Tuesday, March 25th, 2014 | Author: Konrad Voelkel

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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Monday, February 24th, 2014 | Author: Konrad Voelkel

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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Thursday, February 13th, 2014 | Author: Konrad Voelkel

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:

- Is the mathematical definition of Kullback-Leibler distance the key to understand different kinds of information?
- Can we talk about the total information content of the universe?
- Is hypercomputation possible?
- Can we tell for a physical system whether it is a Turing machine?
- 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?
- Can we create or measure truly random numbers in nature, and how would we recognize that?
- Would it make sense to adapt the notion of real numbers to a limited (but not fixed) amount of memory?
- Can causality be deﬁned without reference to time?
- Should we re-deﬁne “life”, using information-theoretic terms?

What do you think?

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Wednesday, April 17th, 2013 | Author: Konrad Voelkel

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 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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Thursday, April 04th, 2013 | Author: Konrad Voelkel

This is about Białynicki-Birula's paper from '72 on actions of reductive linear algebraic groups on non-singular varieties, in particular Gm-operations on smooth projective varieties. I give a proof sketch of Theorem 4.1 therein and explain a little bit how Brosnan applied these results in 2005 to get decompositions of the Chow motive of smooth projective varieties with Gm-operation. Wendt used these methods in 2010 to lift such a decomposition on the homotopy-level, to prove that smooth projective spherical varieties admit stable motivic cell decompositions. Most of this blogpost consists of an outline of the B-B paper.

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Tuesday, April 02nd, 2013 | Author: Konrad Voelkel

There are a lot of fundamental groups floating around in mathematics. This is an attempt to collect some of the most popular and sketch their relations to each other.

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