One of the most basic problems in statistics is how to estimate the expected value of a distribution, based on a sample of independent random draws. When the goal is to minimize the length of a confidence interval, the usual empirical mean has a sub-optimal performance, especially for heavy-tailed distributions. In this talk we discuss some estimators that achieve a sub-Gaussian performance under general conditions. The multivariate scenario turns out to be more challenging. We present an estimator with near-optimal performance. We also discuss how these ideas extend to regression function...

# PACM/Applied Mathematics Colloquium

For more information about this seminar, contact Amit Singer

**Please click on colloquium title for complete abstract.**

##### Mean estimation: median-of-means tournaments

ICREA & Pompeu Fabra University, Spain

##### Symmetry methods for quantum variational principles and expectation value dynamics

Inspired by previous works by Kramer & Saraceno and Shi & Rabitz, this talk exploits symmetry methods for the variational formulation of different problems in physics and chemistry. First, I will use symmetry methods to provide new variational principles for the description of mixed quantum states, in various pictures including Schrödinger, Heisenberg, Dirac (interaction) and Wigner-Moyal. Then, after discussing Ehrenfest's mean-field model, I will modify its symmetry properties to provide a new variational principle for expectation value dynamics in general situations. Upon...

University of Surrey, Guilford, UK

##### The mathematics of charged liquid drops

*Cyrill Muratov is Professor of Mathematical Sciences at New Jersey Institute of Technology. He received his M.Sc. in Applied Mathematics and Physics from Moscow Institute of Physics and Technology, followed by a Ph. D. in Physics from Boston University and postdoctoral training in Applied Mathematics at the Courant Institute. His main research interests lie in understanding the emergence of complexity from basic constitutive laws in problems of science and engineering, using a combination of rigorous mathematical analysis, formal asymptotics and numerical simulations.*

NJ Institute of Technology

##### Self-similar structure of caustics and shock formation

Bristol University