Which completion of the configuration space of $n$ distinct points in $\mathbb{R}^d$ is better suited for numerical analysis?

Question

(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)

Answer 1

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

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