Weekly Bulletin

In this talk we will see a brief history of sharp Fourier restriction theory and some recent developments related to Fourier restriction estimates on spheres. We will discuss the problem of finding sharp constants for such inequalities, as well as the questions of existence and classification of extremizers of these estimates.

Empirical risk minimization may lead to poor prediction performance when the target distribution differs from the source populations. This talk discusses leveraging data from multiple sources and constructing more generalizable and transportable prediction models. We introduce an adversarially robust prediction model to optimize a worst-case reward concerning a class of target distributions and show that our introduced model is a weighted average of the source populations' conditional outcome models. We leverage this identification result to robustify arbitrary machine learning algorithms, including, for example, high-dimensional regression, random forests, and neural networks. In our adversarial learning framework, we propose a novel sampling method to quantify the uncertainty of the adversarial robust prediction model. Moreover, we introduce guided adversarially robust transfer learning (GART) that uses a small amount of target domain data to guide adversarial learning. We show that GART achieves a faster convergence rate than the model fitted with the target data. Our comprehensive simulation studies suggest that GART can substantially outperform existing transfer learning methods, attaining higher robustness and accuracy. Short Bio: Zijian Guo is an associate professor at the Department of Statistics at Rutgers University. He obtained a Ph.D. in Statistics in 2017 from Wharton School, University of Pennsylvania. His research interests include causal inference, multi-source and transfer learning, high-dimensional statistics, and nonstandard statistical inference.

We consider random walks on the homogeneous space SL_2(R)/SL_2(Z) induced by the action of a Zariski-dense subgroup consisting of matrices with algebraic entries. I will present a recent joint work with Timothée Bénard where we showed the following. The random walk equidistributes in law unless it is trapped in a finite orbit. Moreover, the convergence is fast unless the starting point is close to a finite orbit or high in the cusp.

We present results pertaining to selected inverse problems associated with seismology on Earth, Mars and Saturn. We focus on geometrical or travel time data originating from the propagation of singularities and the spectra corresponding with normal modes. For terrestrial or rocky planets we highlight recent insights with generic anisotropic elasticity, and for gas giants we reveal the accommodation of the equations of state all the way up to their boundaries. We briefly touch upon whether information on uniqueness of inverse problems is encoded in the data.