DIMACS REU 2021
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Kenneth Blakey ScB Mathematics Student, Brown University kenneth_blakey[at]brown[dot]edu |
Morse flow trees are combinations of gradient trajectories. They were introduced by Fukaya-Oh and Ekholm to capture the symplectic topology of submanifolds of cotangent bundles. We'll look at the questions like: Are moduli spaces of Morse flow trees compact for very general defining data? What happens when the submanifolds are immersed? Are the moduli spaces of flow trees cobordant for different choices?
Any math below is written as quick summaries of already known results. No unpublished results will be shown in detail. And all math is subject to typos - if any arise feel free to email me.
Morse theory
is the study of the topology of a manifold via studying smooth functions on that manifold. In particular, let $M$ be a smooth manifold. A function $f\in C^\infty(M)$ is a Morse function if all critical points $p$ of $f$ are non-degenerate, i.e, that $\det(\nabla^2 f\vert_p)\neq0$ (the determinant of the Hessian of $f$ at $p$). A simple example is the $n$-sphere equipped with the height function. From here we may define the (psuedo)-gradient vector field $X\in\mathcal{X}(M)$ of $f$ and study its flow lines. In particular, if the pair $(f,X)$ is
Morse-Smale,
we can now study the topology of $M$ quite deeply.
I spent some time looking at Morse Theory and Floer Homology reading topics such as: Morse functions, Morse charts, (pseudo)-gradient flows, (un)stable submanifolds, and Morse homology.
Symplectic topology/geometry
is the study of symplectic manifolds. In particular, let $M$ be a smooth manifold. A closed 2-form $\omega\in\Omega^2(M)$ is symplectic if $\omega_p$ is non-degenerate for all $p\in M$. Then $(M,\omega)$ is called a symplectic manifold. One can then show that, by non-degeneracy, all symplectic manifolds are even-dimensional. A simple example is $(\mathbb{R}^{2n},\{x^i,y^i\})$ with the standard form $\omega=\sum_i dx^i\wedge dy^i$. More interesting examples arise when considering the cotangent bundle of $M$, $T^*M$, equipped with the symplectic form $\omega=-d\theta$ where $\theta$ is the tautological one-form. Another quick note is that, unlike (semi)-Riemannian geometry which introduces curvature, Darboux's theorem shows there are no local invariants of a symplectic manifold.
I also spent some time looking at Introduction to Symplectic Geometry and Lectures on Symplectic Geometry reading topics such as: linear symplectic geometry, symplectic (sub)manifolds, isotopy extension theorems, and almost complex structures.
This week I read through Maxine Calle's Bachelor of Arts Thesis Morse Theory and Flow Categories. Specifically, I looked through section 2 titled The Flow Category. The idea is, for a Morse function on a compact manifold $M$ and $a,b\in\operatorname{Crif}(f)$, to reparameterize the gradient flow $\varphi$ by its "height-reparametrization"
$$
\gamma=\varphi\circ h^{-1}:(f(b),f(a))\rightarrow M\;\mbox{where}\;h=f(\varphi)
$$
and extend such that $\gamma(f(b))=b,\gamma(f(a))=a$. Then one defines the space of broken (height-parameterized) flow lines,
$$
\overline{\mathcal{L}}(a,b)=\bigcup_{c_i\in\operatorname{Crit(f)}}\mathcal{L}(a,c_1)\times\cdots\times\mathcal{L}(c_{k-1},b),
$$
where each $\mathcal{L}(c_{i-1},c_i)$ is the space of (height-parameterized) flow lines viewed as a subspace of $\operatorname{Map}([f(b),f(a)],M)$ with the compact-open topology. This can be shown to be a compactification of $\mathcal{L}(a,b)$. Doing this allows one to define the composition of flow lines in a natural and continuous manner: $(\gamma_1,\gamma_2)\mapsto\gamma_1\circ\gamma_2$. And finally one can then construct the category $\mathcal{C}_f$ internal to Top: with objects $(\mathcal{C}_f)_0$ critical points of $f$ under the subspace topology, homsets $(\mathcal{C}_f)_1$ such that $\operatorname{Hom}_{\mathcal{C}_f}(a,b)=\overline{\mathcal{L}}(a,b)$, and composition $(\mathcal{C}_f)_1\times_{(\mathcal{C}_f)_0}(\mathcal{C}_f)_1\xrightarrow{\circ}(\mathcal{C}_f)_1$ given by composition of flow lines.
Also, I prepared and presented the following expository presentation on Morse homology to my REU group. (Note the difference in the considered topology on $\overline{\mathcal{L}}(a,b)$). Morse homology is very helpful to know since this weekend I intend to start reading Combinatorial Floer Homology and Floer homology's tagline is that its an infinite-dimensional analogue of Morse homology. I'll be giving an expository presentation on what I've read to my REU group and a presentation of the problem I'll be working on to the entire DIMACS REU next week.
On Tuesday Sangjun and I presented the following presentation that gives a broad overview of our individual projects.
Also, I had a complete change of plan. I decided it would be best to read the original analytic theory of Floer cohomology from the book Lagrangian Intersection Floer Theory. I am almost to the section on Floer cohomology.
I prepared and presented the following expository presentation on the Maslov index to my REU group on Tuesday. Specifically, it was on a generalization of the usual Maslov index to symplectic bundle pairs $(\mathcal{V},\lambda)$ over a compact oriented surface with boundary $(\Sigma,\partial\Sigma)$. The definitions are very nested so it's best to just read the file instead of me typing any more on this.
I prepared and presented the following expository presentation on the Novikov covering to my REU group on Thursday. This one is a bit easier to explain. We fix a pair of compact Lagragians $(L_0,L_1)$ inside a symplectic manifold $(M,\omega)$ and consider the space of paths
$$
\Omega(L_0,L_1)=\{[0,1]\xrightarrow{\ell}M:\ell(0)\in L_0,\ell(1)\in L_1\}.
$$
By fixing a path $\ell_0$ and considering the connected component that contains $\ell_0$, $\Omega(L_0,L_1;\ell_0)$, we may assume the pair $(L_0,L_1)$ are connected. We then quotient the universal cover of $\Omega$ by something called $\Gamma$-equivalence - which yields a cover $\tilde{\Omega}(L_0,L_1;\ell_0)$. We do this so that, by definition, the deck transformation group $\Pi(L_0,L_1;\ell_0)$ of the cover
$$
\tilde{\Omega}(L_0,L_1;\ell_0)\rightarrow\Omega(L_0,L_1;\ell_0)
$$
is abelian. This will be useful later when constructing the Floer cohomology.
Over the weekend I plan to continue reading about the action functional and hopefully the construction of Floer cohomology in some specific cases - still using Lagrangian Intersection Floer Theory.
The beautiful thing about doing research is that a plan for the week tends to change.
I prepared and presented the following expository presentation on Floer cohomology. Basically, we define the action functional $\tilde{\Omega}(L_0,L_1;\ell_0)\xrightarrow{\mathcal{A}}\mathbb{R}$ by
$$
\mathcal{A}[l,w]=\int w^*\omega.
$$
It follows that $d\mathcal{A}=-\pi^*\alpha$ (the action 1-form). This shows that the critical points of $\mathcal{A}$ are the constant paths (read: Lagrangian intersection points). Once we define a suitable $L^2$-metric on $\tilde{\Omega}(L_0,L_1;\ell_0)$ using a compatible family of almost complex structures, the gradient flow lines between critical points of $\mathcal{A}$ will be strips connected the Lagrangian intersection points.
Once we define a way to index the critical points we may construct Floer cohomology. The complex will be some free module over the intersection points and the coboundary operator $d$ counts strips. Also, this complex should be invariant under Hamiltonian isotopy. However, in general, $d^2\neq0$. So we need some conditions on various things in the construction.
Following papers by Haniya Azam, Christian Blanchet and Mohammed Abouzaid I did an expository presentation on Floer cohomology of higher genus surfaces. For a closed connected surface
$\Sigma$ with genus greater than one and an oriented immersed pair $(\gamma_1,\gamma_2)$ of curves that are unobstructed and admissible we can define Floer cohomology combinatorially.
We give intersection points a sign based on orientation and define the Floer complex as a $\mathbb{Z}_2$-graded $\mathbb{Z}$-module with basis the intersection points. The grading is based on the sign of the intersection point. The coboundary operator $d$ then counts bigons (read: strips) between intersection points and assigns a weight based on orientation (and marked points). This theory is invariant under isotopy, not only Hamiltonian, and is isomorphic to the analytical theory.
Over the weekend I plan to read Tobias Ekholm's paper on Morse flow trees and present on that next week.
I actually never ended up writing a presentation on Ekholm's paper (and more than likely won't). In a gist: let $M$ be a manifold and give the 1-jet space $J^1(M)=T^*M\times\mathbb{R}$ its standard contact structure $\xi$. Let $L\subset J^1(M)$ be Legendrian and $\Pi_{\mathbb{C}}(L)$ its Lagrangian projection. After giving $M$ a Riemannian metric $g$ and some setup we can define a Morse flow tree.
The flow tree $\Gamma$ has vertices that are either fold points or critical points of local function differences of the Lagrangian and the edges are parametrizations of flow lines of the Legendrian such that the cotangent lifts are "compatible" in some sense.
If we let $M_{\Gamma}$ be the moduli space of flow trees for a fixed $M$ and $L$ we have the natural questions of if the space is compact, Hausdorff, a manifold, etc. Over the week I found a way to give a topology to $M_{\Gamma}$ using a suitable function space.
Over the weekend I've been trying to show compactness using this function space. I've been running into quite a few topological problems. But hopefully I can figure them out over the next week.
With the help of numerous conversations with Dr. Woodward, I have shown the compactness of $M^i_\Gamma$, the moduli space of flow trees with at most $i$ edges. But this is simply the first step in hopefully a much larger theorem.
The next idea is to show compactness of the moduli space of flow trees with a fixed number of incoming and outgoing punctures - which will most likely require the above result. I have a few ideas and am formalizing them a bit before I talk about them with the rest of the group. So hopefully this will be the way to go and there will be a proof by the end of the week.
I've spent the week attempting to prove the full theorem that we'd like to prove. Unfortunately, it seems like with the time available it won't be possible.
However, I've started to try and come up with conditions, what I call the energy conditions, on a moduli space of flow trees that hopefully gets us closer to an answer.
It seems the original conjecture we had was a little misguided. In particular, the results I thought I proved aren't actually correct. I realized a mistake in the proof and while attempting to fix it I realized the conjecture isn't actually true - it fails when reduced to Morse theory. I've since reformulated the conjecture and am starting from scratch.
This week was very busy. So far I've been able to lay the ground work for, what I hope, is how I'll be able to prove the new reformulated conjecture.
This week marks the end of the program. I gave my final presentation on the work that has been done and what more work is required to finish this problem.
I plan to work on the problem for a bit post DIMACS. Hopefully I make some more progress on the problem and it gets solved one day. If so I'll post about it on this website then.
Simply so all links are in one place
