#point-free geometry

[rabraha3-blog]

Point-free geometry XX: towards mushy regions and evolution

Should we refer to it as pf-geometry instead of point-free geometry? Or perhaps give it another name?

So a crazy thought, related to statistics and Bayes' theorem. Let us recall our point-free geometry as a labeled graph. On the edges, the connections relating two regions, is placed a number. It could be perhaps real or complex, etc, but it represents a probability, in a sense. We don't consider whether two regions are intersecting or not, they can "kind of" intersect. And this "degree" is given by this number. See fuzzy set theory for more.

Now this is just an idea. Instead of labeling each edge -- which represents the relationship between two regions -- with a number, what if we label it with a probability distribution function (PDF)? In the operator approach, a "measurement" will choose a value from this distribution. A measurement of one region will collapse the PDF into a number on each edge connected to that region and modify the PDFs connected to all the regions on the other side of these edges.

Time evolution will modify the PDFs of all the edges. I imagine updating the PDFs using something akin to Bayes' theorem. What would the additional information be? Each PDF is book-ended by two regions. These two regions have other PDFs connected to them in general. Perhaps the data needed for evolution via Bayes' theorem could be the moments of these adjacent PDFs. Would this lead to a consistent evolution of the space? I'll have to think on it more. Basically this means the regions which at all influence our region of interest will affect the evolution of the region. If, for instance, we assume Gaussian PDFs on all the edges, perhaps a very wide PDF would imply a fairly low effect, since the most probable number chosen during a "measurement" will be lower (look at the normalization!).

Of course we CAN'T assume a Gaussian PDF, as we are restricted to the unit interval (give or take complex dimensions). So then if the PDF is peaked at or near zero, then there is little effect. on the neighboring region. But it's interesting. A "measurement" would then turn the PDF into a delta distribution, so for a short time, there would be a rather certain effect on neighboring regions. Evolution would proceed accordingly. There are probably inconsistencies flying all around here. But it is my first attempt at making the regions mushy, and evolution and certain operator actions come for free! Of course, we need to properly define the operator which performs these actions, but ... well, that's for later.

[rabraha3-blog]

Point-free geometry XVI: graphs

Sometimes the sets of regions and their respective intersection sets are cumbersome to carry around. What if we don't want to work with them; is there another picture we can use? This is math -- of course there is another framework!

As the title of the post implies, we will be using graphs.

Notice that the lines do NOT have arrows (for now). We can use this with the same definition as before. Only now instead of considering two related sets, we have one single object representing our entire space.

A graph is what we see above: a bunch of dots with lines going between some of the dots. Notice the lines need not go between nearest neighbors. The dots are called vertices and the lines edges. Notice for now, the edges do not have a direction (indicated by an arrow head) nor are the vertices or edge labeled yet. That may or may not come later.

For now we define: vertices are regions and edges connect two regions which intersect each other. So in this one structure, we have all the information necessary. For now, the fact that two regions intersect is a binary relationship. Not only that but the intersection sets both contain both regions; the two regions are in the mutual intersection set of the two regions. Bah, words. Hence the non-directionality of the edges.

Because, for now, the edges represent a binary relationship between regions, we can probably write these graphs in 2-dimensions, or a pseudo-2-dimensions, allowing for edged to hop over and under each other. Fixed regions of particularly nice shapes can likely be written as completely planar graphs.

So then ideas coming off this:

Once we fuzzify the space, edges will be labeled with a number to indicate a sort-of probability. Might other labels mean something too?

Functions will act on vertices (regions) -- they leave the relationships (edges) unchanged.

An operation acting on edges must be included. Think of it as a "time" evolution, if necessary -- it will change the relationships between regions, so it shuffles regions around and/or changes the regions' shapes.

A general function, then, will be a combination of those two functions, perhaps written as the pair: f = (fv, fe); v = vertices and e=edges. Notice we don't care about the lengths of the edges of the graph -- we only care about the relationships, the graph itself.

I'll end with another thought. There are people working on similar ideas: using the combinatorial information of a graph to get geometrical information. We see this in quantum gravity: spin networks and spin foams in loop quantum gravity and especially in group field theory. This is related to the "pre-geometry" that Wheeler referred to -- fundamental theories should derive the geometrical information of spacetime and thus cannot refer to the geometry of spacetime. The theory has to create the geometry. Might these structures be what I'm looking for?

If so, then we will want to look closely at these structures and then find relationships between them and noncommutative geometry. I'm sure some very capable people are already doing just that, if they didn't already find examples. But I don't think people have shown any general correspondence.

Oh and, happy holidays! I can be (moderatly) productive again now that the strain of the past semester is behind me!

[rabraha3-blog]

Point-free geometry XV: A specific example

This idea (point-free geometry) has been kicking around for six months already!

Any physicist, especially experimentalist, will know this example well. Perhaps not in such language, but still... When I say propagation of error, what comes to mind? During an undergraduate physics education this is beaten into our heads (at least it was at Maryland). We have measurements, which in principle can be real numbers but in practice are merely rational numbers. Associated to each measurement is an error. Any quantity utilizing these measurements have their own associated error. This error propagates through the relationship between this quantity and the measurements. Fine.

How does this relate to point free geometry? Generically in this framework, we have to assume the measurement uncertainties are random errors, and thus define a normal distribution around the measurement. The standard deviation of this Gaussian is the reported error. Notice the symmetric-ness. There exist other frameworks which treat non-symmetric errors, but we won't consider them for now.

So then functions lie on a space defined by the possible measurements. In principle, we can say these measurements can be any real number. For instance I could provide whatever voltage I want through a resistor of any resistance (not negatives...). The bottom line is the space of possible measurements is a point-free space! Each point isn't just a point; it comes with additional structure: the associated Gaussian which defined the error. So really we have a fuzzy point-free space. A region is the Gaussian associated to a measurement.

We can refer to each region with an n-tuple of real numbers because the possible measurements are supposed to be able to take the value of any real number. So the number of elements in the set of regions has the same cardinality as the set $\mathbb{R}^n$: we can put the two sets in one-to-one correspondence. So there is nothing stopping us from referring to each region by n real numbers, so long as we remember there is additional structure: the errors.

Cool. More than that, we know how to deal with functions on this point-free space. If we have an experiment to determine the acceleration due to gravity using a dropping ball, our measurements might be the position and velocities at various times. We really only need the positions. For one dimensional motion (straight down) we have a two dimensional point-free space: (t,z). We have errors in each direction. We picture these error bars, then by drawing a surface in this space which defines the 1 sigma region around a measurement. It will look like an ellipse. But the region actually permeates the entire space. Technically, our measurements consist of a one-dimensional subspace of this space, since we have the pairs $(t, x(t))$. So, the ball is released from rest. Then:

\[ x(t) = \frac{1}{2}g t^2\]

Therefore,

\[ g = \frac{2 x(t)}{t^2} \]

And we have a recipe for propagating the errors through, giving us not only $g$, but also $\sigma_g$. This function technically takes $g \colon \mathcal{R} \to \widetilde{\mathbb{R}}$, where $(t,x) \in \mathcal{R}$ and $\widetilde{\mathbb{R}}$ is a point-free real line, i.e. a one-dimensional point-free space where the regions are infinitesimally separated. Earlier posts on this subject cover a possible definition of distance which will likely work here.

This actually defines all possible accelerations $g$ and our measurements select out one. But our argument is more general than physics. Excellent.

We've seen an example of a point-free (and fuzzy) geometry and how to deal with functions defined on it. This example consists of a densely defined set of regions (dense in the sense of points in $\mathbb{R}^n$ for instance) -- two nearest neighbor regions are separated by an infinitesimal distance. The region shapes are fixed with the membership function given by a Gaussian.

Notice: I've done the process of embedding this point-free space into a space WITH points. That's really the only way to make sense of statements like "Gaussians around a measurement," for it requires the notion is the Gaussian at a specific point, not spread over an entire region. It was kind of natural, though. Perhaps I'll try to translate this into a truly point-free language next. No reference to points at all... We also need to relax the requirement of fixed regions. What will it mean geometrically to have "mushy" regions?

[rabraha3-blog]

Point-free geometry (XIV): Brainwave on curvature and further dynamics

We can picture these regions like a fluid! With that, we have to go back to curvature, defining a curved point-free space, in analogy to a Riemannian space.

We have a notion of density of regions, sort of given by the number of elements in each intersection set. I just imagined that if the number of elements in each intersection set was constant over each region, then we should recover a flat space. Curvature can emerge as a result of a difference in the number of elements in the intersection sets between two regions.

How does this make sense to me? We can *sort of* treat general relativity as a medium, at least to explain light deflection by matter. In any rate, the physical distance traversed by light in a given amount of time (time with respect to a local observer) will change, increase for positive curvature and decrease for negative curvature with respect to the distance traveled on flat space.

Now, we will have to slightly modify our definition of distance, right? My hope is that it is close, we'd have to hop, probably, to the nearest neighbor, or something. I don't know, I'll have to think about it. But the difference in the "sized" of the intersection sets really should correspond to curvature. And notice that there is no set number of elements in the intersection set that will give a flat space, it all depends on the neighboring regions. Everything is relative.

And then we can move onto a dynamical point-free space. We'd need to add on some additional operation here. This one is conceptually easy. This operation will take an element from the intersection set of one region, remove it from there, and place it into the intersection set of another region. Of course, the devil is in the details.

Call the region we want to move $R_m$ (m for move). There will likely be multiple regions with $R_m$ in its intersection sets; these would clearly have a non-trivial mutual intersection set, having at least $R_m$ populating it. So we don't have to remove $R_m$ from all intersection sets, but just one. Now the question is for which region do we add $R_m$ into its intersection set? It should move $R_m$ to a region inside the intersection set of the original region, but it cannot intersect this original region. In other words, it has to move it somewhere "close." I'll think on it and try to come up with a more precise definition later.

The operation $M$ acts on the intersection sets $\mathcal{I}_n$ for all regions $n$ (dare I call this the intersection bundle?). It basically shifts regions around by a small distance, as small a distance that we can get away with. So we can basically imagine these regions flowing around. I imagine this as making up a dynamical point-free space, much like general relativity describes a dynamical spacetime.

I dare not get my hopes up, though. How can noncommutativity of functions arise from this construction? Perhaps it does require the fuzzification process, after all. Or maybe it simply cannot emerge from this construct. But if I can manage to define this dynamical operator precisely, perhaps I can start working on a point-free analogue to general relativity. So there are still places to go with this. I'll continue to look into this idea. Just ... a bit more slowly now that I am teaching.