| 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
This week 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.
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