Monday, November 9, 2020
11/09/2020 - 2:30pm
Abstract: Spectral clustering is a popular unsupervised learning technique for finding meaningful structure in large datasets. A weighted graph is constructed on the dataset, encoding the similarities between the data points. A graph Laplacian operator is then defined on this graph whose spectral geometric content reveals the number and shape of clusters in the data set. In this talk I will present some spectral analysis of graph Laplacians in the continuum limit where the number of vertices of the graph goes to infinity. In the first part I will discuss how the different normalizations of the graph Laplacian will affect the spectrum of the continuum operator and introduce a notion of a balanced normalization that has desirable qualities in large data settings. In the second part of the talk I will focus on a specific choice of the graph Laplacian and present some results on the consistency of spectral clustering by first studying the continuum limit operator and extending its properties to discrete approximations.
11/09/2020 - 4:00pm
In this talk, we consider random walks arising from the action of SL_d(Z) on the d-dimensional torus. Quantitative equidistribution in law was first obtained by Bourgain, Furman, Lindenstrauss and Mozes. In this talk I will present a recent progress where the proximality assumption in their result is relaxed. This is based on a joint work with Nicolas de Saxcé.
11/09/2020 - 4:30pm
For a real reductive group G, the set of unipotent representations is a finite set of irreducible representations that satisfy certain desired properties and are supposed to be “building blocks” for all unitary representations of G. In 1982 Arthur proposed a definition of a special unipotent representation, which have been studied extensively in past years. However, this notion does not include some interesting unitary representations, such as a metaplectic representation. We restrict our attention to the case of a complex semisimple group G. In an ongoing project with Ivan Losev and Lucas Mason-Brown we propose a new definition of unipotent representations that extends the set of special unipotent ones. Using this definition we were able to classify unipotent representations and prove their key properties including some that were not known for special unipotent representations before. In this talk, I will explain our construction of unipotent representations, and time permitting sketch proofs of the main results.
https://yale.zoom.us/j/99433355937 (password was emailed by Ivan on 9/11, also available from Ivan by email)