Abstract: In this talk, I will introduce the nilpotent cohomological Hall algebra COHA(S,Z) of coherent sheaves on a smooth quasi-projective complex surface S that are set-theoretically supported on a closed subscheme Z. This algebra can be viewed as the "largest" algebra of cohomological Hecke operators associated with modifications along a subscheme Z of S. When S is the minimal resolution of an ADE singularity and Z is the exceptional divisor, I will describe how to characterize COHA(S,Z) in terms of the Yangian of the corresponding affine ADE quiver Q (based on joint work with Emanuel Diaconescu, Mauro Porta, Oliver Schiffmann, and Eric Vasserot, arXiv:2603.03386). More generally, I will discuss nilpotent COHAs arising from Bridgeland stability conditions on the bounded derived category of nilpotent representations of the preprojective algebra of Q, following joint work with Olivier Schiffmann and Parth Shimpi (arXiv:2511.08576).

We show that various classical theorems of linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a general result that involves a tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing and interpreting the…

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.