Tyler Derr – Department of Computer Science, Vanderbilt University

The field of deep learning on graphs has seen rapid development over the recent years, where graph neural networks (GNNs) have shown great promise in learning node embeddings for link prediction (LP). While numerous studies aim to improve the overall LP performance of GNNs, fewer studies have investigated their node-level LP performance variability. In this talk, we aim to demystify which nodes receive better LP from the perspective of their local topology. Despite the widespread belief that low-degree nodes exhibit poorer LP performance (e.g., recently joined users in a social/e-commerce network), our empirical findings provide nuances to this viewpoint and prompt our development of a better metric, topological concentration (TC), based on the intersection of the local subgraphs of each node with the ones of its neighbors. We show that TC has stronger correlation with node-level LP performance compared to other node-level topological metrics, such as degree and subgraph density. This provides a better way to identify and understand nodes receiving poor predictions, which could signal a higher risk of user churn. We also discuss some other recent related work and future directions.

Darren Creutz, Vanderbilt University
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Mike Neamtu, Vanderbilt University

Convergence of Matrix Products, Subdivision, Refinement Equations, and Cascade Networks

In this talk, we will discuss the following question: Given two square matrices of the same dimension, we consider their arbitrary repeated products and ask under which conditions these products converge to a continuous matrix function. This topic arises from various areas of approximation theory, including subdivision methods for generating curves and surfaces in computer-aided geometric design, refinement equations in multi-resolution analysis, and cascade algorithms. The talk is based in part on joint work with my former student, Diana Sordillo.

Istvan Kadar, Princeton University

The existence of multi black hole solutions in asymptotically flat spacetimes is one of the expectation from the final state conjecture. In this talk, I will present preliminary works in this direction via a semilinear toy model in dimension 3. In particular, I show 1) an algorithm to construct approximate solutions to the energy critical wave equation that converge to a sum of solitons at an arbitrary polynomial rate in (t-r); 2) a robust method to solve the remaining error terms for the nonlinear equation. The methods apply to energy supercritical problems.

Alex Wright- University of Michigan

Given a surface, the associated curve graph has vertices corresponding to certain isotopy classes of curves on the surface, and edges for disjoint curves. Starting with work of Masur and Minsky in the late 1990s, curve graphs became a central tool for understanding objects in low dimensional topology and geometry. Since then, their influence has reached far beyond what might have been anticipated. Part of the talk will be an expository account of this remarkable story. Much more recently, non-trivial examples of totally geodesic subvarieties of moduli spaces have been discovered, in work of McMullen-Mukamel-Wright and Eskin-McMullen-Mukamel-Wright. Part of the talk will be an expository account of this story and its connections to dynamics.
The talk will conclude with new joint work with Francisco Arana-Herrera showing that the geometry of totally geodesic subvarieties can be understood using curve graphs, and that this is closely intertwined with the remarkably rigid structure of these varieties witnessed by the boundary in the Deligne-Mumford compactification

Stephane Jaffard- University Paris Est Creteil

Many types of signals display a very oscillatory behavior in the neighborhood of singularities. It is for example the case for gravitational waves, fully developed turbulence, or brain data. A major issue is to detect such behaviors (referred to as “oscillating singularities” or “chirps”) which are the signature of important physical phenomena. We will show how a multivariate multifractal analysis based on wavelet methods allows to meet these challenges.

Misha Perepelitsa, University of Houston

Abstract: TBA

Jean-Francois Paquet, Vanderbilt University

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Dick Canary

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