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Daily Overview |
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AG1: Algebraic Geometry
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Noetherianity up to Sym and GL 1: University of Bern, Switzerland; 2: University of Michigan, USA; 3: Eindhoven University of Technology, The Netherlands Work by Daniel Cohen from the 1980s implies that, if X is an affine scheme of finite type over a Noetherian ring, then the infinite Cartesian product XN (N={the natural numbers}) is Noetherian up to the infinite symmetric group Sym: every descending chain of Sym-stable closed subschemes is eventually constant. A similar result holds for GL-varieties, infinite-dimensional affine varieties with a suitable action of the infinite general linear group, albeit only for reduced subschemes. Both of these results have led to much follow-up work and applications. I will discuss some of this work and then move on to the following "least common multiple" of both results above: if X is a GL-variety, then XN is Noetherian up to the product Sym x GL. This theorem is joint work with Chiu, Danelon, Eggermont, and Farooq—and it is waiting for good applications! Iarrobino scheme: a self-dual analogue of the Hilbert scheme of points University of Warsaw, Poland Hilbert schemes of points are classical moduli spaces with rich geometry. Together with their cousins, the Quot schemes, they appear in many applications, in particular to enumerative geometry. In the talk I will present recently constructed self-dual analogue of the Hilbert scheme and the corresponding version for the Quot scheme. This scheme is a certain generalisation of the permutohedral variety, it has connections to both classical enumeration and enumeration a la June Huh, as well as algebraic aspects, such as Iarrobino's symmetric decomposition. Representation theory over bands 1: Brown University, United States; 2: Durham University, United Kingdom This talk presents an overview of ongoing research into representation theory over bands, which are combinatorial generalisations of fields, with applications in matroid theory and tropical geometry. A fundamental departure from the classical theory arises from the observation that natural general linear objects in this setting cannot carry group structure; they are instead monoids. To construct a general linear monoid, we study the multiplication of bimatroids and matrices over bands, investigate their interrelations, and construct their moduli spaces as combinatorial analogues of the determinantal variety. These constructions provide the foundation for defining band analogues of classical matrix groups and their representations. As a proof of concept, we conclude with a study of Schur functors in this new setting. | ||
