This thesis develops the theory of sheaves and cosheaves with an eye towards applications
in science and engineering. To provide a theory that is computable, we focus on a …

These notes--originating from a one-semester class by their second author at the University
of Minnesota--survey some of the most important Hopf algebras appearing in combinatorics …

The seeds of this book have been planted in the far east, where I wrote lecture notes for
international schools in Tianjin, China in 2007, and in Bangkok, Thailand in 2011. I then …

Abstract Lieb-Schultz-Mattis (LSM) theorems provide powerful constraints on the emergibility
problem, ie whether a quantum phase or phase transition can emerge in a many-body …

Abstract The Quantum Max Cut (QMC) problem has emerged as a test-problem for
designing approximation algorithms for local Hamiltonian problems. In this paper we attack …

Variational algorithms require architectures that naturally constrain the optimisation space to
run efficiently. In geometric quantum machine learning, one achieves this by encoding group …

Deep neural networks can solve many kinds of learning problems, but only if a lot of data is
available. For many problems (eg in medical imaging), it is expensive to acquire a large …

Graph convolutional learning has led to many exciting discoveries in diverse areas.
However, in some applications, traditional graphs are insufficient to capture the structure …

Let $\mathcal {U} $ be a braided tensor category, typically unknown, complicated and in
particular non-semisimple. We characterize $\mathcal {U} $ under the assumption that there …

Unitary equivariance is a natural symmetry that occurs in many contexts in physics and
mathematics. Optimization problems with such symmetry can often be formulated as …