Chromatic Homotopy Theory – II

Let’s pick up from where we were before. Just to recall, in the previous post, we showed that every complex-oriented cohomology theory gives rise to a formal group law over . In addition, formal group laws are classified by the Lazard ring , that is isomorphic as a graded ring to , so every complex […]

Chromatic Homotopy Theory – I

In this post, I will talk about chromatic homotopy theory, which you can call “chromotopy”. This post assumes familiarity with spectra and the notions of Chern classes of line bundles; the reader who is not familiar with this notion is referred to (the very brief) http://www.mit.edu/~sanathd/papers/htpy.pdf. Recall that there is a universal line bundle over such that […]

MichelFest: Minkowski & Hausdorff Dimensions

I left off the last post on An Introduction to Fractal Geometry discussing the issue of dimension for fractals. Namely, using the defined box dimension, we saw that the Cantor Set and the Sierpinski Carpet have non-integer dimension. We also saw some issues with the Box Dimension, specifically with density, and thus will seek different notions of […]

MichelFest: An Introduction to Fractal Geometry

So I’m currently at The Summer School on Fractal Geometry and Complex Dimensions at Cal Poly San Luis Obispo. The event is a conference/summer school held in honour of the 60th birthday of Michel Lapidus of U.C., Riverside, a French Mathematician, my mentor, editor of The Journal of Fractal Geometry, and one of the world’s leading fractal […]

Shor’s algorithm

Fix an odd composite integer that is not the power of a prime. Shor’s algorithm is a probabilistic algorithm used to find factors of . It is the following. 1. Let . If , then is a nontrivial factor of . 2. If , then , so has finite order – finding this order is […]

A primer on group representations

In this post, we will give a brief introduction to the representation theory of general (finite) groups. Definition: A representation of a group is a homomorphism , where is called the dimension of the representation. A representation is faithful if the map is injective. A representation can also be viewed as the action of on […]