Weekly Log

Week 1 (June 5-9):

This week I read select chapters of Bott/Tu's Differential Forms in Algebraic Topology for a refresher on deRham cohomology. I also started reading Geiges' An Introduction to Contact Topology.

Week 2 (June 5-9):

This week I read select chapters of Bott/Tu's Differential Forms in Algebraic Topology for a refresher on deRham cohomology. I also started reading Geiges' An Introduction to Contact Topology.

Week 3 (June 12-16):

This week I began reading about Liouville vector fields, Contact structures on odd spheres, and Gray Stability and the Moser trick. I mostly consulted chapters 1&2 of Geiges, but also consulted Eilashberg's notes on Lie differentiation and this excellent blog post by Mathews on a more geometric intuition of Liouville and Contact geometry.

I then began reading about fillings. I started with chapter 5 of Geiges, and then moved on to this paper by Etnyre.

After, I began learning about Legendrian Contact homology. Specifically, I worked to understand the Chekanov-Eliashburg Differential Graded Algebra (DGA) for the Lagrangian Projection of a given Legendrian knot. I read Ghiggini's notes on the subject and cross-referenced Etnyre and Ng's paper on Legendrian Contact Homology in $\mathbb{R}^3$. While I understood the combinatorial computations involved, next week, I'll need to read more on J-holomorphic curve theory to understand the Augmentations in Chekanov's DGA.

Finally, I read chapter 5 of Wendyl's Lecture Notes on Bundles and Connections to learn about Riemmanian connections on fibre bundles.

Week 4 (June 19-23):

I began this week by computing the augmentations of the DGA $(\mathcal{A},\partial)$ for the right-handed trefoil explained in Example 3.17 of Ghiggini. These five augmentations correspond to five exact Lagrangian Fillings. To understand the explicit fillings in this example, I read the paper Legendrian Knots and Exact Lagrangian Cobordisms by Ekholm, Honda, and Kálmán, as well as slides by Ng on a related talk. I aso referenced this PhD thesis by Pan to further understand the augmentations of $(2,n)$ Torus knots. On Thursday, we met with Chindu Mohanakumar to clarify the idea of a Saddle Cobordism, which is a key step in the resolution of the torus knots and thus construction of the augmentations.

After our meeting, I began reading the paper Floer Cohomology and Disk Instantons of Lagrangian Torus Fibers in Fano Toric Manifolds by Cho and Oh. For background, I read this survey and this blog post to better understand Blashke products, as well as these slides to and this paper to understand the Maslov index.

Week 5 (June 26-30):

This week, I began by continuing to read Cho Oh. During our meeting on Tuesday, Prof. Woodward explained the proof of why the Clifford torus $\mathbb{T}^2\subset \mathbb{S}^5$ admits no exact Lagrangian filling. Essentially, the obstruction is that all smooth connected $1$-manifolds are either diffeomorphic to $\mathbb{S}^1$ or the interval: ie there is either 0 boundary points, or $2$. He also explained more of the nitty-gritty details of the project: finding (or disproving the existance of) an exact Lagrangian filling of the first Hirzebruch surface. He sketched why he doesn't think a filling exists. The gist of the argument was as follows: take $\Lambda$ to be our Legendrian and $L$ an exact filling. Then, consider $H_1(\Lambda)$ and $H_1(L)$, where elements of $H_1(\Lambda)$ are broken loops in $\Lambda$ capped by some Reeb chord in $L$. Because the dimension of the kernel of the map from $H_1(\Lambda)\rightarrow H_1(L)$ should be $1$, the four elements of the $H_1(\Lambda)$ appearing from the four edges of the Hirzebruch surface must be related -- which is unlikely. Of course, this whole argument is sketchy, so I'm hoping to prove this myself later. We're also hoping to extend this result to higher dimensions, such as $\mathbb{T}^3\subset\mathbb{S}^7$

During the second part of the week, I looked at Legendrian Contact homology with coefficients, and refreshed my memory of some of the toric geometry I read last summer. Specifically, I read parts of an in-preparation paper by Prof. Woodward, this book by de Silva on moment polytopes, the paper The Contact Homology of Legendrian Submanifolds in $\mathbb{R}^{2n+1}$ by Ekholm, Etnyre, and Sullivan, and the paper on Contact Homology of Legendrian Knots in Five-Dimensional Circle bundles by Asplund.

Week 6 (July 3-7):

This week, I began thinking about homological obstructions to exact fillings of the 3-torus $\mathbb{T}^3$. To do so, I began by looking at examples from this paper by Rizell and Golovko, as well as Lemma 3.2.4 from Professor Woodward's paper with Wehrheim. For our purposes this lemma states that given $L$ a filling of $\Lambda=\mathbb{T}^3$ and $\iota_*: H(L)\rightarrow H(\Lambda)$ the pullback of the inclusion map then $\text{dim}(\iota_*(H(L)))=\frac{1}{2}\text{dim}(H(\Lambda))$. Since $H(\lambda)$ is $6$-dimensional, to find an obstruction to a filling, we must find $4$ relations in the (combined) first and second (co)homology groups of $\mathbb{T}^3$. During our two meetings, we worked out that $\mathbb{T}^2$ does have a filling, while $\mathbb{T}^3$ doesn't. For the rest of the week, I'll consider $\mathbb{T}^n$, which we conjecture has no exact filling for $n\ge 3$. In particular, I'll look for some combinatorial characterization of the relations in (co)homology.

Week 7 (July 10-14)

This week, I thought about the minimal and maximum number of relations present in $H_1(\mathbb{T}^n)$. Given a basis $e_1,\dots,e_n,-e_1,\dots,-e_n$ representing Maslov index 2 Holomorphic disks and a set of pairings $\{e_i,e_j\}$ (cobordisms), what are the possible dimensions of the span of the set of pairings? We conjecture that it is in the set $\{\lceil \frac{n}{2}\rceil,\dots,n\}$.

I also spent a lot of time this week reading and attempting to understand Gromov Compactness and the relevant theory of J-holomorphic curves. I've learned bits and pieces of it throughout the summer, but I'm nowhere close to an actual understanding of the theory.

Week 8 (July 17-21)

This week, I finished the proof of the $H_1$ case and moved on to thinking about the $H_{n-1}$ case. I also wrote my talk for Friday, which you can see here. I decided that a technical talk detailing everything I did and my full results would be impossible given my audience's current background and the time constraints, so I wrote the talk to be as expository and `intuitive' as possible. I wasn't able to be super-productive research-wise, but I think I know how to prove what I need to for the $H_{n-1}$ case.

On Wednesday, Prof. Woodward, Soham and I went to the Holomorphic Methods in Low Dimensional Topology Conference hosted at Princeton. I listened to talks by Ali Daemi, Robert Lipshitz, Laura Starkston--Dr. Lipshitz' was particularly interesting, as he discussed torsion in Legendrian Contact homology. Unfortunately due to the REU talk schedule, I wasn't able to attend on Thursday or Friday.

Week 9 (July 24-28)

This week I wrote up my final paper and packed.

Presentations

• Intro Presentation
• Final Presentation

Additional Information

• My Mentors
  • Prof. Christopher Woodward
  • Dr. Yuhan Sun
  • Soham Chanda
• My Personal Website

Acknowledgements

• This work was carried out while the author, Jemma Schroder, was a participant in the 2023 DIMACS REU program at Rutgers University, supported by NSF award 2105417.
• The Pac-Man animation above was found here.