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Using the Fredholm setup of Farajzadeh-Tehrani [Peking Math. J. (2023), https://doi.org/10.1007/s42543-023-00069-1], we study genus zero (and higher) relative Gromov-Witten invariants with maximum tangency of symplectic log Calabi-Yau fourfolds. In particular, we give a short proof of Gross [Duke Math. J. 153 (2010), pp. 297–362, Cnj. 6.2] that expresses these invariants in terms of certain integral invariants by considering generic almost complex structures to obtain a geometric count. We also revisit the localization calculation of the multiple-cover contributions in Gross [Prp. 6.1] and recalculate a few terms differently to provide more details and illustrate the computation of deformation/obstruction spaces for maps that have components in a destabilizing (or rubber) component of the target. Finally, we study a higher genus version of these invariants and explain a decomposition of genus one invariants into different contributions. 

Our previous papers introduce topological notions of normal crossings symplectic divisor and variety, show that they are equivalent, in a suitable sense, to the corresponding geometric notions, and establish a topological smoothability criterion for normal crossings symplectic varieties. The present paper constructs a blowup, a complex line bundle, and a logarithmic tangent bundle naturally associated with a normal crossings symplectic divisor and determines the Chern class of the last bundle. These structures have applications in constructions and analysis of various moduli spaces. As a corollary of the Chern class formula for the logarithmic tangent bundle, we refine Aluffi’s formula for the Chern class of the tangent bundle of the blowup at a complete intersection to account for the torsion and extend it to the blowup at the deepest stratum of an arbitrary normal crossings divisor. 

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). 

Millimeter-wave (mmWave) communications is a key enabler towards realizing enhanced Mobile Broadband (eMBB) as a key promise of 5G and beyond, due to the abundance of bandwidth available at mmWave bands. An mmWave coverage map consists of blind spots due to shadowing and fading especially in dense urban environments. Beam-forming employing massive MIMO is primarily used to address high attenuation in the mmWave channel. Due to their ability in manipulating the impinging electromagnetic waves in an energy-efficient fashion, Reconfigurable Intelligent Surfaces (RISs) are considered a great match to complement the massive MIMO systems in realizing the beam-forming task and therefore effectively filling in the mmWave coverage gap. In this paper, we propose a novel RIS architecture, namely RIS-UPA where the RIS elements are arranged in a Uniform Planar Array (UPA). We show how RIS-UPA can be used in an RIS-aided MIMO system to fill the coverage gap in mmWave by forming beams of a custom footprint, with optimized main lobe gain, minimum leakage, and fairly sharp edges. Further, we propose a configuration for RIS-UPA that can support multiple two-way communication pairs, simultaneously. We theoretically obtain closed-form low-complexity solutions for our design and validate our theoretical findings by extensive numerical experiments.