Mathematics as Metaphor - Curtis McMullen (Harvard University)

TL;DR

Mathematics is both a part of nature, governing the laws of the universe, and an artifact of human culture, influenced by language and communication.

Transcript

hello hi I'm Steve kerkhof I'm the chairman of the math department here at Stanford and I'd like to welcome you to tonight's lecture by professor Curtis McMullen who's a Cabot professor of mathematics at Harvard this public lecture is one of a series that sponsored by the math department and the Stanford math Research Center it's also a very specia... Read More

Key Insights

• 👮 Mathematics exists in both the natural laws of the universe and as a cultural artifact.
• 💨 Tiling provides a useful way to understand and represent mathematical concepts, such as surfaces and knots.
• ❓ Quantum topology and the Jones polynomial offer insights into the relationship between quantum physics and mathematics.

Questions & Answers

Q: What is the difference between positive and negative curvature in geometric space?

Positive curvature is characterized by objects, such as spheres, where the sum of angles in a triangle is greater than 180 degrees. Negative curvature is where the sum of angles in a triangle is less than 180 degrees, resulting in hyperbolic surfaces.

Q: Can the idea of quantum topology be applied to other mathematical concepts, such as surfaces?

Quantum topology focuses specifically on the study of knots and braids, while surfaces can be analyzed using other mathematical tools. However, there may be potential connections between quantum topology and the study of higher-genus surfaces that have not yet been fully explored.

Q: Is mathematics a human invention or a discovery?

Mathematics is a product of human culture and language. While it may describe fundamental truths about the universe, it relies on human interpretation and communication to convey these concepts effectively.

Q: How does the figure-eight knot relate to its quantum polynomial and hyperbolic volume?

The figure-eight knot can be represented by a surface with negative curvature, and its quantum polynomial, known as the Jones polynomial, serves as an invariant that can be used to determine its hyperbolic volume. The computation of the polynomial involves making modifications to the knot and analyzing the resulting changes in its structure.

Summary & Key Takeaways

• Mathematics exists in the laws of the universe, such as general relativity and quantum mechanics, as well as in human culture as a means of communication and understanding complex systems.

• The Platonic solids, such as the cube and dodecahedron, have recurring presence in mathematics and are associated with the elements of nature and the structure of the solar system.

• The concept of tiling, or arranging shapes to fill a space, can be used to represent surfaces of different geniuses, such as the torus and surfaces with handles.

• Knot theory and quantum topology provide insights into the nature of three-dimensional space and the interaction between quantum physics and mathematics.