Asymptotic metrics for SU(N)-monopoles with maximal symmetry breaking

We compute the asymptotic metrics for moduli spaces of SU(N) monopoles with maximal symmetry breaking. These metrics are exponentially close to the exact monopole metric as soon as, for each simple root, the individual monopoles corresponding to that root are well separated. We also show that the estimates can be differentiated term by term in natural coordinates, which is a new result even for SU(2) monopoles.Comment: 26 pages, AMS-Latex; comparison of metrics done differently and in greater detail; version submitte

Going Deeper with Spectral Embeddings

To make sense of millions of raw data and represent them efficiently, practitioners rely on representation learning. Recently, deep connections have been shown between these approaches and the spectral decompositions of some underlying operators. Historically, explicit spectral embeddings were built from graphs constructed on top of the data. In contrast, we propose two new methods to build spectral embeddings: one based on functional analysis principles and kernel methods, which leads to algorithms with theoretical guarantees, and the other based on deep networks trained to optimize principled variational losses, which yield practically efficient algorithms. Furthermore, we provide a new sampling algorithm that leverages learned representations to generate new samples in a single step

Economic Networks: Theory and Computation

This textbook is an introduction to economic networks, intended for students and researchers in the fields of economics and applied mathematics. The textbook emphasizes quantitative modeling, with the main underlying tools being graph theory, linear algebra, fixed point theory and programming. The text is suitable for a one-semester course, taught either to advanced undergraduate students who are comfortable with linear algebra or to beginning graduate students.Comment: Textbook homepage is https://quantecon.github.io/book-networks/intro.htm