DIMACS
DIMACS REU [2018]

General Information

me
Student: [Froylan Maldonado]
Office: [419]
School: [San Diego City College]
E-mail: [froylan.mal@gmail.com]
Project: [Geometry and Combinatorics of Matroids]

Project Description

In the works...

Weekly Log

Week 1:
Met with my mentor and all the other participants in the REU. I discussed my project with Dr. Tarasca and got the general direction where the project might lead. Started reading through June Huh's paper on matroids.
Week 2:
This week was mostly filled with meeting with my mentor, Dr. Tarasca, and getting familiar with material that is needed to understand matroids. Also doing a lot of proofs to understand the paper our group is working on.
Week 3:
This week was more focused on counting independent sets in a projective plane. Had to learn a lot of new material and understand what the question was asking. Spent more time learning about matroids as well.
Week 4:
Most of my time went into understanding matroids as an abstract object and not as something that is finite. Had to spend more time learning new material and applying previous examples to matroid properties.
Week 5:
This week was mostly spent on working on the chow ring of matroids and understanding how to identify flats for different matroids. Wrote code in SageMath, and Macauly2 to determine the 'basis' of the chow ring of the matroid K_4. The code is abstracted to the point where I would just need the flats of a matroid and it can make the chow ring for it.
Week 6:
I rewrote all my code in Macaulay2 since SageMath doesn't do anything useful with the quotient ring. I also realized that there already exist a package for matroids in Macaulay2 some now I'm able to get the Chow Ring of any matroid. I also wrote some more code to get the amount of degree i generators, i being from zero to rank of the matroid.
Week 7:
This week was mostly spent on working on my presentation and meeting with my mentor to see what would be a good approach to the presentation.
Week 8:
Did the chow ring of the complete graph of 4, and the chow ring of K_4 with different edges deleted to see how the degree i generators of the chow ring change. Also found out that the degree i generator numbers actually have a name; betti numbers. So I got the betti numbers for different graphs and now have an idea how the numbers are represent the combinotorial structure of a matroid. Now I'm going to work on higher rank matroids to see how the betti numbers are made and to see if I can get a general formula for betti numbers on matroids constructed from graphs.
Week 9:
Worked on the final paper and tried to find another pattern with the second Betti number.

Presentations

Additional Information