Critical phenomena | Nonlinear systems | Cellular automaton rules | Dynamical systems

The Abelian sandpile model (ASM) is the more popular name of the original Bak–Tang–Wiesenfeld model (BTW). BTW model was the first discovered example of a dynamical system displaying self-organized criticality. It was introduced by Per Bak, Chao Tang and Kurt Wiesenfeld in a 1987 paper. Three years later Deepak Dhar discovered that the BTW sandpile model indeed follows the abelian dynamics and therefore referred to this model as the Abelian sandpile model. The model is a cellular automaton. In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile. This slope builds up as "grains of sand" (or "chips") are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope. Bak, Tang, and Wiesenfeld considered process of successive random placement of sand grains on the grid; each such placement of sand at a particular site may have no effect, or it may cause a cascading reaction that will affect many sites. Dhar has shown that the final stable sandpile configuration after the avalanche is terminated, is independent of the precise sequence of topplings that is followed during the avalanche. As a direct consequence of this fact, it is shown that if two sand grains are added to the stable configuration in two different orders, e.g., first at site A and then at site B, and first at B and then at A, the final stable configuration of sand grains turns out to be exactly the same. When a sand grain is added to a stable sandpile configuration, it results in an avalanche which finally stops leading to another stable configuration. Dhar proposed that the addition of a sand grain can be looked upon as an operator, when it acts on one stable configuration, it produces another stable configuration. Dhar showed that all such addition operators form an abelian group, hence the name Abelian sandpile model.The model has since been studied on the infinite lattice, on other (non-square) lattices, and on arbitrary graphs (including directed multigraphs). It is closely related to the dollar game, a variant of the chip-firing game introduced by Biggs. (Wikipedia).

Ahmed Bou-Rabee (U Chicago) -- Scaling limits of sandpiles

The Abelian sandpile is a deterministic diffusion process on the integer lattice which produces striking, kaleidoscopic patterns. I will discuss recent progress towards understanding these patterns and their stability under randomness.

From playlist Northeastern Probability Seminar 2021

This is one of my all-time favorite differential equation videos!!! :D Here I'm actually using the Wronskian to actually find a nontrivial solution to a second-order differential equation. This is amazing because it brings the concept of the Wronskian back to life! And as they say, you won

From playlist Differential equations

Ian Alevy (U Rochester) -- The Limit Shape of the Leaky Abelian Sandpile Model

The leaky abelian sandpile model (Leaky-ASM) is a growth model in which n grains of sand start at the origin in Z2 and diffuse along the vertices according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighb

From playlist Northeastern Probability Seminar 2020

Alison Etheridge: Spatial population models (4/4)

Abstract: Mathematical models play a fundamental role in theoretical population genetics and, in turn, population genetics provides a wealth of mathematical challenges. In these lectures, we focus on some of the models which arise when we try to model the interplay between the forces of ev

From playlist Summer School on Stochastic modelling in the life sciences

Kazuya Kato - Logarithmic abelian varieties

Correction: The affiliation of Lei Fu is Tsinghua University. This is a joint work with T. Kajiwara and C. Nakayama. Logarithmic abelian varieties are degenerate abelian varieties which live in the world of log geometry of Fontaine-Illusie. They have group structures which do not exist in

From playlist Conférence « Géométrie arithmétique en l’honneur de Luc Illusie » - 5 mai 2021

Alison Etheridge: Spatial population models (1/4)

Abstract: Mathematical models play a fundamental role in theoretical population genetics and, in turn, population genetics provides a wealth of mathematical challenges. In these lectures, we focus on some of the models which arise when we try to model the interplay between the forces of ev

From playlist Summer School on Stochastic modelling in the life sciences

Alison Etheridge: Spatial population models (3/4)

Abstract: Mathematical models play a fundamental role in theoretical population genetics and, in turn, population genetics provides a wealth of mathematical challenges. In these lectures, we focus on some of the models which arise when we try to model the interplay between the forces of ev

From playlist Summer School on Stochastic modelling in the life sciences

Alison Etheridge: Spatial population models (2/4)

From playlist Summer School on Stochastic modelling in the life sciences

Every Group of Order Five or Smaller is Abelian Proof

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Every Group of Order Five or Smaller is Abelian Proof. In this video we prove that if G is a group whose order is five or smaller, then G must be abelian.

From playlist Abstract Algebra

Frogs on Trees by Christopher Hoffman

PROGRAM :UNIVERSALITY IN RANDOM STRUCTURES: INTERFACES, MATRICES, SANDPILES ORGANIZERS :Arvind Ayyer, Riddhipratim Basu and Manjunath Krishnapur DATE & TIME :14 January 2019 to 08 February 2019 VENUE :Madhava Lecture Hall, ICTS, Bangalore The primary focus of this program will be on the

From playlist Universality in random structures: Interfaces, Matrices, Sandpiles - 2019

Abelian networks and sandpile models (Lecture 3) by Lionel Levine

PROGRAM :UNIVERSALITY IN RANDOM STRUCTURES: INTERFACES, MATRICES, SANDPILES ORGANIZERS :Arvind Ayyer, Riddhipratim Basu and Manjunath Krishnapur DATE & TIME :14 January 2019 to 08 February 2019 VENUE :Madhava Lecture Hall, ICTS, Bangalore The primary focus of this prog

From playlist Universality in random structures: Interfaces, Matrices, Sandpiles - 2019

Abelian networks and sandpile models by Lionel Levine

PROGRAM :UNIVERSALITY IN RANDOM STRUCTURES: INTERFACES, MATRICES, SANDPILES ORGANIZERS :Arvind Ayyer, Riddhipratim Basu and Manjunath Krishnapur DATE & TIME :14 January 2019 to 08 February 2019 VENUE :Madhava Lecture Hall, ICTS, Bangalore The primary focus of this program will be on the

From playlist Universality in random structures: Interfaces, Matrices, Sandpiles - 2019

Smallest singular value and limit eigenvalue... by Arup Bose

PROGRAM :UNIVERSALITY IN RANDOM STRUCTURES: INTERFACES, MATRICES, SANDPILES ORGANIZERS :Arvind Ayyer, Riddhipratim Basu and Manjunath Krishnapur DATE & TIME :14 January 2019 to 08 February 2019 VENUE :Madhava Lecture Hall, ICTS, Bangalore The primary focus of this program will be on the

From playlist Universality in random structures: Interfaces, Matrices, Sandpiles - 2019

Regularity properties of LSD for certain families of random patterned matrices by Anish Mallick

From playlist Universality in random structures: Interfaces, Matrices, Sandpiles - 2019

Hydrodynamics and chaos in spin chains: connections to KPZ by Abhishek Dhar

From playlist Universality in random structures: Interfaces, Matrices, Sandpiles - 2019

High trace methods in random matrix theory (Remote Talk) - Lecture 2 by Charles Bordenave

From playlist Universality in random structures: Interfaces, Matrices, Sandpiles - 2019

Recurrence-Transience transition and Tracy-Widom growth in the Rotor-router mode by Rahul Dandekar

From playlist Universality in random structures: Interfaces, Matrices, Sandpiles - 2019

Representation theory: Abelian groups

This lecture discusses the complex representations of finite abelian groups. We show that any group is iomorphic to its dual (the group of 1-dimensional representations, and isomorphic to its double dual in a canonical way (Pontryagin duality). We check the orthogonality relations for the

From playlist Representation theory