Channel Title: Spectral Collective Channel Description: <video> How many times do you ACTUALLY need to shuffle? Description: [CLARIFICATIONS AND CORRECTIONS] Clarification at 2:50 – "probability mass" just means the probability of a particular outcome. Clarification at 6:41 – This is also because the events which bear witness to the distance being high after only a few shuffles are complicated, like the one described in the skit, and so they can’t really be seen in the gameplay of most games (I don’t have a good source for this unfortunately). Also, the point of this message is not to say that you shouldn’t shuffle more if you want to, but that you should not be too upset if someone else only shuffles a few times in a casual context. [CREDITS] Vilas Winstein: black nails, script, animation, voiceover, editing. Anisah Awad: blue and green nails. Special thanks to Alistair Sinclair for teaching me about this subject. Card Decks used in the video: "Playing Arts: Edition One" (http://www.playingarts.com) "Wildflowers of the Rocky Mountains" (http://www.adventurepublications.net) "Bees & Butterflies" (http://www.adventurewithkeen.com) [SOURCES] Bayer, Dave, and Persi Diaconis. "Trailing the dovetail shuffle to its lair." The Annals of Applied Probability (1992): 294-313. Martin, Aigner, and M. Ziegler Günter. "Chapter 31: Shuffling Cards." Proofs from the book–Sixth Edition (2018): 219-228. [SECTIONS] 0:00 Introduction 1:31 What Is Shuffling? 3:19 Total Variation Distance – Introduction 4:22 Total Variation Distance – Definition 1 5:00 Total Variation Distance – Definition 2 6:13 Skit (Don't Think Too Hard) 7:07 Total Variation Distance – Definition 3 8:20 Coin Flip Example – Introduction 8:50 Coin Flip Example – Definition 1 9:03 Coin Flip Example – Definition 2 9:31 Coin Flip Example – Definition 3 (Independent Coupling) 10:25 Coin Flip Example – Definition 3 (Optimal Coupling) 11:48 Total Variation Distance – Review 12:27 Plan for the Rest of the Video 13:23 Reversing the Shuffle 14:45 Top-To-Random Coupling – Introduction 15:09 Top-To-Random Coupling – Description 16:05 Top-To-Random Coupling – Analysis 18:23 Skit (Don't Shuffle This Way) 18:41 Riffle Shuffle – Model Description 20:03 Riffle Shuffle – Coupling Description 21:42 Riffle Shuffle – Coupling Analysis 23:37 Paper by Bayer & Diaconis 24:14 Conclusion This video is an entry in the πth Summer of Math Exposition, #SoMEpi #SoMEπ </video> <video> The Longest Increasing Subsequence Description: 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 </video> <video> What is the limit of a sequence of graphs?? | Benjamini-Schramm Convergence Description: This is an introduction to the mathematical concept of Benjamini-Schramm convergence, which is a type of graph limit theory which works well for sparse graphs. We hope that most of it is understandable by a wide audience with some mathematical background (including some prior exposure to graph theory), but to get the most out of the video it is helpful to know some probability theory and general topology. Made by: Caio Alves, Aranka Hrušková, and Vilas Winstein. Music: Jiná Geometrie (Different Geometry), composed by Peter Graham and performed by Aranka Hrušková. Animations made in Blender and Mathematica (with the MaTeX package). Edited in kdenlive. References: Benjamini, I., & Schramm, O. (2011). Recurrence of distributional limits of finite planar graphs. In Selected Works of Oded Schramm (pp. 533-545). Springer, New York, NY. Curien, N. (2017). Random Graphs the local convergence point of view https://www.imo.universite-paris-saclay.fr/~curien/cours/cours-RG-V3.pdf Lovász, L. (2012). Large networks and graph limits. Vol. 60. American Mathematical Soc., 2012. </video> <video> The Canopy Tree: An Example of a Graph Limit Description: In this video, we compute the Benjamini-Schramm graph limit of the sequence of complete finite binary trees. If you don't already know what Benjamini-Schramm convergence is, please watch our previous video, which gives an introduction to the concept. </video> <video> Percolation: a Mathematical Phase Transition Description: —————SOURCES———————————————————————— Percolation – Béla Bollobás and Oliver Riordan Cambridge University Press, New York, 2006. Sixty Years of Percolation – Hugo Duminil-Copin https://www.ihes.fr/~duminil/publi/2018ICM.pdf Percolation – Geoffrey Grimmett volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1999. —————NOTES————————————————————————— Note at 10:42 – The uniqueness of the infinite cluster is known for the d-dimenional lattice since the works of Aizenman, Kesten and Newman - [Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation (1987)] and Burton and Keane - [Density and uniqueness in percolation (1989)]. It does not hold in general: when the graph in question is a regular tree for example, there are always infinitely many clusters during the supercritical phase. The two last results shown here are only known for site percolation (in which vertices are open or closed instead of edges) in the triangular lattice, where a scaling limit for the boundaries of critical clusters was proved to exist (more on that in the third note). It is believed that these results are universal, that is, valid in great generality for planar percolation processes near criticality. The third result is from an appendix by Gábor Pete in the paper [Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome? (2017)] by Ahlberg and Steif. Consider an n by n box, and the event where there exists a left-right crossing of said box. Recall the uniform coupling from the video: intuitively, the result is saying that the point at which this crossing emerges in the uniform coupling is with high probability inside an interval of size n^{-3/4} around 1/2. The fourth result is saying that the average size of the cluster of the origin (or any other given point) goes to infinity as we let p approach the critical parameter like a specific power of the distance between p and p_c. This power is called a critical exponent. The existence of these exponents was proved by Smirnov and Werner in the paper [Critical exponents for two-dimensional percolation (2001)]. Note at 10:52 – Hugo Duminil-Copin has several major contributions to the study of processes arising in statistical physics, including Bernoulli percolation. Among his works on Ising and Ising-like processes we can cite [Random Currents and Continuity of Ising Model’s Spontaneous Magnetization (2015)] with Aizenman and Sidoravicius and [Sharp phase transition for the random-cluster and Potts models via decision trees (2019)] with Raoufi and Tassion. Note at 12:38 – In the triangular lattice site percolation, Stanislav Smirnov proved the conformal invariance of crossing probabilities at criticality (see https://www.unige.ch/~smirnov/papers/icmp-final.pdf for an overview), which led to the proof of the existence of scaling limits of exploration curves as Schramm–Loewner evolution processes. See [Critical percolation in the plane (2009)] by Smirnov. This provided a deep understanding of the critical phase in the triangular lattice site percolation, which to this day is not extended to the square lattice. Note at 17:52 – It is not at all obvious that the probability of being connected to infinity is continuous above criticality. This result can be proved in the d-dimenional hypercubic lattices using the uniqueness of the infinite cluster, and more generally it was proved for transitive graphs (intuitively, graphs in which all vertices look the same) by Häggström, Peres and Schonmann in [Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness (1999)]. —————SECTIONS——————————————————————— 0:00 Introduction 1:37 Definition – Bernoulli Percolation 5:23 Definition – Uniform Coupling 7:56 Exploration – High-Resolution Square Grid 9:40 Exploration – Questions and Kesten's Theorem 10:58 Exploration – Ising Model 11:54 Exploration – Critical Percolation 12:50 Exploration – Three-Dimensional Cubic Lattice and Beyond 14:13 Proof – Theorem Statement 15:14 Proof – Simplifications 16:29 Proof – Definition of Critical Parameter 18:41 Proof – Critical Parameter is Greater Than Zero 20:44 Proof – Duality Definition 21:56 Proof – Critical Parameter is Less Than One 25:16 Proof – Summary and Idea for Kesten's Theorem 26:11 Conclusion —————CREDITS———————————————————————— Caio Alves – writing, 3D animation Aranka Hrušková – writing, clarinet Vilas Winstein – writing, 2D animation, editing, voice-over Special thanks to Anisah Awad, Gábor Pete, Jyotsna Sreenivasan, Angie Zavala This video is an entry in the second Summer of Mathematics Exposition (#SoME2) The photographs used in this video are licensed under the Creative Commons Attribution-ShareAlike license: https://creativecommons.org/licenses/by-sa/4.0/deed.en </video> <video> Mathematical Animation with Manim — Lecture Recording Description: Lecture by Vilas Winstein at the Alfréd Rényi Institute of Mathematics on 2022/02/16. </video>

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