We will discuss the non-stationary version of the Central Limit Theorem for random products of SL(2, R) matrices. The stationary versions were obtained previously by Tutubalin, Le Page, and Benoist-Quint. In all the previous works the assumption that the random matrices are identically distributed was used in a crucial way. We will explain how the recent results on the rate of growth of non-stationary products of random matrices can be used to overcome this restriction. The talk is based on a project joint with A. Gorodetski and G. Monakov. 

AI today can pass the Turing test and is in the process of transforming science, technology, society, humans, and beyond.
Surprisingly, modern AI is built out of two very simple and old ideas, rebranded as deep learning: neural networks and
gradient descent learning. When a typical feed-forward neural network is trained by gradient descent, with an L2 regularizer
to avoid overly large synaptic weights, a strange phenomenon occurs: at the optimum, each neuron becomes "balanced"
in the sense that the L2 norm of its incoming synaptic weights becomes equal to the L2 norm of its outgoing synaptic weights. We develop a theory that explains this phenomenon and exposes its generality. Balance emerges with a variety of activation functions, a variety of regularizers including all Lp regularizers, and a variety of networks including recurrent networks. A simple local balancing algorithm can be applied to any neuron and at any time, instead of just at the optimum. Most remarkably, stochastic iterated application of the local balancing algorithm always converges to a unique, globally balanced, state.

Random multiplicative functions are probabilistic models for important arithmetic functions in number theory, e.g. Mobius function, Dirichlet characters. In this talk, I would like to introduce the topic and emphasize some recent developments. Part of the talk is based on joint works with Angelo, Harper, and Soundararajan.