| Name: | [Pejmon Shariati] |
|---|---|
| Email: | Pejmon.Shariati (at) tufts.edu |
| Office: | [RM 419, Hill Center] |
| Home Institution: | [Tufts University] |
| Project: | [Geometry and Combinatorics of Matroids] |
Matroid Theory is becoming the current hot topic in the mathematics world and my project is to gain a better understanding of them so that I may help solve some of the still unsolved conjectures.
My goal for this summer is to become fluent in understanding the correlation constant of a matroid, a field, or graph. With that in mind, it would be ideal to determine new boundaries for the correlation constant and find some that are greater than 1. My team's overall goal is to start and hopefully solve Rota's unimodality conjecture.
This week I was assigned to focus on the correlation constant from June Huh's "Combinatorial Applications of the Hodge-Reimann Relations."I then proceeded to study basic concepts in Graph Theory with respect to complete graphs. In addition, I was trying to gain a better understanding of matroids from Hayley Hillman's "Matroid Theory."
I backtracked to an earlier section of June Huh's paper that focused on mixed discriminants and permanents. Together with my team members we used Cauchy's eigenvalue interlacing theorem to formulate a rough proof of certain theorems to help give us a better understanding of the paper.
With respect to my own section concerning the correlation constant I studied lectures on real analysis to familiarize myself with certain definitions like supremum and infimum. I calculate the correlation constant of an incidency matrix representing the complete graph K4. After becoming familiar with how to calculate the correlation constant my goal now was to see how it changes with respect to the field or if the field affects it at all.
This week I dived deeper into graph theory and studied incidency matrices and directed graphs. While keeping up with my own section I wanted to continue to develop my overall understanding of matroids and began reading James Oxley's "Matroid Theory."
Oxley mentions references a lot of concepts in abstract algebra so I began to watch videos on rings and groups. To further educate myself I started reading an introductory textbook in abstract algebra.
My objectives for the forthcoming weeks are to: 1.Design a program to figure the incidencey/adjacency matrix for a complete graph Kn, or to do it by hand and determine a pattern. 2.Determine the correlation constant for any complete graph Kn 3.Prove the correlation constant inequality in Huh's paper
This week I tried to find a common pattern to find the incidency matrix for each compelte graph but soon realized that there was not anything feasible. I decided to find a pattern concerning adjacency matrices which I discovered are primarily used for computing purposes. I found a pattern but when calculating the correlation constant for each matrix I found it to be 1 for every graph so it was not very helpful.
I then turned to Huh's recently published paper "Correlation Bounds for Fields and Matroids" where I began to study the S8 matroid which is referenced in Oxley's book. A correlation constant was given for this matroid so I began to calculate it just to confirm it was correct.
In addition to all of these things I am studying Huh's new paper and continuing to read Oxley and Hillman. My goal is to finish proving the inequality by next week
After trying to calculate the correlation constant for S8 and other matorids I kept reaching incorrect results. Worried that I might have made a mistake I dodownloaded and familiarized myself with the online math program Sage. After inputting my matorids into Sage I noticed that I had made no errors so something else was wrong. I decided to study the rest of "The Correlation Constant of a Field" by Benjamin Schroter and "The Correlation Bounds for Fields and Matroids."
For the rest of the week I had to organize all my work into a presentation to show the rest of the students and mentors in the REU what I have been researching.
This week I came across a breakthrough in my research. When calcualting the total number of bases for S8 and the matroid M 4,2 I always ended up with a number larger than what was accurate. While researching these matroids a little more closesly I realized that when they are represented over the field modulo 2, not only are their entires 0's and 1's but when adding the separate column vectors to see if they are linear combinations of each other I neglected to apply modular arithmetic. After inputting this parameter in sage I found the correct number of bases for each of my previous bases and was able to correct previous correlation constants that I calculated via modulo 2.
I finished Hayley Hillman's paper and plan to finish the other readings I have left. For next week, I will prepare to write my final report for the REU 2018 program.
This week I wrote my final report on all my research for the 2018 REU program.