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Spacetime and Quantum Mechanics, Particles and Strings, Polytopes and Binary Geometries

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CARW03 - Interdisciplinary applications of cluster algebras

The past decade has seen the emergence of surprising new connections between the real-world physics of elementary particle scattering processes, and the notion of “positive geometries”, linking combinatorics, algebra and geometry in interesting new ways, and involving a number of appearances of total positivity and cluster structure.  These ideas provide, in a number of examples, a different starting point for conceptualizing this physics, where the fundamental principles of spacetime and quantum mechanics are not taken as primary, but instead emerge from a more primitive mathematical rubric. In this talk I will illustrate these ideas in their simplest setting, showing how the most basic particle scattering amplitudes are determined by “surfacehedra”, certain generalizations of associahedra and finite-type cluster polytopes,  associated with any surface. I will also describe how the combinatorial relationships captured by the facets of these polytopes remarkably admit a “curvy” realization in terms of “binary geometries”, which provides a global description of the boundary structure of teichmuller space, and has the physical interpretation of generalizing from particle to string scattering”. 

This talk is part of the Isaac Newton Institute Seminar Series series.

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