General Information

---

Project Description

The project will study the extent to which Lagrangian tori in moduli spaces of polygons are displaceable, using the technique of Dusa McDuff.

---

Weekly Log

Week 1:

Dan and I met with our mentor Professor Chris Woodward and graduate student Doug Schultz. To gain the requisite background information on the project we started reading the first two chapters of Ana Cannas da Silva's Lecture's on Symplectic Geometry. The readings consisted of the basic definitions inovlved in Symplectic Manifolds and how one can use certain techniques to find Lagrangians Submanifolds. In addition it was necessary to refer often to John M. Lee's Introduction to Smooth Manifolds.

Week 2:

This week we read through to Chapter 3 of da Silva's lecture series, which was a more in depth look at Lagrangian Submanifolds. In addition we were able to start really getting an idea of what a differential form is.

Week 3:

This week we skipped to reading Chapter 18 of the lecture series as this chapter introduces Hamiltonians and their use in symplectic geometry. Here a lot of reference information from Lee's Intro to Smooth Manifolds was needed with respect to vector fields on manifolds.

Week 4:

This week we read Chapter 21 and 22 and continued to reread the previous material as our understanding of the reference material grew. These chapters expanded on the material of Chapter 18 and introduced the concepts of Lie Group Actions and Moment Maps in the context of symplectic geometry.

Week 5:

This week we kept trying to further our understanding of Chapter's 18, 21, and 22. As we get a better grasp of the background material, its amazing how often we have to look back to what we have already read as things start to make more sense. In addition, we started reading about Symplectic Reduction, as concept mentioned in Chapter's 23 and 24. Actually this tool of symplectic reduction is the last thing we need from da Silva's book, so from now on we will be reading papers.

Week 6:

We have moved on from reading da Silva's lecture series to reading published papers. We read this paper by Jean-Claude Hausmann and Allen Knutson that gives a construction for the moduli space of polygons that our research problem takes place in. After speaking with Doug, we finally started to understand what our research question was. I began to see how symplectic reduction is used throughout the paper to talk about symplectic geometry in the context of polygons.

Week 7:

This week we read through sections of Dusa McDuff's paper Displacing Lagrangian Toric Fibers via Probes which detail the construction and implementation of probes. This tool of Lagrangian displacing probes is actually the main and almost only tool we will be using in answering our question. Everything else up to this point was for the sake of being able to understand the background and theory behind this technique. As well, Daniel and I started working through different examples of moduli spaces with different side lengths to try and understand how the space worked and how to use this probing technique.

Week 8:

This week we attempted to (and did!) solve our problem. Namely, we were able to better map out the points at which one is not able to use the probing technique in this space in the case that the polygon in question has two sides with the same side length.

Week 9:

This week we wrote our paper and made our poster. Also, we discussed with Professor Woodward and Doug Schultz how this work may potentially be expanded upon in the future.

---

Additional Information

• My Mentors
  • Dr. Chris Woodward
  • Doug Schultz