Abstract:

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.

  Abstract:

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.

  Abstract:

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.

  Abstract:

Let \(\Gamma\) be a discrete group. A subgroup \(N\) is called confined if there is a finite set \(F\) in \(\Gamma\) which intersects every conjugate of \(N\) outside the trivial element. For example, a nontrivial normal subgroup is confined. A discrete subgroup of a semisimple Lie group is confined if the corresponding locally symmetric orbifold has bounded injectivity radius. We proved a generalization of the celebrated NST: Let \(\Gamma\) be an irreducible lattice in a higher rank semisimple Lie group G. Let \(N<\Gamma\) be a confined subgroup. Then \(N\) is of finite index. The case where \(G\) has Kazhdan's property (T) was established in my joint work with Mikolaj Frakzyc. As in the original NST, without property (T) the problem is considerably harder. The main part is to prove a spectral gap for \(L_2(G/N)\). This is a joint work with Uri Bader and Arie Levit.

  Abstract:

Let \(G\) be a countable group acting by isometries on a metric space \((M, d)\), and let \(\mu\) denote a probability measure on \(G\). The \(\mu\)-random walk on \(M\) is the random process defined by \[Z_n=X_n \dots X_1 o,\] where \(o \in M\) is a fixed base point, and \(X_i\) are independent \(\mu\)-distributed random variables. Studying statistical properties of the displacement sequence \(D_n:= d(Z_n, o)\) has been a topic of current research. In this talk, which is based on a joint work with Amin Bahmanian, Behrang Forghani, and Ilya Gekhtman, I will discuss a central limit theorem for \(D_n\) in the case that \(M\) is the horospherical product of Gromov hyperbolic spaces.