Two weeks later...

Still really bad at this. Oh well!

Midterm presentation went really well on Friday. A lot of this week is going to be writing up things (parts of my matlab code, mostly), and getting TikZ to work (a package for LaTeX that enables easy graphs and diagrams). I'm also going to try to meet with Professors Li and Torczon sometime soon, since Professor Phillips has gone with way of Professor Day and abandoned me (but only for a week; he still loves me more). I also need to get the size 4 data working for my MATLAB code, since the data set is waaay too big to manually analyze. I'm actually kind of hoping that'll be the last big MATLAB thing I have to do, since the coding stuff is tedious and kind of boring. We'll see how it goes.

I'm really bad at this...

Okay, going to try to do better.

Things I'm working on now:

1. Proving things. Like that the maximum topological entropy of a size n symbol space is equal to log(n), or that raising a graph to a higher edge doesn't change the entropy.
   
2. Playing with Laplacians. Professor Phillips says that Laplacian matrices might offer something different than adjacency matrices for looking at symbol spaces, and that relative topological entropies should remain the same.
3. Playing with higher edge graphs more. Because they're cool, and might prove useful down the road.
   
4. Continuing to work on the matrices on edges stuff. This is a very broad topic, so I need to narrow this down further into a few things I can actually do, but I'm going to do the other stuff first.
5. Keep up with the blog.
   

Catch-Up Post

This post is entry is hopefully going to briefly explain my research area, and also serve as a log for what I've done up to now.

I'm studying symbolic dynamics, specializing in the area of topological entropy. Where normal dynamics sets up a system of rules that describes which states follow from which other states, and then studies the long-term behavior of the system, symbolic dynamics does mostly the same, except on a finite number of symbols instead of a possibly infinite number of different states. The total set of symbols available is known as the alphabet A, and for my purposes, I denote symbols by nonnegative integers. So a size r alphabet would be the set of symbols A = {0, 1, ..., r-1}. Symbolic dynamics studies bi-infinite sequences of symbols (infinite in both directions, so no definite starting or ending point; each sequence is a 'point'), with rules dictating which symbols can follow or precede which other symbols; the total set of all allowed sequences is known as the shift space.

The most generic case is when there are no rules, so all possible infinite combinations of symbols are allowed; this is known as the full shift of A. All shift spaces are subsets of the full shift of some A. For another example, let's say A = {0, 1}, and let's add the rule that no two 1's can be next to each other. So the shift space is the set of all sequences of 0's and 1's that do not have two (or more) 1's next to each other. This is obviously a subset of the full shift of {0, 1}. In the case that the shift space can be described by listing a finite set of forbidden blocks (as in the above case, where the forbidden set F = {11}), it is known as a shift of finite type. It's possible and useful to represent shifts as directed graphs, usually letting the symbols be the vertices of the graph, and the allowed transitions of symbols as the edges. There's also a notion of the shift map, which acts on sequences in the shift space, and maps a shift space to itself. All the shift map does is take an infinite sequence of symbols and move everything to the left once; for example, the sequence ...abcabcabc... would go to ...bcabcabca..., which kind of mirrors a real dynamical system progressing in time.

That's a rough outline of symbolic dynamics, which I'm studying. I'm also studying the specific notion of entropy in symbolic dynamical systems. Topological entropy is a way of categorizing how complex or chaotic a system is; the higher the level of entropy, the crazier the system. In the context of symbolic dynamics, entropy is related to how many length n-paths there exist through the directed graph as you increase n; the more restrictions there are on symbol progression, the lower the shift space's entropy. There are a variety of ways to compute entropy, which I may cover later.

I'll cover more topics as I need to, but that's more than enough of an overview for now.

My current projects:

1. Prove that the maximum topological entropy for a shift space is equal to the natural log of the size of its alphabet, and this entropy is only achieved for the full shift of that size. This should be easy to do, I just need to get the hang of the language and techniques for proving things in the context of symbolic dynamics
2. Prove that isomorphic graphs have the same topological entropy. Same as above.
3. Prove that higher edge graphs have the same topological entropy as their original graphs. That doesn't make much sense; I'll clarify later.
4. Continue coming up with interesting examples where removing different edges of graphs result in different levels of topological entropy (lower than the original, obviously). Then I'll hopefully be able to identify something interesting about the removed edges or the resulting graphs that will indicate why the entropy decreased as it did, relative to the other resulting entropies. The ultimate goal here is to determine what types of systems result in what levels of entropies.
   
5. Continue reading books and papers.
6. Learn how to use Tex (a mathematical writing processor thing).

That's all for now.

Intro Post

Cue the obligatory introduction post!

My name is Brian Paljug, I'm a rising senior at the College of William & Mary in Virginia, and I study mathematics. I'm currently doing research for the CSUMS program ("Computational Science Training for Undergraduates in the Mathematical Sciences") in the fields of symbolic dynamical systems and topological entropy. What do those things mean? I'm still learning myself, but I'll elaborate further in future blog entries. This journal was specifically created to serve as a record of my research progress for this summer in the CSUMS program (David's idea! now the whole internet knows!), but I kind of like the idea of using it as a general record for future research/academic progress, as well. We'll see; I'm ridiculously bad at following through on things I mean to do.

That's all for now; I'll hopefully provide a more substantial entry tomorrow about what I'm researching.