More Like this

1. Decategorified Heegaard Floer theory and actions of both E and F

   https://doi.org/10.4171/QT/204  

   Manion, Andrew (January 2024, Quantum Topology)

   We define larger variants of the vector spaces one obtains by decategorifying bordered (sutured) Heegaard Floer invariants of surfaces. We also define bimodule structures on these larger spaces that are similar to, but more elaborate than, the bimodule structures that arise from decategorifying the higher actions in bordered Heegaard Floer theory introduced by Rouquier and the author. In particular, these new bimodule structures involve actions of both odd generatorsEandFof\mathfrak{gl}(1|1), whereas the previous ones only involved actions ofE. Over\mathbb{F}_2, we show that the new bimodules satisfy the necessary gluing properties to give a 1+1 open-closed TQFT valued in graded algebras and bimodules up to isomorphism; in particular, unlike in previous related work, we have a gluing theorem when gluing surfaces along circles as well as intervals. Over the integers, we show that a similar construction gives two partially defined open-closed TQFTs with two different domains of definition depending on how parities are chosen for the bimodules. We formulate conjectures relating these open-closed TQFTs with the\mathfrak{psl}(1|1)Chern–Simons TQFT recently studied by Mikhaylov and Geer–Young. 

   more » « less
2. On Stability of Metric Spaces and Kalton’s Property Q

   https://doi.org/10.1093/qmath/haae050  

   Baudier, F; Schlumprecht, Th; ZSÁK, A (December 2024, The Quarterly Journal of Mathematics)

   ABSTRACT The first named author introduced the notion of upper stability for metric spaces in F. Baudier, Barycentric gluing and geometry of stable metrics, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM  116 no. 1, (2022), 48 as a relaxation of stability. The motivation was a search for a new invariant to distinguish the class of reflexive Banach spaces from stable metric spaces in the coarse and uniform category. In this paper we show that property Q does in fact imply upper stability. We also provide a direct proof of the fact that reflexive spaces are upper stable by relating the latter notion to the asymptotic structure of Banach spaces. 

   more » « less
3. Twistors, Self-Duality, and Spin^c Structures

   https://doi.org/10.3842/SIGMA.2021.102  

   LeBrun, Claude (November 2021, Symmetry, Integrability and Geometry: Methods and Applications)

   The fact that every compact oriented 4-manifold admits spin^c structures was proved long ago by Hirzebruch and Hopf. However, the usual proof is neither direct nor transparent. This article gives a new proof using twistor spaces that is simpler and more geometric. After using these ideas to clarify various aspects of four-dimensional geometry, we then explain how related ideas can be used to understand both spin and spin^c structures in any dimension. 

   more » « less
4. A quantization of coarse spaces and uniform Roe algebras

   https://doi.org/10.2140/pjm.2025.338.163  

   Braga, Bruno M; Eisner, Joseph; Sherman, David (January 2025, Pacific Journal of Mathematics)

   We propose a quantization of coarse spaces and uniform Roe algebras. The objects are based on the quantum relations introduced by N. Weaver and require the choice of a represented von Neumann algebra. In the case of the diagonal inclusion ell_infty(X) subset B(ell_2(X)), they reduce to the usual constructions. Quantum metric spaces furnish natural examples parallel to the classical setting, but we provide other examples that are not inspired by metric considerations, including the new class of support expansion C*-algebras. We also work out the basic theory for maps between quantum coarse spaces and their consequences for quantum uniform Roe algebras. 

   more » « less
5. Deformation Theory of Log Pseudo-holomorphic Curves and Logarithmic Ruan–Tian Perturbations

   https://doi.org/10.1007/s42543-023-00069-1  

   Farajzadeh-Tehrani, Mohammad (May 2023, Peking Mathematical Journal)

   In a previous paper (Farajzadeh-Tehrani in Geom Topol 26:989–1075, 2022), for any logarithmic symplectic pair (X, D) of a symplectic manifold X and a simple normal crossings symplectic divisor D, we introduced the notion of log pseudo-holomorphic curve and proved a compactness theorem for the moduli spaces of stable log curves. In this paper, we introduce a natural Fredholm setup for studying the deformation theory of log (and relative) curves. As a result, we obtain a logarithmic analog of the space of Ruan–Tian perturbations for these moduli spaces. For a generic compatible pair of an almost complex structure and a log perturbation term, we prove that the subspace of simple maps in each stratum is cut transversely. Such perturbations enable a geometric construction of Gromov–Witten type invariants for certain semi-positive pairs (X, D) in arbitrary genera. In future works, we will use local perturbations and a gluing theorem to construct log Gromov–Witten invariants of arbitrary such pair (X, D). 

   more » « less