| Name: | Peter Kilway |
|---|---|
| Home Institution: | University of Notre Dame |
| Email: | pkilway[at]nd.edu |
| Project: | Vortices in Geometry and Physics |
| Project Mentor: | Paul Feehan |
Project Description
Let M be an orientable Riemannian manifold and E a vector bundle over M. The Yang-Mills Equations are a collection of PDEs for a connection A on the vector bundle E which are important in modern particle physics. The Yang-Mills Equations are: $$d_A * F_A = 0$$ Here, \(F_A\) is the associated curvature 2-form of the connection A, * is the Hodge star operator on 2-forms, and \(d_A\) is the differential associated to A. Solutions to special cases of the Yang-Mills Equations can be used to produce invariants which can distinguish smooth structures, starting with the Anti-Self-Dual Equations for a connection on a vector bundle E over a simply-connected orientable Riemannian 4-manifold M, which are given by: $$*F_A = -F_A$$ The Seiberg-Witten Monopole Equations for a connection on a vector bundle over a complex Kahler surface: [define later] And then the Ginzburg-Landau Vortex Equations for a connection on a Hermitian vector bundle E over a complex Kahler manifold: [define later]
One method of finding solutions is via a method of "gluing" which is outlined below: [To be elaborated on] Although the gluing construction has been done for Anti-Self-Dual connections and Seiberg-Witten monopoles, it has not been done for vortices. The goals of this project are twofold: 1. Write a more comprehensible account of the use of the gluing construction on ASD-connections to produce new solutions and determine the moduli space of ASD-connections. 2. Gain novel insight into the prospect of using the gluing construction to produce new solutions to the Ginzburg-Landau Vortex Equations in some degree of generality. Note that the second project goal does not play any significant role in the first 6 weeks of the program.Weekly Research Log
Week 1: 5/27-5/30
Before and during the week, I read sections 2.1.1, 2.1.2, and 2.1.3 of Donaldson and Kronheimer's book on 4-manifolds in order to aquire important information about Connections on Vector Bundles, the Curvature Form of a Connection, and Anti-Self-Dual Connections on simply-connected orientable Riemannian 4-Manifolds. I met with Paul Feehan on Thursday. Over the course of this meeting, we discussed the basics of mathematical gauge theory, reviewing, refining, and elaborating on the material I had read beforehand. We also defined the energy functional on connections over a vector bundle, and stated the Yang-Mills Equations for a connection on a vector bundle. At the end of the meeting, we briefly discussed the practice of handling PDEs on manifolds and discussed the various equations that I will attempt to understand. These equations are: The Anti-Self-Dual Equations for a connection on a vector bundle over a simply-connected orientable Riemannian 4-manifold, the Seiberg-Witten Monopole Equations, and (special cases of) the Vortex Equations for a Connection on a Hermitian vector bundle over a Kahler manifold. At the present moment I only know the Anti-Self-Dual Equations and will be focusing on studying them for the immediate future. Next week I hope to parse through section 4.2 of Donaldson and Kronheimer's book(which I will call "the 4-manifolds book" as convenient). I hope to establish a routine involving cooking for myself, going on walks around campus, and exercising regularly over the course of week 2, and my non-academic priorities all involve this goal. My roommates and I have begun hosting board game nights.
Week 2
The start of this week began with a somewhat jarring transition into the more analytic side of the required material. I specifically focused on Fredholm operators, elliptic differential operators and their Green's functions, and the Sobolev embedding theorem, which are all ingredients of the gluing of Anti-Self-Dual connections found in section 7.2 of Donaldson and Kronheimer's book. In addition to this, I met with Dr. Feehan on tuesday, during which time he outlined the gluing construction with a focus on the clarity of the ethos of the procedure rather than all the smaller analytic details. After our meeting I gave a short presentation which introduced some motivation for my project to the other DIMACS participants. I also managed to parse through section 4.2 of the 4-manifolds book, which was a goal I set for myself last week. The next goal I have for the project is reading through section 7.2 of the 4-manifolds book, which is the first major milestone of the project. This will not be easy, as the section is extremely information-dense and difficult to read. I expect a lot of friction but hope to understand it well by the end of week 3. Dr. Feehan says that by becoming familiar with the techniques found in that section, I'll have a better working knowledge of the tools used to solve these problems and be prepared to try to gain insight into gluing vortices. Every evening, my roommate Kyle and I go for a somewhat long walk through campus. Although these walks were previously walking through a part of campus that we knew somewhat well, we started discovering new sectors of campus. I found a path through campus that looks really creepy at night. We want to walk through it past midnight by the end of this week. Maybe Arham will tag along. Hopefully we don't die.
Week 3
Luckily nobody died whilst walking late at night! This week marked my transition from the "I don't know what I'm doing" phase to the "I don't know what I'm doing but I'm doing something anyways" phase of the project. I emailed Dr. Feehan about my troubles with reading section 7.2 of Donaldson and Kronheimer's book and he explained that the section is confusing even for senior researchers in the field, and that the first project goal is to write a more comprehensible version of that section to help those that come after. During our meeting he gave an outline of the gluing construction for ASD-connections and instructed me to read certain portions of section 7.2.2. I have begun writing an introduction to the account we hope to write. Now at the end of this week, it includes a more detailed account of most of section 7.2.1, with a few statements missing. I hope to finish this after asking Dr. Feehan some questions on Tuesday, and turn my focus towards the confusing parts of section 7.2.2 in the meantime. On Friday, we had a culture day presentation and decided to organize a pasta cookoff. The details of this may or may not be disseminated via next week's report. Also, on Monday, a group of DIMACS participants, including myself, headed to Hill Center 114 to watch a movie. After that, we visited the Hill Center lobby ice cream vending machine. Edward tried purchasing an ice cream bar with apple pay. The vending machine took a full minute to process the transaction, subsequently declined, and switched into cash only mode. I guess the Hill Center vending machines don't like broke boys.
Week 4
I ended up being very wrong about how difficult parsing through section 7.2.1 would be. There is a lot of detail and justification left out, and I've had to manually go in and patch the exposition together to make it rigorous and coherent. I am still working on this problem at the time of writing it. Luckily, the rest of section 7.2 consists of largely straightforward analysis, so the pace of the project should pick up after that point. On Wednesday, the DIMACS participants got together and ran a pasta cookoff. I was one of the three judges. The Czech group ended up winning, and my roommates took the runner-up position. The best part about it was that I got to eat really tasty pasta for free. One would think that nobody in this program would be able to cook well because we're all just a bunch of basement-dwelling math and computer science nerds, but such a notion ended up being wildly inaccurate that night.
Week 5
I spent all of my research time this week rewriting section 7.2.1. At this point I am almost done supplying the missing detail, and I have one missing proof left. Next week I hope to move on to the relatively straightforward analysis in section 7.2.2 and beyond, so the pace of the project should pick up then and shouldn't take more than 2-3 weeks.
Project References
- Donaldson, S. K., & Kronheimer, P. B. (1990). The geometry of four-manifolds. Clarendon Press.