We’re just about halfway done with the summer now! It really doesn’t feel like it’s been a month and we’re already beginning to prepare for the end of the program. This entails a big write up of all of the things we’ve done so far, which is still incomplete at 11 pages. The plan at the end is to take this super detailed document and prune it down to the most important sections to give us the basis for our paper.
In addition to our paper, Kayleigh and I will give a short presentation on our research to the other STEM scholars at the end of the program. All of the STEM scholars presented quick “chalk talks” on a difficult topic from our research in our Monday meeting in preparation. Kayleigh and I decided to do ours together. The intention of a chalk talk is to give an interactive presentation with the presenter drawing and writing on a chalkboard (or whiteboard, in our case) as they speak. The presenter should have an outline, but also be prepared for interruptions and questions.
Our presentation included explanations of what functions and dynamical systems were, as well as how our function operated. We gave some example plots of our function and related elements of the plots to real world population dynamics so that the pictures would be more palatable. In one of our meetings with Dr. Bayless we had spoken about how mentioning math can scare people away from a topic quickly. Tangible examples and clear definitions are good ways of avoiding intimidation. One of the members of our presentation group even explicitly noted that the population dynamics reference made our function much clearer. The person that went before us presented on how the oxidation of certain elements in cells caused the creation of dangerous ions. The person after us hadn’t actually started her research yet, but presented from the paper her work would be based off of. The paper was about attempts to make new bacteria that could produce ethanol. I thought that this last presentation was interesting because I was surprised at how much I remembered from biology. I could easily recognise how little I would’ve been able to understand without at least an intro bio class. It made me realise how daunting science presentations to the general public can be, because the presenter may not know how much experience the audience will have.
In this week’s presentation to Dr. Bayless I was able to give a proof analogous to Lemma 2.6 on -swift points from (1), and I would say it went very solidly OK for a first run. Dr. Bayless and Kayleigh both had lots of questions for my definition and proof, and I incorporated most of them into what I’ve included in my upload. One thing that was helpful for their understanding was my taking a moment to draw a simplified picture of one of the concepts from the proof, although this was not included in the Beamer itself. It was also pointed out that longer pauses between lines of a proof would be helpful, and I think incorporating short diagrams and rephrasings would accomplish that. Dr. Bayless gave two other tips for math presentations. One is that clearly signaling when you are moving on to a new section is a good way to ensure that your audience is moving with you. The other is that when giving presentations it is often more useful to include an example for a claim and refer people to the proof on another source than it is to walk an audience through an entire proof.
In the Beamer this week we proved Lemma 2.6 in full for periodic points and each had our own section. My section included my current definition for an -swift point:
Let there be some point . Define to be the set of all iterates of and let be cyclically ordered. For and we say is -swift if the following conditions are met:
- the open arc contains exactly points, all of which are periodic and are not in .
I also went through a step-by-step process to choosing a and for which a given orbit and -swift point are possible. I think this will be useful for explaining the function and possibly help lead to a proof that an -swift point implies . As mentioned above, I proved the majority of Lemma 2.6 for -swift points, although after a meeting with Dr. Bayless today I think I will split the proof up into several Lemmas and change notation. Kayleigh presented this week on new definitions for her proofs and what happens when a point is placed between points of an existing orbit.
The Beamer presentation for this week is here!
Adamaszek, Michal, Henry Adams, and Francis Motta. “Random Cyclic Dynamical Systems.” Eprint ArXiv:1511.07832 (2015): n. pag. Web.