DIMACS
DIMACS REU 2016

General Information

me
Student: Daniel Gallagher
Office: CoRe Building 5th floor
School: Boston College
E-mail: gallagdm@bc.edu
Project: Displaced Lagrangians in the Space of Polygons
Partner: George Jeffreys

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:
I met with Professor Chris Woodward and Doug Schultz, my advisors for the summer, and I started reading through the first two chapters of Ana Cannas da Silva's Lecture's on Symplectic Geometry and the chapter on differential forms from John Lee's Introduction to Smooth Manifolds. The second chapter from da Silva's book was required substantial outside reading on cotangent bundles to understand everything.
Week 2:
Continued reading through the second and third. Chapter three is on Larangian submanifolds of symplectic manifolds, one of the key objects of study.
Week 3:
Continued reading through chapter three and began reading chapter eighteen. Chapter eighteen is on Hamiltonian vector fields and I regularly referenced the corresponding chapters from Lee on vector fields.
Week 4:
Chapters twenty-one and twenty-two required substantial outside study on Lie Theory. These chapters deal with moment maps and are defined in terms of Lie algebras and Lie groups. I used Lee as my main reference point; however, because I do not have any background with Lie theory, this week's material was very slow going.
Week 5:
I continued going over chapters twenty-one adn twenty-two, but I also moved forward to Symplectic reduction.
Week 6:
I read a paper by Jean-Claude Hausmann and Allen Knutson that constructed the moduli space of polygons. The paper was difficult to traverse upon the first several passes, but I began to get a feel for how the space is shaped and how objects move through it after speaking with George and Doug. I began to see how symplectic reduction is used throughout the paper to generalize to a moduli space.
Week 7:
I read through the sections of Dusa McDuff's paper Displacing Lagrangian Toric Fibers via Probes which detail the construction and implementation of probes. In conjuction with the paper, George and I started working through different examples of moduli spaces with different side lengths to understand how the space worked.

Additional Information