A place for me to talk about some of my recent mathematical musings.
[02-18-25] Morse-Bott homology
In preparation for a talk at the grad student Differential Geometry Seminar at UW, I've been digging in to some different directions coming from the Morse theory that I studied at the end of my undergrad. Namely, I've been reading about Morse-Bott functions, and how they pose a sort of "non-isolated" form of Morse theory/handle decomposition. Namely, in Bott's paper Nondegenerate critical submanifolds, the isomorphism \(H_k(M^{p+\epsilon}\cap U,M^{p-\epsilon}\cap U)\cong H_{k-\lambda}(\{p\})\) is extended to the situation in which we have a space \(S\) consisting of a disjoint union of finitely many connected submanifolds of critical points; that is, \(H_k(M^{p+\epsilon}\cap U,M^{p-\epsilon}\cap U)\cong H_{k-\lambda}(S)\). (Note: our choice of \(p\in S\) here is arbitrary; we impose the condition that \(\text{Hess}_f(p)\) is nondegenerate in the normal direction to \(S\) for all \(p\).)
A paper that I've found to be of interest (ableit, it may not be the source of material that I will cover in my talk) is Morse-Bott homology, authored by A. Banyaga and D. Hurtubise. I found great interest in Morse homology (i.e., part 1 of Audin and Damian's book) during my first exposure to it, and the extension to orbifolds in Cho and Hong's paper Orbifold Morse-Smale-Witten complexes was a similar positive learning experience. Banyaga and Hurtubise's paper is again an extension of classical results for Morse functions on Riemannian manifolds to a more general setting, and I look forward to working through this paper.
[11-27-23] More general structure than expected?
As mentioned in my previous post, we can construct product structures via the use of Gromov-Witten invariants. However, note the following theorem, due to (at least in part, I believe) Gromov:
Theorem. Symplectic manifolds always have almost-complex structures which are compatible with the symplectic structure. Further, there exists an infinite-dimensional space of compatible almost-complex structures. This space is contractible however, and leads to the fact that all of these almost-complex structures are in fact homotopy equivalent.
So, in the case of manifolds, up to homotopy, the crucial data which allows for the application of Gromov-Witten theory to deform cohomology products does not come from symplectic structure, rather the almost-complex structure on the manifold. This extends to Lagrangians: it is known that a Lagrangian submanifold will be totally-real with respect to the almost-complex structure on the manifold. This motivates the following:
If we consider a symplectic orbifold \(\mathcal{X}\) and a Lagrangian suborbifold \(\mathcal{L}\subset\mathcal{X}\), then the data required to extract the exotic product described in the previous post on \(H^\bullet(I_{\mathcal{X}}\mathcal{L})\) is dependent on some almost-complex/totally-real structure instead of symplectic/Lagrangian structure.
[11-22-23] Chen-Ruan product analogue for “Lagrangian suborbifolds”
Consider a symplectic manifold \((X,\omega)\) whose cohomology \(H^\bullet(X)\) is equipped with the Poincaré pairing \(\langle\cdot,\cdot\rangle\). Gromov-Witten theory is a powerful tool in symplectic geometry which connects to mathematical physics, namely Type IIA string theory. The main point of study are the Gromov-Witten invariants, which "count" genus \(g\) pseudo-holomorphic curves with \(n\) marked points, mapping into \(X\). That is, we consider stable maps \(f: \mathbb{C} P^1\to X\). It turns out that the set of these functions forms a moduli space, which allows us to define the Gromov-Witten invariants:
Definition. Consider the stable maps \(\mathbb{C} P^1\to X\) of genus \(g\), where \(\mathbb{C}P^1\) has \(n\) marked points. Then we can form a moduli space \(\mathcal{M}_{g,n}(X)\) of such curves. Note that for our use, we only consider genus 0 curves with 3 marked points; that is, \(\mathcal{M}_{0,3}(X)\). We define the evaluation maps on the moduli space, \(\text{ev}_i:\mathcal{M}_{0,3}(X)\to X\), by sending a stable map to the value of the stable map at a marked point, \([f:(\mathbb{C}P^1,x_1,x_2,x_3)\to X]\mapsto f(x_i)\).
This moduli space and its evaluation maps allow us to define the Gromov-Witten invariants. Consider three cohomology classes \(\alpha,\beta,\gamma\) on \(X\). Then the Gromov-Witten invariants are given as $$ GW_{0,3}(X)=\int_{\mathcal{M}_{0,3}(X)} \text{ev}_1^*\alpha\wedge\text{ev}_2^*\beta\wedge\text{ev}_3^*\gamma. $$
Interestingly, if we identify the Poincaré pairing with the Gromov-Witten invariants of \(X\), $$ \langle\alpha\star\beta,\gamma\rangle=GW_{0,3}(X), $$ it turns out that the product \(\star\) which satisfies this equality is the quantum product on the quantum ring \(H^\bullet(X)[[q]]\). If we now restrict to just constant curves by setting \(q=0\), this quantum product reduces to the standard cup product on \(H^\bullet(X)\). We can do the same construction for a Lagrangian submanifold \(L\subset X\), but we instead consider a "disk" with three marked points instead of a projective sphere \(\mathbb{C}P^1\) as the source for the pseudo-holomorphic curves.
I spent the majority of the Summer of 2023 working with orbifolds; an extension of the ideas mentioned above would be to a symplectic orbifold \(\mathcal{X}\). It turns out that when similar processes as above are performed on \(\mathcal{X}\), we require the use of the cohomology of the inertia orbifold \(I\mathcal{X}\). Amazingly, when we identify $$ \langle\alpha\star\beta,\gamma\rangle=GW_{0,3}(\mathcal{X}) $$ and restrict to \(q=0\), we retrieve the Chen-Ruan product on \(H^\bullet(I\mathcal{X})[[q]]\)! This is a remarkable result, as this product structure is not immediately observable when one studies the cohomology of orbifolds.
We also would like to study a Lagrangian suborbifold \(\mathcal{L}\subset\mathcal{X}\) as modeled in the manifold case. However, the notion of a "Lagrangian suborbifold" is not well defined. However, in a 2023 preprint of Chen, Ono, and Wang, the idea of the dihedral twisted sector \(I_{\mathcal{X}}\mathcal{L}\) is introduced -- as the notation may suggest, this acts as a sort of Lagrangian "relative" to \(\mathcal{X}\).
So, we conjecture that another exotic cohomology product could be constructed on \(I_{\mathcal{X}}\mathcal{L}\) by restricting \(H^\bullet(I_\mathcal{X}\mathcal{L})[[q]]\) to \(q=0\).
In an alternative approach towards such a result, techniques from this paper suggest the need to construct a double inertia orbifold \(II_{\mathcal{X}}\mathcal{L}\) and an obstruction class in its \(K\)-theory. This would provide an explicit description of this conjectured product structure on the cohomology of the dihedral twisted sector.