| Name: | Ava Ostrem |
|---|---|
| Email: | ava.ostrem (at) rutgers.edu |
| Office: | CoRE 448 |
| Home Institution: | Rutgers University - New Brunswick |
| Project: | Compactness in the Mathematical Universe |
| Mentor: | Prof. Filippo Calderoni |
About My Project
Large cardinals axioms postulate the existence of combinatorial properties of infinity. These axioms are far beyond the usual axioms of mathematics and expand the burden of the mathematical universe. A well-studied consequence of large cardinal axioms is compactness. Compactness denotes the extent to which mathematical structures are determined by their local behavior. We will examine some famous compactness properties for algebraic structures and investigate new ones.
Research Log
Week 0: May 27-May 30
I'm going to start counting the weeks at zero to keep in the set theory theme. We moved in to the dorms on Tuesday and had orientation on Wednesday. I started reading Chapter II of Eklof and Mekler's Almost Free Modules for some set theory background. I also set up my website on Friday and felt very accomplished for 'shelling into the server'.
Week 1: June 2-June 6
I kept reading Chapter II of Eklof and Mekler and also worked on some exercises from the textbook to get familiar with the concepts. Specifically, the exercises were about clubs, stationary sets, and some generalizations of Fodor's lemma. I also gave a short presentation on my research topic to the other REU students. I met with Prof. Calderoni twice and we discussed some of the exercises I worked on and a proof of the compactness theorem for free groups.
Week 2: June 9-June 13
This week we worked on generalizing a compactness theorem for free groups to direct sums of cyclic groups. I read through some of Fuch's Abelian Groups to learn more about abelian groups and find relevant theorems. On Friday, we finished the proof of our result. I started reading some papers in preparation for our next task: "A Characterization of Free Abelian Groups" by Juris Steprans, "Coutable Abelian Groups with a Discrete Norm are Free" by John Lawrence, and "Discretely Normed Abelian Groups" by Frank Zorzitto.
Week 3: June 16-June 20
This week we started thinking about compactness theorems for $\omega_1$-strongly compact cardinals. I fleshed out a proof by Magidor that if $\kappa$ is $\omega_1$-strongly compact, then every $\kappa$-free group is free. For the long weekend, I went to Asbury Park with some other REU students.
Week 4: June 23-June 27
This week I focused on the paper "On $\omega_1$-strongly compact cardinals" by Bagaria and Magidor. Specifically, I focused on the reflection principle. I tried to use the reflection principle to prove that if $\kappa$ is $\omega_1$-strongly compact, then all $\kappa$-$\Sigma$-cyclic groups are $\Sigma$-cyclic. On Friday, we realized that one of our previous proofs was incorrect. :(
Week 5: June 30-July 4
We came to the conclusion that using the reflection principle probably wouldn't work to prove that if $\kappa$ is $\omega_1$-strongly compact, then all $\kappa$-$\Sigma$-cyclic groups are $\Sigma$-cyclic. We tried coming up with a few $L_{\omega_1,\omega}$ formulas to characterize $\Sigma$-cyclic groups, but none of them were in a format where we could apply the reflection principle. However, we were able to fix the previous proof that if $\kappa$ is weakly compact, then all groups of size $\kappa$ which are $\kappa$-$\Sigma$-cyclic are $\Sigma$-cyclic.
Week 6: July 7-July 11
After many failed attempts, we tried a new method to prove the $\omega_1$-strongly compact compactness theorem for $\Sigma$-cyclic groups. In Eklof and Mekler, they use an ultraproduct construction to prove the $\omega_1$-strongly compact compactness theorem for free groups. We were able to generalize the strategy they used in the book to $\Sigma$-cyclic groups.
References & Links
Here are the papers and books I have read for my project:- Almost Free Modules, Paul C. Eklof and Alan H. Mekler
- Abelian Groups, László Fuchs
- "A Characterization of Free Abelian Groups," Juris Steprans
- "Coutable Abelian Groups with a Discrete Norm are Free," John Lawrence
- "Discretely Normed Abelian Groups," Frank Zorzitto