UBC Math Department Colloquium: Danny Ofek

Let X be your favorite type of algebraic structure. For example, X could be the class of quadratic forms, algebras or algebraic varieties. To estimate the complexity of X, you might consider the following natural question: how many algebraically independent parameters are needed to define an object of type X up to isomorphism? The answer to this question is called the essential dimension of X and denoted ed(X). 

Mathematicians have tried to compute ed(X) for various X for a long time, and while the exact number is usually unknown, there are some general tools to estimate it based on the theory of algebraic groups and algebraic stacks. In this talk, I will highlight some of these tools and demonstrate how they can be used to investigate the complexity of classical algebraic structures, such as field extensions, division algebras and quadratic forms. Only basic knowledge of abstract algebra will be assumed.