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The Longest Increasing Subsequence

CLARIFICATIONS/ERRATA: In the limit shape theorem, the probability should tend to 1, not 0. Technically, the random matrix shown in the video is a GOE matrix (it has real entries) not a GUE matrix (which has complex entries). The Tracy-Widom distribution being discussed here corresponds to GUE, not GOE, but I found that a matrix of complex numbers wasn't as nice to look at so I cheated a bit with the animation. Not every optimization problems over independent random noise is expected to have Tracy-Widom fluctuations, but many of the models that are expected to have such fluctuations can be phrased as optimization problems. -------------------------------------------------------------------- This video assumes that the viewer has taken a course in probability theory. I hope it is still somewhat understandable and enjoyable even without that prerequisite, but for brevity I needed to take for granted many of the concepts usually covered in such a course. For more details, check out "The Surprising Mathematics of Longest Increasing Subsequences" by Dan Romik. The cover of this book was used in the video with the author's permission. This video was made by Vilas Winstein, with music by Aranka Hrušková. It is an entry in the third Summer of Math Exposition (#SoME3). -------------------------------------------------------------------- Chapters: 0:00 Introduction 1:36 Heuristic 6:21 Proof Idea 13:18 Fluctuations 16:14 Conclusion