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Hypermatrices (higher-dimensional matrices) are important because they broaden the scope of matrix concepts such as spectra and group actions. Hypermatrix algebras also illuminate subtle aspects of matrix algebra. In this article we describe an instance where the transition from matrices to hypermatrices shines a light on subtle details of invariant theory and leads to new insights.
First, we give an informal and elementary introduction to the idea of a moduli space, and then to moduli spaces of Riemann surfaces and their Deligne-Mumford-Knudsen compactifications. Second, we discuss some tropical geometry, assuming no prior knowledge of the subject, and build up to some recent results on moduli spaces that were obtained using tropical techniques.
“I have been riveted for decades by the breathless bulletins from cosmologists describing the latest twist to their model of space and time at the largest possible scale. This article is partly to share the beauty of this theory with my colleagues but also to express how thoroughly relativity shakes up our deep psychological conviction of the reality of an external physical time.”
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AMS President Jill Pipher to Congress: "No Longer Secure: Cryptography in the Quantum Era"
Mathematics in the Theory and Practice of Auctions
“How would you, like, start on the concept of algebra?”
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