My project involves investigating the Vietoris Rips Complex and began with emphasis placed on finding a strategy to discover links between 1-cycles using the persistent homology of the complex. Since then, the focus has transitioned to finding the bounds for the lifespan of a cycle under the Vietoris-Rips Complex.
- Week 1:
- I will be presenting my initial question on friday, June 8th. Thus far, we have been developing the Vietoris Rips complex and how we may apply it to various data sets.
- Week 2:
- I have been using the Perseus program. I mean to first, discretize functions that display the behavior I am searching for, and then test the resulting finite data set in order to gain experience with the program and investigate the overall behavior.
- Week 3:
- This past week I have been investigating a more formal treatment of simplicial homology in order to better understand how the Perseus program calculates homology.
- I have also begun searching for real questions to pursue in the comng weeks. One thread I plan on following is whether it is possible to determine links by investigating the lifespans of different cycles using Perseus. And, if not immediately possible, under what conditions can the question be answered with confidence.
- Week 4:
- In order to force the issue of a certain barcode necessitating a specific kind of complex, I have been investigating maximal lifespan for a 1-cycle under the Vietoris Rips Complex. Furthermoe, I have been verifying theoretical results with computational examples, as well as continuing to study a more formal treatment of Simplicial Homology and Cohomolgy.
- Week 5:
- I have begun compiling my final write up and am in the process of editting the material to ensure accuracy, as well as typing up theoretical and computational verification of basic results.
- Furthermore, I will also be continuing my investigation of maximal lifespan on 1-cycles searching for a strategy of going from general 1-cycles in 2-space to convex cycles with the hope that we will be able to generalize from n-space into 2-space first.
- week 6:
- As well as continuing to augment and edit my final writeup, I will also be using the material to produce a concise final presentation. Moreover, I hope to develop a strategy for altering planar sets of vertices that will eventually show that maximal lifespan on a 1-cycle occurs with a regular polygon.
- week 7:
- I will take some time away from my research in order to produce a final presentation that unfortunately will inevitably fail to cover all results and abandoned strategies I have investigated over the last few weeks. What I do plan on covering are some conditional statements and defintions I have produced in order to investigate a more relevant set of 1-cycles, followed by a justification for one of the succesful strategies.
- week 8:
- During this final week I will be finalizing my Writeup for the REU. The majority of the writeup will consist of what I have thus far compiled as actual mathematical work, but a short response to the actual experience will also have to be included.
- I will also be continuing my research by investigating a new approach to finding a proof to go from uniform convex polygons to the standard k+1 polygon.