Algebra 1; algebra 2 is recommended but not required; patience with some longer computations, visual thinking, curiosity about the shape of the universe.
Did you ever wonder if the universe has a shape, and what it might be? In this course, participants learn how to think about the shape and internal structure of the cosmos from a mathematical perspective, culminating in an appreciation for various theoretical models of the 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 may look 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, 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, exploring advanced concepts from topology and theoretical cosmology 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.
Course participants gain, in addition to 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 passionate about transmitting knowledge between the math research community and the rest of the world, for the benefit of both parties.