Mathematics

Robust Mean Estimation (part 2)

Greg Terlov (UIUC Math )

Abstract: One of the most fundamental problems of statistics is to estimate the expected value of a random variable given $n$ i.i.d. samples. It is natural to do so by considering the empirical mean, however when one works with heavy-tailed distributions empirical mean would give less than ideal accuracy and confidence intervals for finite $n$ and thus one has to consider other estimators. In the last talk we saw that taking median-of-mean yields promising results in 1 dimension but the construction depends on the given confidence interval. In this talk we will discuss this dependency and show that it cannot be improved without further assumptions on the distributions. Following that we will discuss how to extend this procedure to a multidimensional setting. These talks are be based on the survey paper by Lugosi and Mendelson https://arxiv.org/abs/1906.04280.

Introduction to large deviation theory

Peixue Wu (UIUC math )

Abstract: In the first part of my talk, I will review large deviation principles and the well-known Gartner-Ellis theorem. The main technique is to make a detailed analysis of the log moment generating function and its Fenchel dual. As an application, we recover some large deviation principles we established before.

Large Deviations Part II

Peixue Wu (UIUC math )

Abstract: I will continue the proof of Gartner-Ellis theorem and show the applications to dynamical Large deviations of Markov processes. Moreover, I will mention some further research problems.

Witten deformation of Laplacian

Kesav Krishnan (UIUC math )

Abstract: I will introduce the notion of Witten deformation of Laplacian, and discuss probabilistic interpretations of the same, I will then discuss some applications to the study of lattice models in statistical mechanics.

witten deformation of Laplacian part II

Kesav Krishnan (UIUC math )

Abstract: I will introduce the notion of Witten deformation of Laplacian, and discuss probabilistic interpretations of the same, I will then discuss some applications to the study of lattice models in statistical mechanics.

An introduction to Pirogov-Sinai Theory

Bob Krueger (UIUC MATH)

Abstract: I will introduce a few spin models on lattices and discuss the usual polymer models. Then I will describe the polymer model at the basis of Pirogov-Sinai Theory. I hope to get to a recent result of Helmuth, Perkins, and Regts which uses Pirogov-Sinai Theory to efficiently sample from these spin models at low temperatures. I assume no knowledge of statistical physics.

An introduction to Pirogov-Sinai Theory

Bob Krueger (UIUC Math )

Abstract: I will introduce a few spin models on lattices and discuss the usual polymer models. Then I will describe the polymer model at the basis of Pirogov-Sinai Theory. I hope to get to a recent result of Helmuth, Perkins, and Regts which uses Pirogov-Sinai Theory to efficiently sample from these spin models at low temperatures. I assume no knowledge of statistical physics.

The ant on a rubber rope paradox

Ting-Yang Hsiao (UIUC Math )

Abstract: In this talk, we consider a puzzle called "Ant on a rubber rope”. It is a stochastic process on a random growing domain. To be more specific, an ant is at the left endpoint of a rubber band, which is 1 kilometer long and the ant crawls along with the rubber band at a pace having an expected value of 1 centimeter per second. After the first second, the rubber band stretches extra length L such that the expected value of L is 1 kilometer. Repeat the steps above. It seems that there is a high probability that this ant will not achieve the right endpoint of the rubber band. In this talk, however, we prove that the ant will reach the right end in finite time almost surely.