# 학과 세미나 및 콜로퀴엄

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Lec 1: Introduction to Markov chain mixing theory
In this lecture, we introduce the Markov chain mixing theory by assuming that the attendances have no background on this field. The precise definition of the mixing time and several basic techniques estimating mixing time will be discussed.

(전체일정: 7/26, 7/27, 8/2, 8/4)
Lec 2: Introduction to cut-off phenomenon
In this second lecture, we introduce the so-called cut-off phenomenon and look at several examples exhibiting this phenomenon. Then, we rigorously prove the cut-off phenomenon for the simplest possible model, the lazy random walk on hypergraphs.

(전체일정: 7/28, 7/29, 8/3, 8/5)
In 2d first-passage percolation, we let (t_e) be a family of i.i.d. weights associated to the edges of the square lattice, and let T = T(x,y) be the induced weighted graph metric on Z^2. If we let p be the probability that a weight takes the value 0, then there is a transition in the large-scale behavior of T depending on the value of p. Specifically, when p < 1/2, T grows linearly, but when p > 1/2, T is stochastically bounded. In these lectures, I will describe some of the standard questions of FPP in the case p < 1/2, and then focus on the "critical" case, where p = 1/2. Regarding this case, I will show some of my work over the last few years including exact asymptotics for T, universality results, and recent work on a dynamical version of the model. The work I will present was done in collaboration with J. Hanson, D. Harper, W.-K. Lam, P. Tang, and X. Wang.
Lec 1: Introduction and the passage time to infinity

(전체일정: 7/28, 7/29, 8/3, 8/5)
In 2d first-passage percolation, we let (t_e) be a family of i.i.d. weights associated to the edges of the square lattice, and let T = T(x,y) be the induced weighted graph metric on Z^2. If we let p be the probability that a weight takes the value 0, then there is a transition in the large-scale behavior of T depending on the value of p. Specifically, when p < 1/2, T grows linearly, but when p > 1/2, T is stochastically bounded. In these lectures, I will describe some of the standard questions of FPP in the case p < 1/2, and then focus on the "critical" case, where p = 1/2. Regarding this case, I will show some of my work over the last few years including exact asymptotics for T, universality results, and recent work on a dynamical version of the model. The work I will present was done in collaboration with J. Hanson, D. Harper, W.-K. Lam, P. Tang, and X. Wang.
Lec 2: Phase transition and the shape theorem

Analyzing group behavior of systems of interacting variables is a ubiquitous problem in many fields including probability, combinatorics, and dynamical systems. This problem also naturally arises when one tries to learn essential features (dictionary atoms) from large and structured data such as networks. For instance, independently sampling some number of nodes in a sparse network hardly detects any edges between adjacent nodes. Instead, we may perform a random walk on the space of connected subgraphs, which will produce more meaningful but correlated samples. As classical results in probability were first developed for independent variables and then gradually generalized for dependent variables, many algorithms in machine learning first developed for independent data samples now need to be extended to correlated data samples. In this talk, we discuss some new results that accomplish this including some for online nonnegative matrix and tensor factorization for Markovian data. A unifying technique for handling dependence in data samples we develop is to condition on the distant past, rather than the recent history. As an application, we present a new approach for learning "basis subgraphs" from network data, that can be used for network denoising and edge inference tasks. We illustrate our method using several synthetic network models as well as Facebook, arXiv, and protein-protein interaction networks, that achieve state-of-the-art performance for such network tasks when compared to several recent methods.

Certain zero range processes on a finite set exhibits metastability. Most of the time nearly all particles of the zero range process are at one single site, and the site of condensate asymptotically behaves as a Markov chain. In this talk, we prove the metastability of zero range processes on a finite set with an approach using the Poisson equation. This approach doesn't need precise estimates of capacities and can be applied for both reversible and non-reversible cases. This talk is based on the joint work with F. Rezakhanlou.

(전체일정: 7/26, 7/27, 8/2, 8/4)
Lec 3: Cut-off phenomenon for mean-field spin systems
In this third lecture, we explain the cut-off phenomenon for the Glauber dynamics of the mean-field Ising or Potts model and general strategy to prove it.

(전체일정: 7/28, 7/29, 8/3, 8/5)
In 2d first-passage percolation, we let (t_e) be a family of i.i.d. weights associated to the edges of the square lattice, and let T = T(x,y) be the induced weighted graph metric on Z^2. If we let p be the probability that a weight takes the value 0, then there is a transition in the large-scale behavior of T depending on the value of p. Specifically, when p < 1/2, T grows linearly, but when p > 1/2, T is stochastically bounded. In these lectures, I will describe some of the standard questions of FPP in the case p < 1/2, and then focus on the "critical" case, where p = 1/2. Regarding this case, I will show some of my work over the last few years including exact asymptotics for T, universality results, and recent work on a dynamical version of the model. The work I will present was done in collaboration with J. Hanson, D. Harper, W.-K. Lam, P. Tang, and X. Wang.
Lec 3: Critical FPP: the Bernoulli case

(전체일정: 7/26, 7/27, 8/2, 8/4)
Lec 4: Information Percolation
In this final lecture, we discuss the celebrated technique known as the information percolation introduced by Lubetzky and Sly. Then, we explain the application of this technique to Glauber dynamics of the Ising model on lattice, and of the Random cluster model. The last result is a joint work with Shirshendu Ganguly.

(전체일정: 7/28, 7/29, 8/3, 8/5)
In 2d first-passage percolation, we let (t_e) be a family of i.i.d. weights associated to the edges of the square lattice, and let T = T(x,y) be the induced weighted graph metric on Z^2. If we let p be the probability that a weight takes the value 0, then there is a transition in the large-scale behavior of T depending on the value of p. Specifically, when p < 1/2, T grows linearly, but when p > 1/2, T is stochastically bounded. In these lectures, I will describe some of the standard questions of FPP in the case p < 1/2, and then focus on the "critical" case, where p = 1/2. Regarding this case, I will show some of my work over the last few years including exact asymptotics for T, universality results, and recent work on a dynamical version of the model. The work I will present was done in collaboration with J. Hanson, D. Harper, W.-K. Lam, P. Tang, and X. Wang.
Lec 4: Critical FPP: the general case