Explore mathematical illusions and logical fallacies in this engaging lecture by Sarah Hart, Gresham Professor of Geometry. Discover seemingly valid "proofs" that lead to absurd conclusions like 1=0, the non-existence of fractions, and all triangles being isosceles. Examine the nature of mathematical proof and its real-world implications. Delve into infinite series, the Riemann series theorem, and learn to spot errors in deceptive mathematical arguments. Gain valuable insights into common logical pitfalls and their consequences in mathematics and beyond through this comprehensive exploration of mathematical paradoxes and reasoning.