\}

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:21):

Yes, making the underlying type is no problem. But now you have to hope that quotient topologies and quotient functions and quotient "I am differentiable" and everything all works fine

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:21):

Glueing and quotienting by a group action are both quotients from the point of view of type theory

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:22):

All I'm saying is that it might be painful but possible. This was what happened in alg geom anyway.

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:22):

And diff geom is harder :-)

view this post on Zulip Reid Barton (May 17 2020 at 14:22):

Both of these constructions are of course very important in general.

view this post on Zulip Kevin Buzzard (May 17 2020 at 14:28):

For me this was a shock. We had two rings A and B, an explicit isomorphism A[1/f]=B[1/g], where of course I'm using equality the way a mathematician does, but then to get the corresponding isomorphism of the locally ringed spaces is not immediate in dependent type theory, and a locally ringed space is a complex object. However I believe the objects you're dealing with are more complex

view this post on Zulip Heather Macbeth (May 17 2020 at 14:31):

Has someone done the quotient of a topological space by an equivalence relation, yet?

view this post on Zulip Reid Barton (May 17 2020 at 14:32):

Yes: https://github.com/leanprover-community/mathlib/blob/master/src/topology/constructions.lean#L45-L49

view this post on Zulip Reid Barton (May 17 2020 at 14:32):

(the second one, for an equivalence relation, but the construction is the same)

view this post on Zulip Patrick Massot (May 17 2020 at 17:12):

I'm sorry I missed the fun discussion. I haven't read very carefully all messages, but I have a couple of comments.

  • Kevin likes to say differential geometry is harder than algebraic geometry but this is of course meaningless (and he knows it)
  • There is no obstruction at all to defining projective spaces in currently mathlib, including the topology in the real or complex cases.
  • There is no obstruction to build a manifold structure on projective space in the real and complex cases. We could do that in at most one day. But it would bring us nothing interesting.
  • There is an obstruction to quickly do it right, because the natural proof requires machinery that would take time to build. This would be interesting.

view this post on Zulip Heather Macbeth (May 17 2020 at 17:17):

Kevin Buzzard said:

You have two copies of affine 1-space, and then you take the disjiont union, make the quotient, and now we have the underlying type

@Patrick Massot , are you saying that doing this for Pn\mathbb{P}^{n} using n+1 charts is not actually that hard?

view this post on Zulip Heather Macbeth (May 17 2020 at 17:18):

Heather Macbeth said:

Putting a manifold structure on the quotient of a manifold by a lie group action might be a challenge. Will there be underlying type theory problems which mathematicians don't usually notice?

The right theorem is, the quotient by a free proper action is a manifold.

And is this what you mean by "the right way"?

view this post on Zulip Patrick Massot (May 17 2020 at 17:24):

What Kevin wrote is not the right. The underlying type should be the natural one, without making any choice. Start with a vector space EE and define the obvious quotient of E{0}E \setminus \{0\} to get P(E)P(E). And then, assuming the base field is real or complex numbers there are two roads. In the first road you define affine charts (which will be very sad because we don't have affine spaces) and check transition maps are real or complex analytic (here I'm not sure whether Sébastien is totally ready, maybe I should say smooth instead of analytic). This is what I call the wrong way, but I'm exaggerating a bit when I write this would bring nothing good. What I call the right way is to setup the quotient manifold infrastructure.

view this post on Zulip Patrick Massot (May 17 2020 at 17:26):

Note that I never mention trying to get exactly n+1n+1 charts (or finite dimensionality). I don't see what good it would do. Affine chart are defined as soon as you fix a hyperplane, and their canonical affine structure is much more apparent if you don't pollute things with coordinates.

view this post on Zulip Heather Macbeth (May 17 2020 at 17:30):

OK, makes sense!

view this post on Zulip Heather Macbeth (May 17 2020 at 17:32):

Even while the quotient infrastructure is not yet set up, people could work on defining Lie groups, Lie group actions, freeness and properness of Lie group actions

view this post on Zulip Heather Macbeth (May 17 2020 at 17:32):

And make a bunch of examples of GL(n)GL(n)-actions and discrete group actions

view this post on Zulip Heather Macbeth (May 17 2020 at 17:33):

(GL(n)GL(n) and discrete groups being the only Lie groups available without the implicit function theorem)

view this post on Zulip Heather Macbeth (May 17 2020 at 17:34):

So that as soon as the quotient manifold infrastructure is done, there are immediately several new examples of manifolds produced

view this post on Zulip Kevin Buzzard (May 17 2020 at 17:36):

Off topic: A poor man's definition of an automorphic form is a function on GL(n,R)GL(n,\mathbb{R}) which satisfies certain differential equations (coming from the centre of the universal enveloping algebra of the Lie alegbra). I feel like we are slowly but surely converging to the point where we might be able to do this.

view this post on Zulip Patrick Massot (May 17 2020 at 17:38):

This is all part of the missing infrastructure, so yes, these are all steps in the right direction

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 18:27):

@Heather Macbeth Merged master into bcf-normed-algebra.

view this post on Zulip Heather Macbeth (May 17 2020 at 18:35):

Thanks, doing the rest now. I discovered that the normed_algebra structure on bounded continuous functions only works when the domain is nonempty!

view this post on Zulip Heather Macbeth (May 17 2020 at 18:36):

It uses this lemma
@[simp] lemma norm_const_eq [h : nonempty α] (b : β) : ∥const α b∥ = ∥b∥

view this post on Zulip Heather Macbeth (May 17 2020 at 18:37):

Is there a way to add a hypothesis to an instance?

view this post on Zulip Kevin Buzzard (May 17 2020 at 18:37):

The instance might never trigger, unless the hypothesis is itself a typeclass.

view this post on Zulip Reid Barton (May 17 2020 at 18:38):

instance [nonempty α] : normed_algebra ...

view this post on Zulip Heather Macbeth (May 17 2020 at 18:58):

Sorry, another question, very much like this one @Yury G. Kudryashov answered for me on Thursday. The following

import algebra.ring
import analysis.normed_space.basic

example {R : Type*} [comm_ring R] [normed_ring R] (a b : R) : a * b = b * a :=
begin
  apply mul_comm,
end

fails with "invalid apply tactic, failed to unify a * b = b * a with ?m_3 * ?m_4 = ?m_4 * ?m_3 ". (I discovered this while trying to get that C(α,R)C(α, R) is commutative if RR is.)

view this post on Zulip Heather Macbeth (May 17 2020 at 18:59):

Is this impossible to fix without implementing a theory of normed_comm_ring which unifies comm_ring and normed_ring?

view this post on Zulip Yury G. Kudryashov (May 17 2020 at 19:00):

I think you're right, you need one more typeclass.

view this post on Zulip Heather Macbeth (May 18 2020 at 04:29):

PR'd, #2722

view this post on Zulip Heather Macbeth (May 18 2020 at 04:32):

I decided not to make normed_comm_ring, that can wait for a future refactor of normed_space.basic


Last updated: May 06 2021 at 18:20 UTC