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  1. ; ; (, Studia Scientiarum Mathematicarum Hungarica)
    Extending work of Saneblidze–Umble and others, we use diagonals for the associahedron and multiplihedron to define tensor products of𝐴∞-algebras, modules, algebra homomorphisms, and module morphisms, as well as to define a bimodule analogue of twisted complexes (typeDDstructures, in the language of bordered Heegaard Floer homology) and their one- and two-sided tensor products. We then give analogous definitions for 1-parameter deformations of𝐴∞-algebras; this involves another collection of complexes. These constructions are relevant to bordered Heegaard Floer homology. 
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    Free, publicly-accessible full text available September 18, 2026
  2. ; ; (, Journal of Symplectic Geometry)
    Auroux, Denis; Biran, Paul; Donaldson, Simon; Mrowka, Tomasz (Ed.)
    We describe a weighted A-infinity algebra associated to the torus. We give a combinatorial construction of this algebra, and an abstract characterization. The abstract characterization also gives a relationship between our algebra and the wrapped Fukaya category of the torus. These algebras underpin the (unspecialized) bordered Heegaard Floer homology for three-manifolds with torus boundary, which will be constructed in forthcoming work. 
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    Free, publicly-accessible full text available January 1, 2026
  3. ; ; (, International Press)
    Gross, David; Yao, Andrew Chi-Chih; Yau, Shing-Tung (Ed.)
    Bordered Floer homology is an invariant for 3-manifolds with boundary, defined by the authors in 2008. It extends the Heegaard Floer homology of closed 3-manifolds, defined in earlier work of Zoltán Szabó and the second author. In addition to its conceptual interest, bordered Floer homology also provides powerful computational tools. This survey outlines the theory, focusing on recent developments and applications. 
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    Accepted Manuscript Full Text Available
  4. ; ; ; (, Journal of the London Mathematical Society)
    Full Text Available 
  5. ; ; (, Groups, Geometry, and Dynamics)
    Full Text Available 
  6. ; (, Forum of Mathematics, Sigma)
    Abstract Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all known examples of continuous functions on geodesic currents, such as nonpositively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length. 
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    Full Text Available