Combinatorics and Geometry of Convex Polyhedra: March 20-24, 2023

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Organized by: Karim Adiprasito, University of Copenhagen and Hebrew University of Jerusalem,
Alexey Glazyrin, University of Texas Rio Grande Valley,
Isabella Novik, University of Washington,
Igor Pak, UCLA.

Convex polyhedra are fundamental structures in geometry playing an intermediate role between the continuous (smooth) and discrete (combinatorial) worlds. Over the past century, much progress has been made in their study from both combinatorial point of view (f-vectors, face lattices, the Hirsch conjecture) and from metric geometry point of view (the Alexandrov-Fenchel theory, the Minkowski-and Steinitz-type theorems). Yet, they continue to represent a major challenge as the role the convexity plays in higher dimensions continues to be mysterious, so much that even the truly new examples are difficult to understand.

In this workshop we intend to advance both the study of combinatorial structures of convex polytopes in high dimensions and the combinatorial geometry of notable combinatorial polytopes. We are especially interested in discussing geometric and enumerative properties of generalized associahedra (introduced by Fomin and Zelevinsky in 2003 in connection with cluster algebras), the volume and triangulations of amplituhedron, and the f-vectors of various classes of polytopes (see above). We plan to have several long talks surveying each subject, in addition to more technical talks discussing the state of the art.