Is the universe infinite or finite? What is the curvature and overall shape of the space we live in, and how might we detect this? Many have considered these questions throughout the ages, and while we still don’t know the answers, a field of mathematics called geometric topology has given rise to a relatively new framework for approaching them.
In this course, participants learn how models for topological spaces relate to theories on the shape of the physical universe. Philosophical discussions are informed by pencil and paper computations, experiments with common household materials, and interactive online games and modules.
We begin by learning from the mistakes of archaic geocentric models so as to build intuition about manifolds — structures that look one way up close but totally different from far away. This sets the stage for a precise definition of dimension, providing the framework for the rest of the course. Students explore notions of curvature, flatness, boundedness, embeddings, and finiteness by constructing models of two-dimensional geometries out of toothpicks, cotton swabs, tape, or whatever works — creativity encouraged! We then apply our findings to the three-dimensional world, with the help of a few video games, such as flying a space ship through hyperbolic space. Finally, we look at the role of fractals in modeling reality, and how they push our notion of dimension to fascinating new horizons.
Participants gain, in addition to early exposure to modern content at the intersection of topology and physical cosmology, an appreciation for rigorous mathematical thinking that is motivated not so much by numbers and quantity as by profound questions about the nature of our world.
Joe Quinn holds a PhD in mathematics from The Graduate Center, CUNY, and publishes in the field of geometric topology. He has taught and designed math curricula for Hunter College, Johns Hopkins Center for Talented Youth, Bronx Community College, Matemorfosis CIMAT (Guanajuato), MoMath, and Bridge to Enter Advanced Mathematics. He is the director of Hypothesis, a New York City based company that creates games and events based on conceptual math and art. Joe is passionate about transmitting knowledge between the math research community and the rest of the world, for the benefit of both parties.
Specific course detail such as hours and instructors are subject to change at the discretion of the University. Not all instructors listed for a course teach all sections of that course.