Explore the fascinating world of character estimates for finite simple groups and their applications in this 45-minute lecture by Michael Larsen. Delve into key theorems, including the Diaconis-Shahshahani theorem, and examine their relevance to mixing time for groups of Lie type. Investigate Waring's Problem for finite simple groups, Ore's conjecture, and Thompson's conjecture. Learn about Fuchsian groups and representation spaces, and discover how characters play a crucial role in these mathematical concepts. Examine various types of character estimates, including centralizer bound, Diaconis-Shahshahani bound, and Murnaghan-Nakayama rule. Analyze the Müller-Schlage-Puchta bound, exponential bounds, and Gluck's bound. Explore highly regular elements and their significance in unbounded rank. Conclude with a proof by amplification, gaining a comprehensive understanding of character estimates and their wide-ranging applications in group theory.