Classification of Division Algebras

Tuesday, January 27th, 2015 | Author:

Root System

I was asked to give a talk about division algebras (purpose and classification). This is a rough overview of it, mostly learned from the book of Springer and Veldkamp "Octonions, Jordan Algebras and Exceptional Groups" and the book of Lam "Noncommutative Rings". The book "Numbers" by Ebbinghaus et al. was also nice to read along for references.

We will look mostly at finite dimensional (not neccesarily associative) division algebras over the real numbers.

Instead of these rather loose notes, you may prefer to look at my texed notes in this pdf file.

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Continuing Integer Sequences

Monday, October 27th, 2014 | Author:

Old Computer

A classic class of "math problem" is the continuation of integer sequences from a finite sample (usually at the beginning). For example:

Continue 2,4,6,8,...

To which the solution is often supposed to be 10,12,14, and so on.

The problem, as any mathematician knows, is that there is not the one solution. In fact, given any set of finite numbers one could just talk about the sequence that starts like that and continues with zeroes only, that's a perfectly valid sequence. Of course, one should really figure out the rule behind the finitely many numbers, but there are always many possible choices. The game is not to find any sensible rule, but to find the rule the designer had in mind. It's more about testing your knowledge of culture than an honest test of mathematical ability (or anything else).

But there is a way to fix this.

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Fundamental Adjunctions

Tuesday, March 25th, 2014 | Author:

adjoint

In this lightheaded post (written long time ago) I want to share with you some fundamental adjunctions that are the "source" of various other adjunctions that pop up all over in mathematics (well, at least all over algebraic topology).

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Magmas, Loops and Monoids

Monday, February 24th, 2014 | Author:

Magma

This is about the mathematical concepts of a "magma", a "loop" and a "monoid", which are descriptions of certain properties that the combining of things may enjoy.

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9 Questions on Information Theory

Thursday, February 13th, 2014 | Author:

Entropy

Back in 2010 I had a series of posts going about questions in information theory that arose from a 2-week seminar with a bunch of students coming from various scientific disciplines (a wonderful event!). Here I picked those that I still find particularly compelling:

  1. Is the mathematical definition of Kullback-Leibler distance the key to understand different kinds of information?
  2. Can we talk about the total information content of the universe?
  3. Is hypercomputation possible?
  4. Can we tell for a physical system whether it is a Turing machine?
  5. Given the fact that every system is continually measured, is the concept of a closed quantum system (with unitary time evolution) relevant for real physics?
  6. Can we create or measure truly random numbers in nature, and how would we recognize that?
  7. Would it make sense to adapt the notion of real numbers to a limited (but not fixed) amount of memory?
  8. Can causality be defined without reference to time?
  9. Should we re-define “life”, using information-theoretic terms?

What do you think?

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Motivic Cell Structure of Toric Surfaces

Wednesday, April 17th, 2013 | Author:

Root System

In this post I'll do a few very explicit computations for motivic cell structures of smooth projective toric varieties coming from the Białynicki-Birula decomposition, namely \mathbb{P}^1, \mathbb{P}^1 \times \mathbb{P}^1, \mathbb{P}^2 and Hirzebruch surfaces. It is a bit lengthy but maybe helpful to anyone who wants to do some explicit calculations with BB-decompositions. I hope you're accustomed to toric varieties, but I won't do anything fancy. You can safely skip the motivic part of this post.

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