| Mentee: | Paige Dote |
| Mentors/Advisors: | Professor Larry Guth and Shengwen Gan |
| Project: | Marstrand's projection theorem solves a relatively intuitive problem: finding the dimension of the projection of a subset of Rn to a subspace. This idea itself makes sense, but to prove it requires some basic Fourier analysis and some relatively difficult machinery like Hausdorff dimension, measure, etc.. This has already been proven (and thus is a theorem), but the goal for this project is to reprove the theorem using a method known as the high-low method, and then apply this approach to an open conjecture. |
| UPDATE: | Howdy y'all! I really liked updating this while I was doing readings, but now that SPUR has started in actuality, I am going to wait to update this site until after the research is concluded. Don't worry though! I am documenting the experience in a notebook/in LaTeX so I won't lose track of what research actually felt like. |
| Materials: | We are using multiple sources over the course of the summer, including but not limited to (this is a running list):
[M] Mattila: Fourier Analysis and Hausdorff Dimension |
| Week 4 | Jun 13 |
I added a preface written to the reader regarding the idea behind the project, given a lot of people were asking what the reading I was doing was actually for i.e. what the project is that I am working on. It was interesting to write this absed off of the first meeting I had with Shengwen, but with the knowledge I now have about Hausdorff dimension. Preface added |
| Intro | Jun 13 |
Intro to Week 4 notes plus to-do list. Includes new exercise from Larry to try. Intro to Week 4 Notes |
| Week 3 | Jun 08-12 |
I took this week off to spend time with friends and family, and decompress after the spring semester. It proved very useful, but as such there are no notes for this week. Week 3 Notes (or lack thereof) |
| Week 3 | Jun 7 |
Discussion from the meeting with Shengwen. Current plan for the rest of the week is to keep reading through [GSW]. There was a slight delay in getting these notes out as I took some time to decompress, I sincerely apologize. Meeting Notes |
| Intro | Jun 6 |
Intro to week 3, and topics to discuss with Shengwen today. Intro to Week 3 notes |
| Week 2 | Jun 5 |
Notes on the introduction and Proposition 2.1 in [GSW]. To-do list included at end to discuss with Shengwen Monday. Propositon 2.1 notes |
| Week 2 | Jun 3 |
Notes on Lemma 3.13 [FO]. Planning to read [GSW] for meeting with Shengwen Monday. Lemma 3.13 notes |
| Week 2 | Jun 1 |
Notes on §4.1. Planning to read [FO] next. Section 4.1 notes |
| Intro | May 30 |
Intro to week 2 and to-do list. Intro to Week 2 notes |
| Week 1 | May 23-29 |
We started by reading §§2.2,2.5,4 of Mattila. This covered the idea of Hausdorff dimension, s-energy, Frostman's lemma, and the proof of Marstrand's projection theorem. We then worked on calculating the Hausdorff dimension on some classic sets. End of Week 1 notes |
| Intro | May 23 |
The notes for this project started today, and an introduction for the week was made.
Intro to Week 1 notes |