Totally nonnegative Grassmannian and Grassmann polytopes
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These are lecture notes intended to supplement my second lecture at the Current Developments in Mathematics conference in 2014. In the first half of article, we give an introduction to the totally nonnegative Grassmannian together with a survey of some more recent work. In the second half of the article, we give a definition of a Grassmann polytope motivated by work of physicists on the amplituhedron. We propose to use Schubert calculus and canonical bases to replace linear algebra and convexity in the theory of polytopes.
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Brave new categorical spectral positive Schubert geometry and the categorical Dual Amplituhedron
The dissertation rewrites positive Schubert geometry via spectral algebraic geometry and differential cohesion to construct a categorical dual to the Amplituhedron with a De Rham volume.
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