University of California, San Diego.
Academic year: 2023-2024.
Monday 3:00-4:00.
APM 7321

Nov 6 Manny Reyes
UC Irvine
  Categories of hypergroups and hyperstructures


A hyperoperation on a set M is an operation that associates to each pair of elements a subset of M. Hypergroups and hyperrings are two examples of structures defined in terms of hyperoperations. While they were respectively defined in the 1930s and 1950s, they have recently gained prominence through various appearances in number theory, combinatorics, and absolute algebraic geometry. However, to date there has been relatively little attention given to categories of hyperstructures. I will discuss several categories of hyperstructures that generalize hypergroups. A common theme is that in order for these categories to enjoy good properties like (co)completeness, we must allow for the product or sum of two elements to be the empty subset, which cannot occur in a hypergroup. In particular, I will introduce a category of hyperstructures called mosaics whose subcategory of commutative objects possess a closed monoidal structure reminiscent of the tensor product of abelian groups. This is joint work with So Nakamura.

Nov 20
Sebastian Pardo Guerra
UC San Diego
  On the lattice of lattice preradicals and some isomorphic structures


Most algebraic structures can be given a lattice structure. For instance, any \(R\)-module defines a lattice where the meet and the join operations are given by the intersection and the sum of modules. Furthermore, any \(R\)-module morphism gives rise to a usual lattice morphism between the corresponding lattices. Actually, these two correspondences comprise a functor from the category of \(R\)-modules to the category of complete modular lattices and usual lattice morphisms. However, this last category does not summon some basic algebraic properties that modules have (for example, the first theorem of isomorphism). With this in mind, we consider the category of linear modular lattices and linear morphisms, where we extend the notions of preradicals, and thus, describe the big lattice of lattice preradicals. In the process, we define some isomorphic structures to such lattice of lattice preradicals.

Nov 27 Eoin Mackall
UC San Diego
  Splitting Severi--Brauer varieties by torsors for genus one curves


We'll be talking about recent results on the problem of splitting Brauer classes by torsors for genus one curves. In its geometric form the question to be asked is: does every Severi--Brauer variety contain a smooth and projective genus one curve? Algebraically, this question is related to the existence of certain finite Galois modules inside the linear algebraic automorphism group of the Severi--Brauer variety. Our goal will be to motivate why this is an intuitive and interesting question, giving some new results along the way.

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