Equidistribution results play an important role in dynamical systems and their applications in number theory.  Often in such applications it is desirable for equidistribution to be effective (i.e. the rate of convergence is known). In this talk I will discuss some of the history of effective equidistribution results in homogeneous dynamics and give an effective result for horospherical flows on the space of lattices. I will then describe an application to studying the distribution of almost-prime times in horospherical orbits and discuss connections of this work to Sarnak’s Mobius disjointness conjecture.

Inverse Problems on graphs encompass many areas of physics, algorithms and statistics, and are a confluence of powerful methods, ranging from computational harmonic analysis and high-dimensional statistics to statistical physics. Similarly as with inverse problems in signal processing, learning has emerged as an intriguing alternative to regularization and other computationally tractable relaxations, opening up new questions in which high-dimensional optimization, neural networks and data play a prominent role. In this talk, I will argue that several tasks that are ‘geometrically stable’ can be well approximated with Graph Neural Networks, a natural extension of Convolutional Neural Networks on graphs. I will present recent work on supervised community detection, quadratic assignment, neutrino detection and beyond showing the flexibility of GNNs to extend classic algorithms such as Belief Propagation.