Monday, April 19, 2021
04/19/2021 - 10:15am
Let G be a semi-simple real algebraic group, $G_\mathbb R$ its real points and $G_\mathbb C$ its complex points. Let $\Gamma$ be a discrete subgroup of the real points $G_\mathbb R.$ The purpose of the talk is to explain a general procedure relating (in some situations) the second variation of the Hausdorff dimension of the limit set of $\Gamma,$ when deforming $\Gamma$ inside the complex group $G_\mathbb C,$ with natural geometries (known as pressure forms) of the real characters $X(\Gamma,G_\mathbb R).$
This is joint work with M. Bridgeman, B. Pozzetti and A. Wienhard
04/19/2021 - 1:00pm
Abstract: The tremendous importance of graph structured data due to recommender systems or social networks led to the introduction of graph convolutional neural networks (GCN). Those split into spatial and spectral GCNs, where in the later case filters are defined as elementwise multiplication in the frequency domain of a graph. Since often the dataset consists of signals defined on many different graphs, the trained network should generalize to signals on graphs unseen in the training set. One instance of this problem is the transferability of a GCN, which refers to the condition that a single filter or the entire network have similar repercussions on both graphs, if two graphs describe the same phenomenon. However, for a long time it was believed that spectral filters are not transferable.
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Zoom Meeting ID: 97670014308
04/19/2021 - 4:30pm
There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio:
Trace(ρ(g)) / dim(ρ),
for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.
Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms”, which is (since the 60s) the main organization principle, and is based on the (huge collection) of “Large” representations. This talk will discuss the notion of rank for the group GLn over finite fields, illustrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks. This is joint work with Roger Howe (Yale). The numerics for this work was carried with Steve Goldstein (Madison) and John Cannon (Sydney).
https://yale.zoom.us/j/92811265790 (Password is the same as last semester)