SUMS: Large -- Curvature: From Euclid to Einstein

This week’s talk is by Cambridge undergrad Tim Large.  

Abstract: 
For such an intuitive geometrical concept, so central to our experience of physical 
objects, defining curvature has always been somewhat elusive.  Trying to pin it
down and answer the question "what does it mean for a surface to be curved" is a
surprisingly difficult task that requires a fair amount of sideways thought and
creativity.  This question has dominated geometry over the last 200 years, and a huge
variety of ways to approach and answer it have been developed.  

In this talk, we’ll look at curvature from a select few perspectives.  In fact we’ll
start over 2000 years ago with Euclid’s axioms of classical geometry, and use them as
the point of departure to think about what geometry - the study of lengths, angles,
areas - is like on curved surfaces such as on a sphere.  We’ll then compare this to the
work of Gauss, who applied the machinery of calculus to give the new concept of Gaussian
curvature, which bears his name.  Finally we’ll briefly discuss how these ideas feed
directly into the thoughts Einstein was having around 1915, and produce the backbone of
general relativity, a modern physical theory where curvature becomes the mechanism for
gravity.  

Apart from knowing that the sum of the angles of a triangle is 180 degrees (or is it!!),
absolutely no prior knowledge of geometry or physics will be needed to follow this talk
- just an open mind.  Hopefully this talk will be a demonstration of how deep and
fascinating simple things can be, and how far ancient ideas can permeate through the
most modern of maths and physics.