For property (T) II$_1$ factors, any inclusion into a tracial von Neumann algebra has spectral gap, and therefore weak spectral gap. I will discuss characterizations of property (T) for II$_1$ factors by weak spectral gap in inclusions. I will explain how this is related to the non-weakly-mixing property of the bimodules containing almost central vectors, from which we also obtain a characterization of property (T).

Typical comparison results in Riemannian geometry, such as for volume or for spectrum of the Laplacian, require Ricci curvature lower bounds. In dimension three, we can prove several sharp comparison estimates assuming only a scalar curvature bound. The talk will present these results, their applications and describe how dimension three is used in the proofs. Joint work with Jiaping Wang.

Decision tree-based models are a popular tool for use in prediction and regression machine learning problems. In this talk, we will provide an overview of decision tree models and ensemble methods, including (but not limited to) random forests and XGBoost. We'll also discuss considerations of building such models and some applications. This talk does not require any background knowledge in machine learning.