Explore the concept of essential dimension in algebraic geometry through this 57-minute lecture by Patrick Brosnan from the University of Maryland. Delve into the numerical measure of complexity for algebraic objects, invented by J. Buhler and Z. Reichstein in the 1990s. Examine how essential dimension relates to the number of parameters needed to define an object over a field, using examples like mu_n torsors. Investigate recent work by Brosnan and Najmuddin Fakhruddin, which recovers and extends results from Farb, Kisin, and Wolfson's 2019 preprint on the essential dimension of congruence covers of Shimura varieties. Learn about a new fixed point theorem and its applications to incompressibility results, including cases involving Shimura varieties of exceptional type. Consider a general conjecture on the essential dimension of congruence covers arising from Hodge theory, and explore its implications for the dimension of the image of the period map.