Pre-seminar title: Introduction to essential dimension
Abstract: The essential dimension of an algebraic structure is the minimal number of independent parameters needed to define it. It is a natural measure of complexity connected to many problems in algebra and geometry. J. Buhler and Z. Reichstein showed that the essential dimension of many different types of objects can be studied in a uniform way using equivariant birational geometry. In this pre-seminar talk, we will explore this geometric perspective, focusing on the example of étale algebras.
Seminar title: Lower bounds on the essential dimension of reductive groups
Abstract: The essential dimension of an algebraic group G is an integer measuring the complexity of G and of its torsors. Often G-torsors classify a class of algebraic objects, in which case ed(G) is the minimal number of independent parameters needed to define a generic object of that type. For example, ed(PGL_n) is the number of parameters needed to define a generic division algebra of degree n.
We introduce a new technique for proving lower bounds on the essential dimension of split reductive groups. As an application, we strengthen the best previously known lower bounds for various split simple algebraic groups, most notably for the exceptional group E8. In the case of the projective linear group PGL_n, we recover A. Merkurjev’s celebrated lower bound with a simplified proof. Our technique relies on decompositions of loop torsors over valued fields due to P. Gille and A. Pianzola