We define a new class of symplectic objects called 'stops', which, roughly speaking, are Liouville hypersurfaces in the boundary of a Liouville domain. Locally, these can be viewed as pages of a compatible open book. To a Liouville domain with a collection of disjoint stops, we assign an A∞-category called its partially wrapped Fukaya category. An exact Landau-Ginzburg model gives rise to a stop, and the corresponding partially wrapped Fukaya category is meant to agree with the Fukaya category one is supposed to assign to the Landau-Ginzburg model. As evidence, we prove a formula that relates these partially wrapped Fukaya categories to the wrapped Fukaya category of the underlying Liouville domain. This operation is mirror to removing a divisor. 373is the open-closed map. Here, the action filtration on Hochschild homology is induced from the action filtration on Floer cochain groups; for a precise definition, see Section 4.3.In particular, any punctured Riemann surface other than C and any cotangent bundle is strongly nondegenerate.Remark 1.1. A nondegenerate Liouville domain F is one for which some element e ∈ HH * (W(F )) satisfies OC(e) = 1. This condition is Abouzaid's generation criterion [2], which says that such an F admits a finite collection of Lagrangians which split-generate W(F ). Ganatra further proved that in this case, OC is an isomorphism [15], which in particular implies that e is unique.An approximate version of Theorem 4.21 can be stated as follows: Theorem 1.2. Let M be a Liouville domain, and let σ be a strongly nondegenerate stop in M . Let B ⊂ W σ (M ) be the full subcategory of objects supported near σ. Then the inclusion W σ (M ) → W(M ) induces a fully faithful functorwhere the quotient is a quotient of an A ∞ category by a full subcategory in the sense of .At the most basic level, Theorem 1.2 shows that the partially wrapped Fukaya category, together with B, really knows at least as much as the wrapped Fukaya category. That is, restricting to the zero-filtered part but remembering the stop does not lose information. At an intuitive level, this happens because the 'nice' presentation of W(M ) giving rise to the stop filtration uses a contact form with a large number of canceling Reeb chords. These chords live at different levels in the filtration, so passing to the zero-filtered part results in them no longer canceling. In a more minimal presentation, these chords could be eliminated geometrically.The key ingredient of the proof of Theorem 1.2 is an auxiliary filtration on W(M ), presented as the trivial quotient A = W(M )/B. The benefit of this quotient presentation is that it naturally contains the category A 0 = W σ (M )/B(σ) as the minimally filtered part, which makes it possible to build a homotopy which retracts A onto A 0 . The homotopy itself requires a filtered version of the annulus trick, which was introduced in [2] and extended in a forthcoming paper by Abouzaid and Auroux. Specifically, one factors the identity operation as a composition of a product and a coproduct, where ...

We define Szegő coordinates on a finitely connected smoothly bounded planar domain which effect a holomorphic change of coordinates on the domain that can be as close to the identity as desired and which convert the domain to a quadrature domain with respect to boundary arc length. When these Szegő coordinates coincide with Bergman coordinates, the result is a double quadrature domain with respect to both area and arc length. We enumerate a host of interesting and useful properties that such double quadrature domains possess, and we show that such domains are in fact dense in the realm of bounded C ∞ -smooth finitely connected domains.1991 Mathematics Subject Classification. 30C35.

For any high-dimensional Weinstein domain and finite collection of primes, we construct a Weinstein subdomain whose wrapped Fukaya category is a localization of the original wrapped Fukaya category away from the given primes. When the original domain is a cotangent bundle, these subdomains form a decreasing lattice whose order cannot be reversed.Furthermore, we classify the possible wrapped Fukaya categories of Weinstein subdomains of a cotangent bundle of a simply connected, spin manifold, showing that they all coincide with one of these prime localizations. In the process, we describe which twisted complexes in the wrapped Fukaya category of a cotangent bundle of a sphere are isomorphic to genuine Lagrangians.

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We prove the surprising fact that the infinity-category of stabilized Liouville sectors is a localization of an ordinary category of stabilized Liouville sectors and strict sectorial embeddings.From the perspective of homotopy theory, this result continues a trend of realizing geometrically meaningful mapping spaces through the categorically formal process of localizing. From the symplectic viewpoint, these results allow us to reduce highly non-trivial coherence results to much simpler verifications. For example, we prove that the wrapped Fukaya category is coherently functorial on stabilized Liouville sectors: Not only does a wrapped category receive a coherent action from stabilized automorphism spaces of a Liouville sector, spaces of sectorial embeddings map to spaces of functors between wrapped categories in a way respecting composition actions. As we will explain, our methods immediately establish such coherence results for most known sectorial invariants, including Lagrangian cobordisms.As another application, we show that this infinity-category admits a symmetric monoidal structure, given by direct product of underlying sectors. The existence of this structure relies on a computation-familiar from the foundations of factorization homology-that localizations detect certain isotopies of smooth manifolds. Moreover, we characterize the symmetric monoidal structure using a universal property, again producing a simple-as-possible criterion for verifying whether invariants are both continuously and multiplicatively coherent in a compatible way.Much of this paper is devoted to the foundational work of rigorously constructing the infinitycategory of stabilized Liouville sectors, where any sector M is identified with its stabilization M × T * [0, 1]. We begin with a verification that our construction is indeed an infinity-category (which relies on new constructions in Liouville geometry), then show that our infinity-category computes the "correct" mapping spaces (employing new simplicial techniques). A significant ingredient is a demonstration that two notions of sectorial maps give rise to homotopy equivalent spaces of maps.