William Thurston's Influence

The impact of American mathematician William Thurston (1946–2012) is profound. A former student remarked, "His influence extends well beyond the vast content of his mathematics. He transformed the way geometers and topologists approach mathematics." — Mathematician Benson Farb, University of Chicago.

To understand Thurston's contributions, we first need a brief overview of geometry and topology. Geometry focuses on measurable aspects such as angles, lengths, and volumes, while topology examines geometric forms under continuous deformation, emphasizing properties that remain unchanged despite distortions.

Topology treats objects as if they are made from elastic material—shapes can be stretched and contorted, but not cut. A classic joke about topologists is that they can't distinguish between a coffee cup and a donut.

Beginning in the 1970s, Thurston ignited a revolution in mathematics with a unified approach to studying three-dimensional shapes through geometry and topology, centered on a novel understanding of 3-manifolds.

3-Manifolds: Understanding the Universe's Shape

In mathematics, a 3-manifold is a topological space that locally resembles three-dimensional Euclidean space. You may have heard that 3-manifolds can represent potential shapes of the Universe. But what does this entail?

To clarify this concept, I consulted low-dimensional topologist Jessica Purcell, a Professor of Mathematics specializing in geometry and topology at Monash University in Melbourne and author of a book on Hyperbolic Knot Theory.

"A 3-manifold is a generalization of a surface," she explains. "All 3-manifolds resemble our Universe at a sufficiently small scale."

To visualize this, consider standing on Earth's surface, where you can move forward, backward, or side to side. In three-dimensional space, you also gain an up-and-down direction. Thus, in a 3-manifold, each point permits movement in three distinct directions.

How does this connect to the Universe's shape? Topologists view Earth as a 2-dimensional manifold because its surface can be manipulated into a sphere with stretching. If Earth is a two-dimensional manifold, then the Universe must be some form of three-dimensional manifold. While mathematicians agree on this, the Universe's exact shape remains uncertain—it could be spherical, toroidal, or something even stranger.

The first video titled "The evolution of geometric structures on 3-manifolds" explores how geometric structures develop within these mathematical frameworks, providing insights into their complexities.

Hyperbolic Geometry: A Deeper Dive

Hyperbolic geometry represents the geometry of negative curvature. Curvature describes how a shape deviates from a flat surface. For instance, a person's head is positively curved, while the neck is negatively curved, illustrating the concepts of hyperbolic spaces.

What would a hyperbolic 3-manifold look like? Extending the two-dimensional example into three dimensions would involve a curvature both inward and outward from a tangent plane at each point, leading us to the definition of a hyperbolic 3-manifold.

Thurston's research focused on the properties of 3-manifolds, which also extended to the study of mathematical knots.

Early Developments in Knot Theory

"For a mathematician, a knot exists in space with closed ends, creating a flexible object that can be twisted and bent," states topologist Stephan Tillmann from the University of Sydney.

Scottish physicist Peter Guthrie Tait (1831–1901) was the first to systematically classify knots. He painstakingly hand-drew numerous knot diagrams, which represent knots projected onto a plane—akin to a shadow cast by a knot on a wall.

Tait categorized knots based on their crossing number, the total number of string crossings, and published his first knot table in 1867, covering knots with up to seven crossings. The daunting task of cataloging beyond this point led him to cease at seven crossings.

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The second video, "The Mystery of 3-Manifolds - William Thurston," provides an insightful look into Thurston's groundbreaking theories and their implications on the understanding of 3-manifolds.

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Continuing with the exploration of Dehn surgery and the implications for hyperbolic knots, the legacy of Thurston persists as mathematicians navigate the complexities of knot theory in the context of modern research.