In this post I want to summarize the list of problems I am currently thinking about. This is not a list of regular mathematical problems, see the disclaimer on style written at the end of the post.
Here is the list:
1. what is “computing with space“? There is something happening in the brain (of a human or of a fly) which is akin to a computation, but is not a logical computation: vision. I call this “computing with space”. In the head there are a bunch of neurons chirping one to another, that’s all. There is no euclidean geometry, there are no a priori coordinates (or other extensive properties), there are no problems to solve for them neurons, there is no homunculus and no outer space, only a dynamical network of gates (neurons and their connections). I think that a part of an answer is the idea of emergent algebras (albeit there should be something more than this). Mathematically, a closely related problem is this: Alice is exploring a unknown space and then sends to Bob enough information so that Bob could “simulate” the space in the lab. See this, or this, or this.
Application: give the smallest hint of a purely relational model of vision without using any a priori knowledge of the (euclidean or other) geometry of outer space or any pre-defined charting of the visual system (don’t give names to neurons, don’t give them “tasks”, they are not engineers).
2. non-commutative Baker-Campbell-Hausdorff formula. From the solution of the Hilbert’s fifth problem we know that any locally compact topological group without small subgroups can be endowed with the structure of a “infinitesimally commutative” normed group with dilations. This is true because one parameter sub-groups and Gleason metrics are used to solve the problem. The BCH formula solves then another problem: from the infinitesimal structure of a (Lie) group (that is the vector space structure of the tangent space at the identity and the maniflod structure of the Lie group) and from supplementary infinitesimal data (that is the Lie bracket), construct the group operation.
The problem of the non-commutative BCH is the following: suppose you are in a normed group with dilations. Then construct the group operation from the infinitesimal data (the conical group structure of the tangent space at identity and the dilation structure) and supplementary data (the halfbracket).
The classical BCH formula corresponds to the choice of the dilation structure coming from the manifold structure of the Lie group.
In the case of a Carnot group (or a conical group), the non-commutative BCH formula should be trivial (i.e. , the equivalent of in the case of a commutative Lie group, where by convention we neglect all “exp” and “log” in formulae).
3. give a notion of curvature which is meaningful for sub-riemannian spaces. I propose the pair curvdimension- curvature of a metric profile. There is a connection with problem 1: there is a link between the curvature of the metric profile and the “emergent Reidemeister 3 move” explained in section 6 of the computing with space paper. Indeed, at page 36 there is this figure. Yes, is a curvature!
Disclaimer on style. I am not a problem solver, in the sense that I don’t usually like to find the solution of an already formulated problem. Instead, what I do like to do is to understand some phenomenon and prove something about it in the simplest way possible. When thinking about a subject, I like to polish the partial understanding I have by renouncing to use any “impure” tools, that is any (mathematical) fact which is strange to the subject. I know that this is not the usual way of doing the job, but sometimes less is more.