Deligne defined symmetric monoidal categories Rep(GL_t), Rep(O_t) and Rep(S_t) (t in C) which interpolate the usual finite dimensional representation categories of GL(n), O(n) and S_n to complex parameters t. Even earlier, Turaev had defined quantized versions of the GL_t and O_t categories. They are also known as the idempotent completions of the oriented skein category (the GL_t case) and Kauffmann category (the O_t case). I will explain why I am interested in these categories and what I would like to know about them. The main result is a classification of thick tensor ideals for Rep(U_q(o_t))$ if $q$ is generic (joint work with Hans Wenzl). If time permits and I feel ambitious, I might say something about the root of unity case.

Abstract: Given a cyclic subgroup G of GL(2,C) acting on C^2, it was first noticed by Wunram in the 80s that there is a correspondence between certain special representations of G and the exceptional curves appearing in the minimal resolution Y of the surface singularity C^2/G. In modern terms, this was reformulated by Ishii and Ueda as…

We investigate the hierarchy between rationality, unirationality, and rational connectedness, discussing the various implications among these concepts. We then examine specific cases, addressing quadric bundles and del Pezzo surfaces over arbitrary fields, as well as complex 3-folds.

We construct affine charts of a smooth projective toric variety which contain its non-
negative points, and which admit a closed embedding into the total coordinate space
of Cox’s quotient construction. We show that such positive charts arise from smooth
subcones of the nef cone. To each positive chart we associate an algebraic moment map,<…