Title: Embeddings as a Problem Solving Technique

Abstract: Whether the graph of roads in Okinawa, or connections in social media, data often comes to as a network. We may wish to study optimization problems in such networks: such as shortest path or decompositions of the space. It turns out, that embedding the network into a linear space is a powerful way of simplifying these problems. We will first see some examples of this. Given the remarkable implications of such embeddings, we then wish to study when and how to construct such embeddings. This turns out to be a rich mathematical question, and I will be giving a mini course at OIST on this as well. In the talk, I will explain some ways embeddings can help you in your problems. I will also show you, with some pictures, some situations where an embedding can be very difficult to construct.

Profile: I am primarily interested in geometric problems which arise in the context of "rough" spaces: embedding problems of metric spaces, the structure of boundaries of hyperbolic groups, the geometry of fractal spaces arising from dynamics, or fractal properties of sub-Riemannian geometries. Since many of these concepts are related to "quasisymmetric mappings", I would describe these questions as being part of the "quasiworld". Especially embedding questions are not only of theoretical interest. They arise naturally in data analysis and the analysis of certain relaxation methods to solve optimization problems. Further, these questions have profound connections to the analysis on such rough spaces and concepts that come from first order analysis of metric spaces. Indeed, these notions can obstruct embeddings, prevent certain geometries from existing, or reveal subtle geometric facts about dynamical systems. Their study also leads to natural invariants that can be used to classify and distinguish spaces from each other. Especially recently, I have come to see that the study of Analysis on Metric spaces can lead to answering some difficult open problems related to geometric group theory. However, an obstacle to making progress in this direction is that many fundamental notions in Analysis on metric spaces remain poorly understood. It is there, that I strive to develop new techniques, concepts and results, that help us answer these open problems. Personal Website

Language: English, no interpretation.

Target audience: General audience / everyone at OIST and beyond.
Freely accessible to all OIST members and guests without registration.

This talk will also be broadcast online via Zoom:

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Meeting ID: 916 4861 5935
Passcode: 677103   

※ Please note that this event may be recorded and the videos uploaded. In addition, photos may be taken during the event. These are intended for publication online (the OIST website, social media, etc.)※