| Name: | Sangjun Ko |
|---|---|
| Email: | sangjun.ko@rutgers.edu (sangjun dot ko at rutgers dot edu) |
| Home Institution: | Rutgers University - New Brunswick |
| Project: | Morse Flow Trees |
| Mentor: | Professor Chris Woodward |
About My Project
Morse Theory is a branch of differential topology that studies the shape of smooth manifolds using smooth, real-valued functions defined on these manifolds and their critical points. Loosely, flow trees are maps tree-like objects in a manifold where each branch is identified with a gradient flow line of some difference of functions; Morse flow trees have an additional condition that each edge is a Morse trajectory of a difference of functions. For my project I plan to study the topology of Morse flow trees: is it compact and Hausdorff, or what conditions can we impose to make it compact and Hausdorff?
Research Log
Week 1: May 24 - May 28
I had already been reading Yukio Matsumoto's An Introduction to Morse Theory a little before the program started, and this week I continued reading chapters 2 and 3 from it. I learned about the basics of Morse Theory: about Morse functions and handle decompositions. For next week I intend to learn some of the basics of Symplectic Geometry from Ana Cannas da Silva's Lectures on Symplectic Geometry; for a more specific goal, I want to get to the definition of a Lagrangian submanifold. I'm slightly worried because I may lack some of the necessary background in differential topology, but I hope that supplementing any gaps with Lee's Introduction to Smooth Manifolds (by my partner Kenny's recommendation!) will be sufficient.
Week 2: May 31 - June 4
This week I spent some more time learning background material, this time in symplectic geometry, by reading the beginning of Ana Cannas da Silva's Lectures on Symplectic Geometry. In particular, I learned about symplectic vector spaces, forms, manifolds, and most importantly Lagrangian submanifolds and some examples of Lagrangian submanifolds. This week was also when we gave an introductory presentation for other participants of the DIMACS REU about what we are doing. For next week I plan to go back to learning Morse theory, this time from Audin and Damian's Morse Theory and Floer Homology about the space of broken trajectories.
Week 3: June 7 - June 11
This week I learned about the space of broken Morse Trajectories (whose topology is defined using a neighborhood basis consisting of a broken trajectory and the entry and exit neighborhoods of each Morse chart) and went through the proof of the fact that it is compact and Hausdorff in Audin and Damian. I also spent some time reading Tobias Eckholm's paper (linked below) on Morse flow trees to try and understand its definition. This is (to little surprise) probably the hardest thing I have had to do so far! I intend to continue parsing this paper to look at what happens at the vertices of these flow trees.
Week 4: June 14 - June 18
This week I continued reading Morse flow trees, in particular about punctures and the types of vertices that are allowed in rigid flow trees. In our discussions, we started thinking about the problem at hand to show compactness for Morse flow trees. Kenny devised a scheme to define a topology for the space of flow trees using function spaces. It seems that our general strategy will be to show compactness using sequences; that is, show that for every sequence of flow trees, there is a convergent subsequence. Currently our "worry" is that we might not have a way to bound the number of edges in a tree, so a sequence of trees might have the number of branches going off into infinity, which is not allowed. So the task is to think about how this could be bound, particularly how the adding of branches at a vertex can be controlled.
Week 5: June 21 - June 25
This week we broke down the problem into three subproblems: given a fixed number of edges, does any sequence of Morse trees, by passing to a subsequence if necessary, converge; is the number of vertices in between folds bounded; and is the number of vertices at fold singularities bounded. Kenny will be working on the first subproblem, and I will be looking at the number of $Y_0$ vertices in between folds. The general strategy will be to consider a sequence of Morse flow trees where the number of $Y_0$ vertices are unbounded and derive a contradiction from it.
Week 6: June 28 - July 2
I spent much of the time rereading Ekholm's paper and thinking about $Y_1$ vertices. At the end of the week I discussed with Kenny and Professor Woodward to consider a different problem involving Morse flow trees. One possible question to ask is if the image of space of Morse flow trees under the Lagrangian projection is a bijection or not.
Week 7: July 5 - July 9
After another discussion I settled on a different problem to work on for the remainder of the program. We first recall that a Morse complex for a Morse function $f$ is defined as $C_k(f):=\left\{\sum_{c\in\operatorname{Crit}_k(f)}a_c c\,\Big|\, a_c\in \mathbb{Z}_2\right\}$, and the differential, if $a$ is a critical point of Morse index $k$, is defined as $\partial C_k(a)=\sum_{b\in\operatorname{Crit}_{k-1}(f)}n_k(a,b) b$, where $n_k(a,b)$ denotes the number of Morse trajectories going from $a$ to $b$ modulo 2. The proof that $\partial^2=0$ in this context can be found in Audin and Damian (below). Now if we were to change the context slightly so that we define the complex for a Morse flow tree $\Gamma$ to be analogous except instead of critical points of index $k$, we are looking at punctures of $\Gamma$ with Morse index $k$, and the differential similarly also, is $\partial^2=0$, and if so, why? To do this we will need to consider if the moduli space of flow trees (in dimension 1) is a manifold, for if this is the case then the boundary points will be even numbers and we will be done.
Week 8: July 12 - July 16
For this week, I worked on a particular example of a flow tree in 1 dimension. In this particular example, we had a puncture that was replaced with a $Y_0$ vertex and let it approach another vertex so that it would start to look like a puncture. I also started to work on the report to submit to DIMACS.
Week 9: July 19 - July 23
I spent most of the week preparing for the presentation on Thursday as well as working on the report to submit to DIMACS. In a conversation with Professor Woodward, we decided that I could consider the case of Morse gluing (that is, slightly perturbing the trajectory so that it is no longer broken) to further work on the problem.
Links
Visit my partner Kenneth Blakey's website here!
Click here to return to the REU main page.
References
Here are some of the references I used during the duration of the program in no particular order--I have also made some references to these in my research log above. I am also providing links to where one may procure the books (I personally like using the tool BookFinder to buy used books). This list will grow longer as the program progresses!
- Y. Matsumoto, An Introduction to Morse Theory, trans. Kiki Hudson and Masahico Saito, American Mathematical Society, Providence, 2002.
- A. Cannas da Silva, Lectures on Symplectic Geometry, Springer-Verlag, Berlin, 2008.
- J.M. Lee, Introduction to Smooth Manifolds, Second Edition, Springer, New York, 2012.
- M. Audin and M. Damian, Morse Theory and Floer Homology, trans. Reinie Erné, Springer-Verlag, London, 2014.
- T. Ekholm, Morse flow trees and Legendrian contact homology in 1-jet spaces, Geometry & Topology 11:1083-1224, 2007.