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(My original post starts here, and ends right before the Edit part. I am keeping it so that the comments and answer make sense, but what I am really interested in is what is in the Edit section.)

My experience has been so far mostly with finite-dimensional spaces. However, there is definitely a need to study infinite-dimensional spaces, even if one is mostly interested in finite-dimensional spaces, as the spaces of functions on finite-dimensional spaces are often infinite-dimensional.

Let us consider the infinite-dimensional polynomial ring $\mathbb{R}[x_1, \ldots, x_d]$. I am interested for instance in the set $\mathcal{I}_n$ of codimension $n$ ideals in that ring. Note that $\mathcal{I}_n$ can be thought of as a subset of the Grassmannian $\mathcal{G}_n$ of codimension $n$ subspaces of $\mathbb{R}[x_1, \ldots, x_d]$.

I am interested in the question of whether or not there is a natural topology on $\mathcal{I}_n$ which makes it into a smooth/continuous manifold. Maybe there is a manifold topology on $\mathcal{G}_n$ which makes $\mathcal{I}_n$ into a manifold?

I think that if one considers topologies related to the Zariski topology, then analogues of my questions above would have a negative answer, because the Hilbert scheme of degree $n$ points on affine $d$-dimensional space $\mathbb{A}^d$ could develop singularities if $d > 2$ (please correct me if I am writing something inaccurate or wrong).

Edit: here is what I am really interested in. In $\mathbb{R}^d$, you can consider the configuration space $C_n(\mathbb{R}^d)$ of $n$ distinct points in $\mathbb{R}^d$. I would like to define some kind of completion of $C_n(\mathbb{R}^d)$ which contains for instance degenerate configurations where $2$ or more of its points collide. The purpose is to enlarge the domain for instance for numerical approximations of the first order directional derivative of a function $f$ of $x_1, \ldots, x_d$ at some point $p$ and in some direction $v \in \mathbb{R}^d$ (i.e. $D_v(f)(p)$). For instance, one may approximate

$$D_v(f)(p) \simeq (f(p_1) - f(p_0))/h$$

where $p_0 = p$ and $p_1 = p + h v$ and $h$ is a small positive number. Note that if we enlarge the domain of the approximation so as to include the case where $h$ goes to $0$, so that $p_1$ and $p_0$ collide, we then get that the approximation is exactly $D_v(f)(p)$.

So I would like to enlarge the domain of numerical approximations so as to depend not just on finitely many distinct points, but to also include limiting configurations where two or more of these points collide.

There are two such approaches which come to my mind (well, that I know of) that could be relevant: the Fulton-Macpherson compactification and the Hilbert scheme of points.

So my question is really, which completion of $C_n(\mathbb{R}^d)$ is well suited for extending the domain of $n$ point numerical schemes? Another question is, has this been done? (Such questions were proposed to me by another Mathematician actually)

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    $\begingroup$ The set $\mathcal I_n$ is the set of real points of a finite type scheme over $\mathbb R$ (the Hilbert scheme of points). As such it has a natural topology, the analytic topology (the Zariski topology is not relevant). I claim that this is the only natural topology to consider. When $d>2$ this will not be a manifold, as you indicated, and the answer to your question is "no". Infinite-dimensional manifolds seem to be a red herring here. $\endgroup$ Jun 15 at 15:45
  • $\begingroup$ @DanPetersen, how is the natural topology defined on the set $\mathcal{I}_n$ please? I have been reading on the internet about Hilbert schemes of points, and I have learned a lot, but still do not know what is the natural topology on it. If you could give an online reference perhaps, or write a few comments, it would help me and perhaps other readers. $\endgroup$
    – Malkoun
    Jun 16 at 0:13
  • $\begingroup$ @Malkoun - The above claim is nonsense - The relevant topology to consider is the Zariski topology. $\endgroup$
    – user122276
    Jun 16 at 8:51
  • $\begingroup$ @Malkoun By "analytic topology" I mean the following construction. If $X \subseteq \mathbb A^n_{\mathbf R}$ is an affine real algebraic variety, then $X(\mathbf R)$ is a closed subset of $\mathbf R^n$ and inherits a topology from the usual euclidean topology. This topology on $X(\mathbf R)$ is independent of choice of affine embedding, and for a general real algebraic variety $X$ one can define a topology on $X(\mathbf R)$ by working locally on affine charts of $X$. I am just describing the real version of the usual complex analytic topology on a complex algebraic variety. $\endgroup$ Jun 16 at 9:38
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    $\begingroup$ I would be slightly more optimistic about the Fulton-MacPherson compactification; more specifically, the version due to Kontsevich (the "real oriented Fulton-MacPherson compactification"), rather than the algebro-geometric version you can find in the paper of Fulton and MacPherson. A useful reference is Sinha, "Manifold theoretic compactifications of configuration spaces". $\endgroup$ Jun 16 at 9:39
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Question: "I am interested in the question of whether or not there is a natural topology on In which makes it into a smooth/continuous manifold. Maybe there is a manifold topology on $G_n$ which makes $I_n$ into a manifold?"

Answer: Given any field $k$ and any finitely generated $k$-algebra $A$, you may use the Chinese Remainder Theorem to construct a class of ideals - the class of $\mathfrak{m}$-squeezed ideals denoted $sqz(\mathfrak{m})$. This class of ideals has the property that any cofinite ideal $I \subseteq A$ is a product $I=I_1 \cdots I_l$ of such ideals. This construction is related to the construction of the Hilbert scheme $Hilb^n(Spec(A))$ of subschemes of $Spec(A)$ of length $n$.

The Hilbert scheme of points is much studied in algebraic geometry/algebra and you will find much litterature on the subject if you type "Hilbert scheme" into you search engine. I believe the Hilbert scheme of points on a smooth surface is smooth. The Hilbert scheme $Hilb^n(X)$ of subschemes of $X$ (assume for simplicity that $X$ is projective over a field) of length $n$ is defined as the unique scheme representing the hilbert functor $hilb^n(X)$, hence $Hilb^n(X)$ is canonically a scheme and has a "universal family". It has a complicated struture and is singular in general. For affine schemes there are complications due to the fact that there is no Hilbert polynomial for such schemes, and one has to define the Hilbert functor using other methods.

The Hilbert scheme was originally defined for a closed subscheme $X$ of $\mathbb{P}^n_S$, where $S$ is a Noetherian scheme, in particular it is defined for $S:=Spec(k)$ with $k$ any field. Hence you may choose $k$ to be the the real numbers. The relevant topology to consider in your case is the Zariski topology, since the Hilbert scheme is a closed subscheme of a grassmannian

$$Hilb_{P(x)}(X) \subseteq \mathbb{G}(m,V)$$

with $V$ a finite dimensional vector space over $k$. The grassmannian $\mathbb{G}(m,V)$ has the structure of a finite dimensional real differentiable manifold and you may use this topology to get an induced topology on $Hilb_{P(x)}(X)$. But the natural topology to consider is the Zariski topology. You may of course consider the open subset $Hilb_{P(x)}(X)_{sm} \subseteq Hilb_{P(x)}(X)$ of "smooth points" as a real differentiable manifold and use methods from analysis. In general it is better to use the Zariski topology and the study of singularities in algebraic geometry.

See also the following link:

https://math.stackexchange.com/questions/4171591/applications-of-the-chinese-remainder-theorem-to-the-study-of-the-hilbert-scheme

If you view $k:=\mathbb{R}$ and $V:=k[x_1,..,x_n]$ as an infinite dimensional vector space over $k$, you may also define the grassmannian $G(n, V)$ parametrizing $n$-dimensional quotients of $V$. This scheme "exists" by constructions in

Grothendieck, Alexander; Dieudonné, Jean A. Éléments de géométrie algébrique. I. (English) Zbl 0203.23301 Die Grundlehren der mathematischen Wissenschaften. 166. Berlin-Heidelberg-New York: Springer-Verlag. IX, 466 p. (1971).

The "scheme" $G(n,V)$ is a separated scheme over $Spec(k)$ which is "not of finite type" over $k$. Hence such a "manifold structure" would be "infinite dimensional". The "dual" $G(n,V^*)$ parametrize n-dimensional sub-vector spaces of $V$. This too is not of finite type over $k$

Your comment: "The purpose is to enlarge the domain for instance for numerical approximations of the first order directional derivative of a function $f$ of $x_1,…,x_d$ at some point $p$ and in some direction $v\in R^d$."

I posted a question (several questions) on MSE and MO related to a possible link between the Hilbert scheme of points, Taylor maps and differential operators:

Applications of the Chinese remainder theorem

Example: Given any maximal ideal $I:=(x-a,y-b)\subseteq A:=k[x,y]$ with $p:=(a,b)\in k^2$ it follows the canonical projection map

$$T: A\rightarrow A/I^{l+1}$$

may be viewed as the Taylor expansion. Given any polynomial $f\in A$ we may write

$$f(x,y)=f(a,b)+ \sum_{k\geq 1}\sum_{i+j=k}\frac{\partial^kf(p)}{\partial_x^i \partial_y^j}(x-a)^i(y-b)^j$$

and the equivalence class $T(f)\in A/I^{l+1}$

is the $l$'th Taylor series

$$T^l(f):= f(a,b)+ \sum_{k=1}^l \sum_{i+j=k}\frac{\partial^kf(p)}{\partial_x^i \partial_y^j}(x-a)^i(y-b)^j$$

of the polynomial $f$.

I'm unsure how you plan to "take derivatives" for "differentiable manifolds with singularities". Taking derivaties in ananlysis involves taking limits. Taking derivatives in algebra does not involve the notion "limit": If $s\in H^0(X,L)$ is a global section of a line bundle $L\in Pic(X)$ and if $x\in X$ is a (closed) point there is an evaluation map

$$ev:H^0(X,L) \rightarrow L_x$$

and a projection map $t^l:L_x \rightarrow L_x/\mathfrak{m}_x^{l+1}L_x$, and the composed map

$$T^l_x: H^0(X,L) \rightarrow L_x/\mathfrak{m}_x^{l+1}L_x$$

may be viewed as the Taylor expansion of a global section $s$ at $x$:

$$T^l_x(s):=s(x)+s'(x)dx+\cdots \frac{s^{(l)}}{l!}dx^l \in L_x/\mathfrak{m}_x^{l+1}L_x.$$

How do you plan to introduce such non-reduced ideals and multiplicities when using the language of differentiable manifolds "with singularities". If you wish to study a parameter space using "analytic techniques" you will have to include "spaces with singularities".

Note that the "Fulton-Macpherson" compactification $X^{[n]}$ is constructed via a sequence of blow up's of the product $X^{\times n}$, hence $X^{[n]}$ is an algebraic variety (it is "smooth" when $X$ is "smooth"). If $X$ is defined over the real numbers it follows $X^{[n]}$ has the structure of a real differentiable manifold. Hence you may study $X^{[n]}$ using algebraic and analytic methods. If $Hilb^n(X)$ is the Hilbert scheme of subschemes of $X$ of length $n$, there is a relation between $Hilb^n(X)$ and the symmetric product $Sym^n(X):=X^{\times n}/S_n$. The Hilbert scheme $Hilb^n(X)$ may be constructed as a blow up of $Sym^n(X)$. If $\mathcal{I}$ is the ideal sheaf of the symmetric product and $J$ the inverse image ideal sheaf of $X^{\times n}$, whose blow up satifies

$$Hilb^n(X) \cong Bl_I(Sym^n(X)),$$

there is a canonical morphism

$$Bl_J(X^{\times n}) \rightarrow Hilb^n(X).$$

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    $\begingroup$ Thank you for your answer. Please note that, while using the Zariski topology and related concepts is very interesting, and indeed related to the Hilbert scheme of points on affine $d$-dimensional space, yet I am trying to study similar things using the manifold topology. I will check out the link though. $\endgroup$
    – Malkoun
    Jun 15 at 13:04
  • $\begingroup$ Thank you for the linked discussion. Yes, I agree that my questions are related to what you call $\mathcal{m}$-squeezed ideals in the linked post. I wonder if the space I am considering has a smooth manifold structure, or if it is singular. So my question is really about describing neighborhoods in what I call $\mathcal{I}_n$ of degenerate configurations of $n$ points (taking multiplicity into account) in $\mathbb{R}^d$ where at least two of the points collide (this can be phrased algebraically too...). $\endgroup$
    – Malkoun
    Jun 15 at 14:48
  • $\begingroup$ @JohannesHahn I just saw your edits. Good that you bring back the answers. I don't know why hm2020 is trolling... $\endgroup$ Aug 6 at 8:53

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