Courses

Please register here for courses and other event activities.


 

TOPICS IN PROBABIITY AND RIGOROUS STATISTICAL MECHANICS

Students who wish to obtain credit for participation in the program should take this class. See the syllabus (in Portuguese) and/or contact the instructors for more details. [REGISTER/TOP]
 

A TÉCNICA DE RENORMALIZAÇÃO DINÁMICA

This course at UFRJ is not officially affiliated with the thematic program, but should be of interest to participants. Click here for more information. Please contact the organizers for information on transportation between UFRJ and IMPA on the days of the course. [TOP]
 

CONCENTRATION INEQUALITIES WITH APPLICATIONS

Concentration inequalities estimate deviations of functions of independent random variables from their expectation. Such inequalities have countless applications and they play a fundamental role in probabilistic combinatorics, the analysis of learning algorithms and statistical procedures. In these lectures we present some of the basic ideas and some useful inequalities. We discuss in detail the so-called "entropy method" for deriving general concentration inequalities. We discuss various applications. [REGISTER/TOP]
 

INTERMEDIATE MARKOV CHAINS

In this course we will discuss some properties of random walks on graphs and some techniques to estimate the mixing time of reversible Markov chains. We will try to follow the plan below.

  • * Random walks on graphs: hitting time, cover time and relation with electrical networks.
  • * Upper bound on the mixing time of Markov chains via stationary times and coupling.
  • * Spectral decomposition and comparison of Markov chains.
  • * (time permitting) Lower bound on the mixing time via bottlenecks.
[REGISTER/TOP]
 

METASTABILITY

A process is said to exhibit a metastable behavior if it remains for a very long time in a state before undergoing a rapid transition to a stable state. After the transition, the process remains in the stable state for a period of time much longer than the time spent in the first state, called for this reason metastable. In certain cases, there are two or more "metastable wells" of the same depth, a situation called by physicists "competing metastable states". In these cases, the process thermalizes in each well before jumping abruptly to another well where the same qualitative behavior is observed. The sticky zero range process is an example of this situation. In a joint work with C. Landim, we presented an approach to derive the metastable behavior of continuous-time Markov chains in which the main tool is the martingale problem. The purpose of this course is (a) to present the martingale problem as a suitable tool to prove convergence (in law) of Markov processes (b) to illustrate the use of some other tools as the trace process and capacities in our approach to metastability (c) to apply our results to the sticky zero range process. [REGISTER/TOP]
 

RANDOMNESS, MATRICES AND HIGH DIMENSIONAL PROBLEMS

This course provides a very short and partial introduction to high dimensional problems in Probability, with applications in Statistics and Compressed Sensing. Our focus will be on problems involving random covariance-type matrices and random matrices with independent entries. Besides proving concentration properties for these matrices, we will also discuss how these two kinds of matrices appear in problems such as Least Squares, Compressed Sensing and Community Detection in Random Graphs. [REGISTER/TOP]
 

HIGH DIMENSIONAL ESTIMATION: FROM FOUNDATIONS TO ECONOMETRIC MODELS

In this mini-course we start with the foundations of modern statistical techniques based on L1-penalization for high dimensional estimation under sparsity assumptions. We will cover rates of convergence for Lasso, sparsity bounds on the selected model, and simple lower bounds on its performance. We then will shift our interests to how further develop these ideas on models that are motivated by Econometric applications. For example, heteroskedastic errors, logistic regression, conditional quantiles, and error-in-variables. Some emphasis will be placed on properly handling the different assumptions induced by the (econometric) data generating process (e.g. approximate sparse models and non-Gaussian errors). We will finish the mini-course by establishing results on the uniform validity of confidence regions (e.g. confidence intervals) for parameters. We will attempt to cover partially linear models, instrumental variables, and Z-estimators. [REGISTER/TOP]
 

FIRST PASSAGE PERCOLATION

First-passage percolation is a model for spatial growth introduced about half a century ago. The model is obtained by assigning i.i.d. non-negative weights to the edges of a graph, such as the square or cubic lattice. The non-negative weights give rise to a random metric structure on the graph. Distances in this metric can be interpreted as travel times of a growing entity. Understanding the long term evolution of this growing entity is the main objective in the area. First-passage percolation has been intensively studied within both the mathematics and physics communities. These studies has led to a rigorous theory for subadditive ergodic processes and far reaching predictions of KPZ theory. Despite the ease of which the model is defined there are many fundamental questions that remain unanswered. We will in these lectures give an introduction to the field and the many open problems that remain. Subjects to be treated include: subadditive ergodic theory and the asymptotic shape; competing growth and the existence of geodesics; columnar defects and inhomogeneous growth; fluctuations and scaling theory. [REGISTER/TOP]
 

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