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Explore the fascinating journey of a traveler moving along straight paths through various geometric worlds in this 54-minute mathematical lecture. Begin with the familiar Euclidean torus world where the traveler encounters subtori, a phenomenon explained by Kronecker's 1884 theorem demonstrating that any line's closure in a Euclidean torus forms a subtorus. Venture beyond into the more complex realms of closed hyperbolic manifolds and hyperbolic manifolds of infinite volume to discover what geometric wonders await. Learn how symmetry and motion of shapes in curved spaces reveal hidden patterns, even in endlessly stretching spaces, through numerous illustrations that bring these abstract geometric concepts to vivid life. Gain insights into the intersection of dynamics, Lie groups, geometry, and number theory from Yale University's Abraham Robinson Professor of Mathematics, whose groundbreaking work has earned recognition including the AMS Satter Prize, Guggenheim Fellowship, and Ho-Am Prize in Science.
