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The possibility of using quantum mechanics to build computers that are both faster and more secure is an exciting technological leap. One of the most promising pathways to a robust architecture, referred to as topological quantum computing, has lead to fascinating research at the frontier of mathematics and physics. Properties of new, exotic states of matter can be understood - and hence implemented in practice - using tools that were developed by pure mathematicians in connection to surfaces and their curvature. This workshop is bringing together international experts to explore the intimate connections between such topological properties of quantum matter and the key resource of quantum computing: entanglement.