The Mathematical and Computational Sciences at Western are represented by four separate departments: Applied Mathematics, Computer Science, Mathematics, and Statistical & Actuarial Sciences. The Department of Mathematics has established research groups in several areas of contemporary mathematics including algebra, analysis and analytic geometry, homotopy theory, and noncommutative geometry.
Quaternion algebras contain quadratic field extensions of the center.
Given two algebras, a natural question to ask is whether they share a common field extension. This gives us an idea of how closely related those algebras are to one another.
If the center is of characteristic 2 then those extensions divide into two types - the separable type and the inseparable type.
It is known that if two quaternion algebras share an inseparable field extension then they also share a separable field extension and that the converse is not true.
We shall discuss this fact and its generalization to $p$-algebras of arbitrary prime degree.
Relative categories are maybe the most naive model for
abstract homotopy theory (just categories with a subcategory of "weak
equivalences"). Barwick and Kan showed that the category of relative
categories has a model structure, Quillen equivalent to the Joyal model
structure on simplicial set, which has infinity-categories as fibrant
objects. We will show that model categories define fibrant relative
categories and also discuss other aspects of the homotopy theory of