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The sl_3 web category is a combinatorial model for the representation theory of the quantum group U_q(sl_3). We study the category from the cellular point of view, exploiting diagrammatic methods and light-ladder bases to compute branching rules, homomorphisms, and composition factors of cell modules when q is a root of unity. We then apply these results to the study of clasps. In the semisimple case, these higher rank analogues of the Jones—Wenzl idempotents can be constructed using Elias’s recursive triple-clasp expansion. When q is a root of unity, however, their construction is considerably more subtle, and we use the aforementioned results from cellularity to obtain formulas for some of the non-semisimple clasps.

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I will describe a method that deforms analytic cycles to algebraic cycles. This will be done using global derived analytic geometry and analytic (Efimov) K-theory. I will conjecture how this is related to the Tate and Hodge conjectures. If time permits, I will speculate about a global arithmetic site. This is joint work in progress with Federico Bambozzi, Jack Kelly, and Devarshi Mukherjee.

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