# Introduction to Noncommutative Geometry. Part I. The roots of NCG in Quantum Theory

** Speaker:** Erik van Erp, Dartmouth College

** Date:** October 12, 2023

** Abstract:** Book I of Euclid’s Elements opens with “A point is that which has no part.” Ever since Euclid, our theories of space have been based on the idea of a dimensionless point. For all their revolutionary impact, non-Euclidean geometries do not challenge this foundation. Likewise, the space-time continuum of general relativity is a “point set”. Philosophically, the concept of space as a continuum of points is intimately related to the principle of locality in physics: causality travels “from point to point”. With the violation of locality in quantum theory, the idea that physical space should be modeled as a point set becomes questionable. Space, at subatomic scales, is stranger than curved spacetime. Noncommutative Geometry is a theory of space that is compatible with quantum theory. If there are no points, how do we define coordinates? geometry? vector bundles? curvature? differential forms? integrals? field theories? This is Part I of a series of introductory talks about NCG. I will discuss the roots of NCG in quantum theory.