# What do Mathematicians Do All Day?

One of the big problems with science and math is that the practitioners are terrible at explaining what they do. Occupy Math will use this post as a way to work on his skills at communicating what on earth he does outside of the classroom. This week in Vancouver is the 2016 IEEE World Congress on Computational Intelligence and Occupy Math is presenting several papers. This week’s post summarizes what they are about. Hopefully this will help students who might like a career in computer science, mathematics, or statistics see what they are getting into.

Occupy Math works on Evolutionary Computation, but with a lot more math mixed into it than most other people working in the field. He’s even published a book on the subject – its a university level text, so don’t anticipate a light read. Evolutionary Computation is any computer algorithm based on Darwin’s theory of evolution. Occupy Math’s specialty is designing evolvable structures that make it easier (faster) for evolution to solve problems.

Evolving Polyomino Puzzles

The first talk is joint work with Occupy Math’s student Lauren Taylor. We devised a system for evolving polyomino puzzles. The system evolves one solved version of the puzzle – but the smallest number of ways to assemble any of the puzzles shown above is thousands. You solve a puzzle by assembling the pieces to form a square. The paper is visible here.

A Curves Fitting Technique

The second talk is on a way to fit multiple different types of curves to data – where evolution not only fits the curves, but splits up the data into the points that belong to each curve. Yes, the plural “curves” in the section heading is not a typo! This paper is joint with Occupy Math’s former student Joseph Brown who thought up this cool idea. Dr. Brown is currently a computer science professor in the Russian Federation at Innopolis University. The paper is visible here.

Finding Networks that Exhibit a Specified Behavior

The third talk is on using evolution to find networks (like the one shown about) that behave in a way that solves a problem or helps answer a question posed by a researcher. This talk is on joint work with Meghan Timmins who is defending her master’s thesis this August. This paper works on creating networks with a specified geometric shape; we can also find graphs that are contact networks that could have led to a particular type of epidemic or design networks that are robust against server failure. The paper is visible here.

An Adaptive Generative Representation for Evolutionary Computation

This paper is joint work with my collaborator in Tasmania, James Montgomery. This paper is on a new way to specify problems for evolution that cuts out huge numbers of useless structures from the space evolution operates in. This lets us solve much harder problems in a given amount of time. The research arose from James’ clever ideas about how to evolve RFID antennae, one of which is shown above in digital and physical form. The physical form uses two symmetric copies of the digital form. The paper is visible here.

The Do What’s Possible Representation

This paper is joint work with my current student Sierra Gillis and former students Andrew McEachern, who is a postdoctoral fellow at Queens University in Kingston, Ontario and Jeffrey Tsang, who is in private industry. This work is another take on the representation that James Montgomery thought up – it uses infinite string generators and possibility filters to solve really large problems really fast. By “infinite” Occupy Math means that you can always ask for the next character in the string – we only use the front part of the string, up until we solve the problem. The picture above is a solution to one of our test problems. The paper is visible here.

Divide the Dollar, a Generalization.

Divide the dollar is a game invented by John Nash. It models making a deal (or not making a deal). Two players bid and, if their bids total no more than a dollar, they get their bids; otherwise they get nothing. Occupy Math and his collaborator Garrison Greenwood specify a family of games that include divide the dollar. The picture above shows how the players we evolved learned to bid. We use the game/agent system to look at how a subsidy for players that bid (roughly) fairly helps increase the number of times a deal is made. Next up? We will model how undependable government subsidies might cause trouble. The paper is visible here.

The Impact of Elite Fraction and Population Size on Evolved Iterated Prisoner’s Dilemma Agents

This paper is joint work with my former student Eun-Youn Kim who is currently teaching at Hanbat National University. It is the next in a long series of papers dissecting a technique for modeling conflict and cooperation. The picture above shows how changing some little details in the simulation setup can radically change the outcome. The agents corresponding to the blue tracks cooperate less than the ones corresponding to the red tracks and the green agents are all over the place. The three colors are three different types of agents – but only slightly different. This is a very standard task for someone in math: checking to make sure other researchers really understand what assumptions they made! The paper is visible here.

Conway Crossover to Create Hyperdimensional Point Packings, with Applications

My collaborator Steffen Graether and I develop a new method of choosing well-spaced-out sets of points in as many dimensions as are needed to solve a problem. The three-dimensional version lets us find the contrasting sets of colors, shown above. Dr. Graether posed a problem of identifying proteins, called dehydrins, via their codon usage, something that needed a 64-dimensional set of points to sort out. The paper is mostly about a cool new way of searching for good point packings, but the applications are nifty too. The mathematical point is that, if we had used a regular grid, we would have needed between 18446744073709551616 and 3433683820292512484657849089281 points to sort out the proteins; the evolutionary point packing did the job pretty well with 127 points. The paper is visible here.

Evolvable Warps for Data Normalization

This paper, joint work with my student Jeremy Gilbert, is the only statistics paper in the batch. It’s something that Jeremy did as a senior research project. What we did was use some really advanced math (group theory) to find a way to let evolution straighten out a data set. The picture above shows the original data and the straightened out data. Why bother? This is a new way of learning a formula for the way the data is distributed and also gives us a cool new visualization to tell us how different two sets of data are. The paper is visible here.

Occupy Math hopes you appreciate the diversity of topics this year’s crop of WCCI papers cover!

If you are within driving distance of Guelph, Occupy Math would be happy to give talks about careers in mathematics to your class or club. Any level of students are okay from primary grades all the way up to graduate school. Talks can be about the nature of math, specific research topics, or even just a series of interactive exercises. Interested? Contact Occupy Math at danwell42@gmail.com and we can try and sort it out.

Occupy Math thinks that a better relationship with mathematics, increased numeracy, and logic help people survive in a difficult world. Through research, Occupy Math and his colleagues create the tools that help civilization weather difficulties. This week’s post looks behind the curtain at the part of math that creates new things – comment or tweet if you have problems that you would like us to think about!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics