We exhibit a distinctly low-dimensional dynamical obstruction to the existence of Liouville cobordisms: for any contact 3-manifold admitting an exact symplectic cobordism to the tight 3-sphere, every nondegenerate contact form admits an embedded Reeb orbit that is unknotted and has self-linking number −1. The same is true moreover for any contact structure on a closed 3-manifold that is reducible. Our results generalize an earlier theorem of Hofer-Wysocki-Zehnder for the 3-sphere, but use somewhat newer techniques: the main idea is to exploit the intersection theory of punctured holomorphic curves in order to understand the compactification of the space of so-called "nicely embedded" curves in symplectic cobordisms. In the process, we prove a local adjunction formula for holomorphic annuli breaking along a Reeb orbit, which may be of independent interest.
Contents2010 Mathematics Subject Classification. Primary 57R17; Secondary 32Q65, 53D35. 1 2 ALEXANDRU CIOBA AND CHRIS WENDLRecall that an oriented 3-manifold is reducible if and only if it is either S 1 × S 2 or M 1 #M 2 for a pair of closed oriented 3-manifolds that are not spheres. This condition is now known to be equivalent to the hypothesis π 2 (M ) = 0 used in [Hof93]: in one direction this follows from the sphere theorem for 3-manifolds, and in the other, from [Hat, Prop. 3.10] and the Poincaré conjecture. Note that both of the above theorems require nondegeneracy of the contact form α, but it is possible for the sake of applications to weaken this condition; see Theorem 1.12 below.UNKNOTTED ORBITS AND NICELY EMBEDDED CURVES 3 1.2. Context. The prototype for Theorems 1.1 and 1.2 is a 20-year-old result of Hofer-Wysocki-Zehnder [HWZ96c], which amounts to the case (M, ξ) = (S 3 , ξ std ) of Theorem 1.1. The result in [HWZ96c] was in some sense far ahead of its time, as it required ideas from both the compactness theory [BEH + 03] and the intersection theory [Sie11] of punctured holomorphic curves, but it appeared several years before either of those theories were developed in earnest. In the mean time the available techniques have improved, and our proofs will make use of those improvements.A weaker version of Theorem 1.
Recently, successful applications of reinforcement learning to chip placement have emerged. Pretrained models are necessary to improve efficiency and effectiveness. Currently, the weights of objective metrics (e.g., wirelength, congestion, and timing) are fixed during pretraining. However, fixed-weighed models cannot generate the diversity of placements required for engineers to accommodate changing requirements as they arise. This paper proposes flexible multiple-objective reinforcement learning (MORL) to support objective functions with inference-time variable weights using just a single pretrained model. Our macro placement results show that MORL can generate the Pareto frontier of multiple objectives effectively.
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