A collection of code and visualisations related to the MATH40008 Poster Project on Concentration Phenomena in Asymptotic Geometry.
What is MATH40008?
MATH40008 is a module all first year maths undergraduates at Imperial College London take after completing their summer exams. The department suggests topics and students choose one. For their chosen topic, they are expected to learn via independent research, construct a poster and give a presentation.
'Concentration Phenomena in Asymptotic Geometry' falls under 'Law of Large Numbers and The Central Limit Theorem'.
What are 'Concentration Phenomena in Asymptotic Geometry'?
Asymptotic Geometry is when we look at what happens to geometric objects in very high dimensions. Concentration refers to the notion that almost all the volume of certain -dimensional regions cluster around an arbitrarily small
-neighbourhood of an
-dimensional region. We discuss two examples:
- The volume of an asymptotic
-Cube concentrates on its intersection with an
-Sphere.
- The area of an asymptotic
-Sphere concentrates on its equators
.
One of the main theorems we showcase in the poster is that in high dimensions almost all the volume of the -Cube concentrates on its intersection with an
-Sphere with radius
. Formally we want to show that
The main idea is that in high dimensional cubes we can think of a point's coordinates as independent and identically distributed (i.i.d) random variables. In order for a point to lie at an extremity these random variables must all cooperate in the same direction. This is unlikely and the Law of Large Numbers stipulates that for large this is impossible. Instead, the collective effort of these random variables will reflect their average. Geometrically speaking, this means points must exist a certain distance from the origin (a sphere in high dimensions). For more details see the poster and the proofs folder in this repository.
Figure 1 - For higher dimensions we can see that more and more of the volume of the cube falls inside the sphere.
Figure 2 - The concentration eventually increases to 1 no matter how slow.
In other words, the sphere contains almost all the volume of the cube.
Figure 3 - In lower dimensions the sphere is inside the cube. For the radius of the sphere is
and hence the sphere exceeds the boundaries of the cube. This is why we consider the intersection of the cube and sphere.