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Noise typically degrades the performance of physical systems. In some cases, however, noise can provide an enhanced effect. For instance, stochastic resonance may be observed in a nonlinear oscillator when a weak signal is boosted by noise. On the other hand, adaptive oscillators are a type of nonlinear oscillator that can learn and store information in dynamic states. Here, noise is shown to increase the learning rate of an adaptive oscillator, using a Duffing adaptive oscillator. To highlight this effect, stochastic simulations are employed, and the Fokker–Planck equation is semi-analytically solved using the cumulant neglect method. As adaptive oscillators can also be used as a powerful physical reservoir computer architecture, this work shows that the Duffing adaptive oscillator can be optimized for fast learning in noisy environments. 

We survey the current state of affairs in the study of thresholds and sharp thresholds in random structures on the occasion of the recent proof of the Kahn–Kalai conjecture by Park and Pham and the fairly recent proof of the satisfiability conjecture for large k by Ding, Sly, and Sun. Random discrete structures appear as fundamental objects of study in many scientific and mathematical fields including statistical physics, combinatorics, algorithms and complexity, social choice theory, coding theory, and statistics. While the models and properties of interest in these fields vary widely, much progress has been made through the development of general tools applicable to large families of models and properties all at once. Historically, these tools originated to solve or make progress on specific difficult conjectures in the areas mentioned above. We will survey recent progress on some of these hard problems and describe some challenges for the future. This survey was prepared in conjunction with a talk for the Current Events Bulletin at the 2024 Joint Mathematics Meetings (JMM) in San Francisco, California.