I just attended my first real mathematical conference this past weekend and gave my first real presentation of my work (read, in a forum not catering entirely to undergraduate research). There are a lot of new things to think about that I picked up at the conference. This post is going to be basically just a long list of stuff that I thought about while at the conference rather stuff that was sparked by the conference. I want to put some of these thoughts down somewhere so that I won't just forget about all of them. Maybe in a year or two I will look back at this post and pick something I had forgotten about up again, Who knows.
I met Thomas Garrity from Williams College and had what I would like to call a conversation with him. This is really rather exciting for me because I do mean conversation, I do not mean that I asked him a question at the end of his lecture and got brief expository answer from him. I mean that there was actually dialogue. That is not to say that I think I was an equal partner in these discussions I fully realize that the flow of information was primarily from one to the other. But what characterizes a conversation (at least one part of the characterization) is that both parties shape the course of the discussion not just one or the other. In a lecture the one person chooses the subject and the other listens and even perhaps asks questions but they do not really have any real control over what is discussed or how. Especially in the atmosphere of conferences and lectures and the like it can be difficult for a lowly audience member to have a real conversation with a high up. When I went to Lisa Randall's lecture on campus during the science and literature symposium I found it essentially impossible to have any more than completely superficial interaction with her. She certainly didn't seem interested with interacting with one of the swarm. Admittedly the mathematics conference was a very different sort of atmosphere since most of the people attending were not physicists.
Over the course of the two days of the conference I talked to Garrity a number of times and there are four things in particular that I will have to give some greater thought.
The most obvious one is what he gave his lecture on, at the end of the lecture he posed an open problem which is does there exist a way to express real numbers as a sequence of integers so that the sequence is periodic if and only if the number being represented is cubic. Cubic here means that somehow it contains a cube root. Continued fractions do it for square roots and decimal expansions are periodic for rational numbers which is basically the impetus for thinking about the question.
More interesting though were the things that we discussed outside of a lecture structure. First off he made me question whether my recent push to find a method of imposing a coordinate structure onto a fractal space is possible at all. I had always assumed that it was of course possible and even had a vague set of arguments why it should be possible. Something I hadn't considered is that many (maybe all?) fractal spaces are not decidable. By that I mean that probably in order to parameterize a space you probably need to be able to decide what points are in the space and what points are not in the space in a finite number of steps. There is no finite time algorithm for computing the mandelbrot set by which I mean that given any general point in the complex plane you cant always in a finite number of steps decide if the point is in the set or not (assuming that the operations on the reals such as addition and multiplication have a finite computational value which you just make 1 step for convenience). This sort of points out to me that I really don't know exactly what parameterization really is. On the superficial level that I was thinking about it a parameterization was simply some function which mapped the set of points of some space onto an ordered set of real numbers. Clearly such a procedure must exist because the cardinality of the sets are the same and so therefore a one to one mapping exists. At the very lowest level that means that you can construct such a map from one to the other in a direct manner just by looking at all the points and making the map one set of points at a time doesn't it? Actually the answer is no, it doesn't mean that you can make such a map because you cant make a list of the points and so if you cant find a way to characterize all of the points in your set then you clearly can't make the map. So in at least one well defined sense of what you mean by "generalized coordinate system" there is a clear reason that there cannot exist a generalized coordinate system for the space of the mandelbrot and probably most fractal spaces. I don't really have any reason to think that this result would generalize to any fractional dimensional space but I do. There might be very nice fractal dimensional spaces that are decidable but I doubt it. It just so happens that this piece of the puzzle fits with what I have been thinking for a while now, that fractal dimensional spaces cannot be handled within the framework of first order logic
This brings us to the next thing we talked about which I will just mention briefly. That is the suggestion by a philosopher of the name Hintikka that the usual first order logic needs to be extended by game theoretic ideas. I don't know if I agree yet or not but it certainly sounds like something that I should look at.
Lastly, he talked about the applications of the partition function to abstract objects in sets instead of to thermal physics. The partition function being something that I am only just really becoming familiar with in my thermal physics course this semester I was of course intrigued. Apparently the properties of the partition function can be very useful in the analysis of completely abstract collections of things.
I also attended a talk on the ordering dimension of partially ordered sets. I personally thought that the talk was not of particularly high quality and the idea was not very exciting since it was merely a slight further generalization of a set of objects called generalized crowns. However the talk did introduce me to the concept of order dimension and made me think a little bit more about what a dimension really is. While the talk didn't actually go into any real detail about the nature of partially ordered sets or posets since they basically assumed that everyone in the audience was familiar with them (bad assumption). I still don't know in a formal way what a partially ordered set is but basically a partially ordered set is one in which there is a partial ordering relation. Lets say you have 5 objects in the set and let @ denote an ordering of some kind of two objects like for instance a @ b shows that in some sense a < b. A partial ordering on the five objects a, b, c, d, e might say something like a @ b, c @ d, e @ d. in which case we would know nothing about the "ordering" of a and d so the set is only partially ordered. Thinking about this made me think of possible connections to second order logics and also more to the point non-integer dimensional spaces. I don't know if this really sticks but it would make sort of intuitive sense that one requirement for a set to have a countable cardinality is for there to exist a complete ordering on the set. This condition is related to being able to write the set as a list since a list constitutes an ordering on the set (the things that come before something on the list is "less" than it and things that come after are "greater" than it). But it is intriguing to notice that having an ordering relation is not the same as the capacity to write a set as a list. For instance the reals have a complete ordering but they cannot be written as a list. I wonder if perhaps things with the cardinality of the power set of the reals cannot have complete orderings. At any rate any set with at least two elements can have a partial ordering on it (as I understand it) since you can just relate two elements and leave all the others unrelated. Seeing how this is related to non integer dimensions is perhaps a bit of a stretch and this post is primarily meant to write down ideas not expound on them in detail (and it is running long as it is) so I will just leave that one alone for now.