Disk Embeddings: Geometry and Applications

• Disk embeddings are advanced methods that embed disk-like domains into geometric or metric spaces, enabling precise analysis in symplectic geometry and machine learning.
• They leverage techniques such as pseudoholomorphic curves, ECH capacities, and toric domain constructions to address complex embedding problems in symplectic topology and manifold calculus.
• Applications span from demonstrating symplectic rigidity and constructing explicit Lagrangian fibrations to driving innovative representation learning models with superior performance in hyperbolic spaces.

Disk embeddings encompass a spectrum of advanced techniques and theories in geometry, topology, symplectic geometry, and representation learning, unified by the general theme of embedding disk-like domains—either topological disks, phase space regions, or abstract metric disks—into target spaces with geometric, dynamical, or categorical structure. Applications arise in symplectic embedding theory, manifold calculus, Lagrangian geometry, and, more recently, machine learning, where “disk embedding” denotes a geometric encoding method for partially ordered sets and graphs in metric or pseudo-metric spaces.

1. Symplectic Disk Embeddings: Foundational Notions

In symplectic geometry, disk embeddings typically refer to symplectic (volume and structure-preserving) embeddings of disks, polydisks, or their products into more complex symplectic manifolds. Given $(\mathbb{C}^n,\omega_0)$ with $\omega_0$ the standard symplectic form, the core objects are:

• The closed $2n$-dimensional ball (symplectic ball)

$$B^{2n}(a) = \left\{z\in\mathbb{C}^n \mid \pi\lVert z\rVert^2 \le a\right\}$$

• The symplectic polydisk

$$P(r_1,\ldots,r_n) = \{z\in\mathbb{C}^n \mid \pi|z_j|^2 \le r_j,\;\forall j\}$$

The primary goal is to determine, for specified domains $X$ (e.g., a polydisk or Lagrangian bidisk) and targets $Y$ (e.g., ball, ellipsoid, toric domain), the existence or nonexistence of a symplectic embedding $X\hookrightarrow Y$. The critical breakthrough in this area lies in the understanding that, unlike in volume-preserving topology, symplectic embedding problems in dimension four (notably for products of disks) display “rigidity,” with existence governed by symplectic capacities and, in subtle cases, pseudoholomorphic curve theory.

For example, it is proved that the symplectic embedding $P(1,2)\to B^4(a)$ exists if and only if $a\ge 3$. Neither Ekeland-Hofer nor ECH capacities suffice to detect this rigidity; sharpness is established only via detailed pseudoholomorphic (finite-energy) foliation and neck-stretching arguments specific to dimension four (Hind et al., 2013).

2. Lagrangian Bidisk, Toric Domains, and Symplectic Capacities

The Lagrangian bidisk

$\omega_0$0

has emerged as a central object in symplectic topology. Its study connects classical ball and polydisk embedding questions to toric geometry via concave toric domains, using action-angle and moment map coordinates. Ramos (see (Ramos, 2015, Achig-Andrango et al., 2 Jun 2025)) showed that the interior of $\omega_0$1 is symplectomorphic to an explicit concave toric domain $\omega_0$2, with the region $\omega_0$3 parametrized in terms of billiard flow coordinates.

This correspondence enables direct computation of symplectic embedding obstructions using ECH (embedded contact homology) capacities, which are computed from the weight expansion (“graph” representation) of $\omega_0$4. For $\omega_0$5, this yields capacity sequences such as $\omega_0$6, establishing sharp obstructions for embeddings into balls, ellipsoids, and polydisks:

• $\omega_0$7
• $\omega_0$8

This capacity-based approach is both necessary and sufficient for embeddings into convex toric domains, thanks to the work of Cristofaro-Gardiner and others (Ramos, 2015, Achig-Andrango et al., 2 Jun 2025).

3. Singular Lagrangian Fibrations and Toric Realizations

Disk cotangent bundles $\omega_0$9 of Riemannian manifolds $2n$0, equipped with their canonical symplectic forms, play an essential role in explicit constructions of toric and almost toric fibrations. Advances detailed in (Achig-Andrango et al., 2 Jun 2025) reveal that certain $2n$1 admit singular Lagrangian ("almost toric") fibrations: $2n$2 where $2n$3 is a momentum map for a Hamiltonian torus action and $2n$4 computes areas in reduced fibers. This procedure produces base diagrams in $2n$5 whose image is a convex region $2n$6. After base change and "nodal trades," these domains become explicit toric domains

$2n$7

For example, the disk cotangent bundle $2n$8 admits a convex base whose maximal inscribed triangle area recovers Gromov width $2n$9 (Achig-Andrango et al., 2 Jun 2025). Similarly, the Lagrangian bidisk can be realized as such a toric domain, unifying billiard flow, moment polytope, and symplectic embedding theory.

4. Poincaré Duality and Embedding Towers

The space of codimension-zero embeddings of a Poincaré duality space $$B^{2n}(a) = \left\{z\in\mathbb{C}^n \mid \pi\lVert z\rVert^2 \le a\right\}$$0 in a disk $$B^{2n}(a) = \left\{z\in\mathbb{C}^n \mid \pi\lVert z\rVert^2 \le a\right\}$$1 is addressed using homotopy-functor (Goodwillie) calculus, as in (Klein, 2014). Given the category $$B^{2n}(a) = \left\{z\in\mathbb{C}^n \mid \pi\lVert z\rVert^2 \le a\right\}$$2 of complements $$B^{2n}(a) = \left\{z\in\mathbb{C}^n \mid \pi\lVert z\rVert^2 \le a\right\}$$3 with appropriate boundary conditions, the realization of this subcategory

$$B^{2n}(a) = \left\{z\in\mathbb{C}^n \mid \pi\lVert z\rVert^2 \le a\right\}$$4

gives the embedding space. A key result states that there exists a tower of spaces $$B^{2n}(a) = \left\{z\in\mathbb{C}^n \mid \pi\lVert z\rVert^2 \le a\right\}$$5 interpolating between Poincaré immersions and “unlinked” embeddings, with each layer described by coefficient spectra from Goodwillie calculus. In the best cases (sectioned $$B^{2n}(a) = \left\{z\in\mathbb{C}^n \mid \pi\lVert z\rVert^2 \le a\right\}$$6, sufficient codimension), the tower converges and computes the space of unlinked Poincaré embeddings.

The first-stage map (Browder–Pontryagin–Thom construction)

$$B^{2n}(a) = \left\{z\in\mathbb{C}^n \mid \pi\lVert z\rVert^2 \le a\right\}$$7

is central, with obstruction theory (encoded in the link class) describing the failure to lift immersions to embeddings.

5. Disk Embeddings in Metric and Quasi-Metric Spaces (Machine Learning)

In representation learning, disk embeddings refer to a geometric framework for embedding nodes of directed acyclic graphs (DAGs) into a quasi-metric space $$B^{2n}(a) = \left\{z\in\mathbb{C}^n \mid \pi\lVert z\rVert^2 \le a\right\}$$8, associating to each node $$B^{2n}(a) = \left\{z\in\mathbb{C}^n \mid \pi\lVert z\rVert^2 \le a\right\}$$9 a formal disk $$P(r_1,\ldots,r_n) = \{z\in\mathbb{C}^n \mid \pi|z_j|^2 \le r_j,\;\forall j\}$$0 interpreted as a closed ball: $$P(r_1,\ldots,r_n) = \{z\in\mathbb{C}^n \mid \pi|z_j|^2 \le r_j,\;\forall j\}$$1 Partial order is encoded by disk containment: $$P(r_1,\ldots,r_n) = \{z\in\mathbb{C}^n \mid \pi|z_j|^2 \le r_j,\;\forall j\}$$2 This "disk inclusion order" allows subsumption, entailment, or hierarchy relations to be encoded as geometric relations among disks (Suzuki et al., 2019).

Special cases include:

• Order Embeddings: As in Vendrov et al., recoverable as disk embeddings in polyhedral quasi-metric spaces.
• Entailment Cones: As in Ganea et al., correspond to spherical disk inclusion via canonical projections.

The "hyperbolic disk embedding" is performed in the Lorentz model of hyperbolic space, using geodesic balls and explicit gradients for Riemannian optimization. The central entailment score is

$$P(r_1,\ldots,r_n) = \{z\in\mathbb{C}^n \mid \pi|z_j|^2 \le r_j,\;\forall j\}$$3

with margin-based loss and Riemannian SGD for training.

Empirically, hyperbolic disk embeddings attain superior F1 scores over standard Poincaré and order-based embeddings (see table below), especially for general DAGs with complex ancestry relations (Suzuki et al., 2019).

Model	WordNet nouns	Reversed WordNet
Euclidean Disk Embedding	72.0%	70.6%
Hyperbolic Disk Embedding	94.2%	88.2%
Spherical Disk Embedding	93.9%	90.1%
Entailment Cones (Poincaré)	93.8%	75.8%
Order Embeddings	84.1%	41.4%
Poincaré Embeddings	85.4%	51.9%

Hyperbolic disk embeddings thus generalize and unify several prior geometric hierarchy-embedding approaches.

6. Methodological Connections and Applications

Disk embeddings serve as a central unifying method across several mathematical and applied domains:

• Symplectic geometry: Classifying embedding possibilities of standard domains, with critical thresholds and obstructions informed by holomorphic curves, symplectic/capacity theory, and almost toric fibration structures.
• Manifold calculus: Encoding and understanding the space of embeddings up to homotopy via functor calculus, explicit towers, and obstruction-theoretic diagrams (Klein, 2014).
• Machine learning: Disk embeddings provide an interpretable, geometrically principled, and empirically strong way of embedding partial orders and generalizations (DAGs, entailment graphs) into metric or quasi-metric spaces, with special power in hyperbolic geometry for representing complex, non-tree-like hierarchies (Suzuki et al., 2019).

A plausible implication is that ongoing advances in singular Lagrangian fibration construction (e.g., (Achig-Andrango et al., 2 Jun 2025)) will continue to supply new explicit models, norm boundaries, and embedding invariants for disk-like or cotangent domains, influencing both classical symplectic embedding theory and geometric representation models in computational contexts.

7. Open Problems and Future Directions

Research directions include:

• Extending explicit Lagrangian fibration and toric domain constructions in dimensions $$P(r_1,\ldots,r_n) = \{z\in\mathbb{C}^n \mid \pi|z_j|^2 \le r_j,\;\forall j\}$$4, and clarifying embedding bounds for cotangent disk bundles of general manifolds (Achig-Andrango et al., 2 Jun 2025).
• Sharpening and comparing symplectic obstructions: For domains where capacity sequences remain incomplete or non-sharp, further development of pseudoholomorphic foliation techniques and intersection theory is ongoing (Hind et al., 2013).
• Deeper interplay between manifold calculus and embedding theory: Full description of the interaction between Goodwillie towers, Poincaré category embeddings, and smooth manifold calculus remains conjectural but promising (Klein, 2014).
• Unification of geometric embedding methods in machine learning: Systematic study of disk embeddings in spaces of varying curvature and their implications for representations in AI is active (Suzuki et al., 2019).

The concept of disk embedding thus threads together fundamental symplectic rigidity, deep homotopy-theoretic classification, and scalable geometric representation methods.