Weekly Log:
Week 1 (January 21st – January 27th)
The first meeting was cancelled due to freezing temperatures. I took the week to decide what I wanted to start diving deep into. I enjoyed learning the basics of set theory from my intro to proofs course. The subject that stood out to me most in set theory is the idea of Induction. I enjoyed learning about the Principle of Mathematical Induction and strong induction.
I also knew that I wanted to combine my research from induction into my honors thesis somehow.
Week 2 (January 28th – February 3rd)
In this week, I solidified my topic interest which is transfinite induction and how it works with the fields surrounding it such as graph theory. I was also given the book An Outline of Set Theory by James Henle. In reading the introduction, I was going to slowly build my proofs up from the very basics up till I reach ordinal and cardinal numbers which is where transfinite induction occurs.
Over the week, I started with the first couple of pages such as the introduction, the history of set theory. I later continued onto the first project which was just basic statements introducing me to the notation and the way the proofs will be solved. The first project just involved subset and proper subset notation as well as negation of universal and existential statements.
Week 3 (February 4th – February 10th)
This week I continued working on the second and third projects of the book. Project two essentially wanted me to take a collection of sets and find which set matched with the description given. In example, (i) x is an element of e. Using this, I had to find which set given to me in the problem is x. This project helped me understand the relationships between the sets. Project three was about writing the axioms of Zermelo-Fraenkel Set Theory using the abbreviations defined in the previous projects. It felt good to re-write from English sentences to mathematical statements since it practiced my understanding of the symbols.
Week 4 (February 11th – February 17th)
I ended up bruising my heel badly this week and had to stay in my room resting. Despite having to miss school and our meetings, I continued to work on the book projects.
This week, I started to work on Project four of the booklet. This project had me prove different theorems regarding sets. Theorem 1.1. goes: If A and B are sets, then A intersection B is a set. As I worked through the first couple of theorems, I realized that the previous theorems helped me remember key components of these bigger proofs.
Aside from working on my projects, I continued to work through my Honors Thesis plan. I am currently working on my bibliography entry and how I will introduce my Thesis to the audience.
Week 5 (February 18th – February 24th)
I was able to make it to the lab meeting this week. I had a couple of questions regarding some of the Theorems in Project four from the book. One of the theorem is 1.7: The sets described in the Union and the Power Set axiom are unique. At first this tripped me up but after receiving help from both Dr. Boney and the other fellows, it became clear.
Essentially the proof involves considering that we can make a new set (called it C prime) and we can show how based on the separation axiom then we can show C is a subset of C’ and C’ is a subset of C, thus C = C’. Hence, the set is unique. The proof tripped me up the first time I thought about it but after explaining it in my own words, it made so much sense. Over Spring Break, I plan on getting Project Five done and maybe starting Project Six.
Aside from proofs, I started writing my Honors Thesis Literature Review. I begun the search for mathematical sources. My professor for the Capstone course is requiring us to write a ten page lit. review for a non-math lover to read and understand exactly what I am trying to explain to them. I still want to explore transfinite induction but I will have to develop the ability to teach somebody who has never heard of set theory.
and how I will introduce my Thesis to the audience.
Week 6 (February 25th – March 2nd)
In lab, I continued to ask questions regarding Project four. I also mentioned a lot about my literature review and asked for advice and opinions. I explained that I want to introduce the idea of set theory and induction to any mathematical level and how I can be efficient with that.
This week was primarily dedicated to working on my literature review for a gigantic amount of time. I continued reading through Henle’s Projects and book. I found extra sources about set theory which I started to read as well. I paused on trying to do the projects continuing from project four as I needed the most amount of time to write my review.
Week 7 (March 3rd – March 9th
I kept reading more books over set theory and the axiom of choice to continue writing my literature review. Not much math aside from reading about math.
I did however meet with my Graduate Mentor and she helped me structure my project which was helpful. She also mentioned the Mathematics Annual Celebration which I was not sure about going to but now I am.
Week 8 (March 10th – March 16th)
SPRING BREAK!!
Week 9 (March 17th – March 23rd)
I was extremely close to finishing my literature review, but then my Thesis Professor mentioned how the review has to be 10 pages double spaced. I had 20 double spaced pages. I essentially had to crush everything in half and make it make sense still. Although it was a struggle, I think it worked out in my favor. I continued reading more books and choosing which would be the best for my interests and goals.
Week 10 (March 24th – March 30th)
After a long time of hard work and reading through various books such as Henle’s An Outline of Set Theory, Jech’s The Axiom of Choice, Enderton’s Elements of Set Theory, I finally submitted my literature review. Although I submitted it simply because of the due date for my research class, I plan on adding much more to it and changing it over the course of the year.
In lab this week, I was not able to provide much as I was too busy studying for both of my midterms for Analysis 1 and Linear Algebra. I however learned from my labmates about what Zorn’s Lemma is and how I can possibly apply it to my research.
Week 11 (March 31st – April 6th)
This week in lab I introduced project five and asked questions regarding it. The fifth project is over Cartesian Products and what set satisfies the products. As defined, A x B is the collection of all ordered pairs, (a,b) such that a is in A and b is in B. Project five states: We must define a set to represent the ordered pair (a,b). It must have the property that (a,b) = (c,d) iff both a=c and b=d. The correct choice from the ones given is: {{a}, {b}} since this set represents both sets of a and sets of b.
In my thesis course, I am now required to write a proposal for my research. I have to include my motivations and qualifications as to why I believe I can conduct this research and explain it to those who are not familiar with it. The proposal includes a mini literature review, a biography of myself, and a methods and results secction.
Week 12 (April 7th – April 13th)
There was no lab meeting this week, so I continued working through my thesis proposal.
This is the week of my interview for the Summer Undergraduate Research over Khovanov Differentiation with a faculty member here at Texas State University. Despite never seeing this topic before, I feel confident that I can be a great addition to the research team.
Week 13 (April 14th – April 20th)
I met with Dr. Boney twice this week. First, we all met as a group in lab. I introduced project six from Henle’s An Outline on Set Theory. Project six is about relations and equivalence classes. The theorem stated, “If A is a relation, then prove that the equivalence class [a]r is also a set.” In lab, I mentioned that I proved it using the definition of relations, but I believe there is another way of proving this.
I also met one on one with Dr. Boney to go over my literature review for my thesis. I was given advice on my set theory notation along with examples. Aside from these minor/fixable parts of my thesis, I also found my final theorem for my thesis project. I will be proving the equivalence between the Axiom of Choice theorem and the Well Ordering theorem using transfinite induction and transfinite recursion.
I also heard back from Dr. Lee about the assistantship and I am hired!
Week 14 (April 21st – April 27th)