What is Algebraic Topology?

Thursday, October 25th, 2012 | Author:

This short article is intended to be read by non-mathematicians who don't quite remember what the term "matrix" or "polynomial" refers to. I'll try to give you an intuitive idea of what a PhD student working on "Algebraic Topology" studies nonetheless.

First of all, studying or researching mathematics is not about remembering formula or calculating some really large numbers. That is a part of mathematics, but it's not what drives it, it is merely a tool that is more and more handed to computer systems.
So, what is mathematics instead? I am not competent to give an answer (and there have been many many different answers to that question in the past) but I can explain you my view on pure mathematics:

Pure Mathematics is the study (by any means) of the statements of which truth can be derived syntactically, i.e. by some computational process. (Yes, I'm quite a syntactic thinker).

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Lagrange and the Supernaturals

Friday, August 03rd, 2012 | Author:

You might know the theorem of Lagrange from group theory: A finite group of order n can have a subgroup of order m only if m divides n. Here the order is just the number of elements, a natural number.

Recently I came across a generalization to profinite groups. How do you make sense of the order of an infinite group? How to say that the order of a subgroup divides the order of the group? The solution is a simple concept called supernatural numbers, which I will explain in this short article. The main part should in principle be accessible to non-mathematicians as well.

Category: English, Mathematics | 4 Comments

Informationstheorie

Tuesday, August 24th, 2010 | Author:

Ich bin hier gerade in Rot an der Rot in einem Seminar über die Rolle der Informationstheorie in den Naturwissenschaften. Mein Vortrag hat heute statt gefunden, es war der vierte von zwanzig und behandelte bedingte Entropie, zusammen mit den notwendigen Voraussetzungen aus der diskreten Wahrscheinlichkeitstheorie, der Begrifflichkeit der Entropie (Information) an sich und zahlreiche Anwendungen.

Vorgetragen habe ich mit Folien (am Beamer), es dauerte etwa 90 Minuten und ging am Ende ein bisschen zu schnell. Ich habe auch noch ca. 90 Backup-Folien mit deutlich mehr Informationen vorbereitet, bevor ich den konkreten Vortrag daraus destilliert habe. Die Folien sind mit der Latex-Beamer-class erstellt.

Beides gibt es natürlich ohne Gewähr, allerdings habe ich (meines Wissens nach) alle (2) Fehler korrigiert, die das Publikum gefunden hat.

Quellen waren die Paper von Shannon und Weaver, das Buch von Brillouin (Science and Information Theory) und meine alten Notizen zur Stochastik aus dem Vordiplom (zusammen mit dem einführenden Buch von Hans-Otto Georgii, das ich immer noch gern zum Nachschlagen verwende). Einzelne Grafiken stammen von Wikipedia, andere habe ich gezeichnet (und auch als public domain in Wikipedia eingebracht). Meine Zeichnungen habe ich mit Inkscape erstellt.

Viel Spaß damit!

Category: German, Mathematics | Comments off

Review – Seife: Decoding the Universe

Monday, August 16th, 2010 | Author:

I just finished Decoding the Universe - How the new science of information is explaining everything in the cosmos, from our brains to black holes, written 2006 by the former mathematics student and now associate professor of journalism Charles Seife, apparently well known for his other books Zero and Alpha&Omega (which I didn't read).
The book in Google Books and a much shorter review I wrote in German on Amazon.de

Category: English | One Comment

Why believing in conspiracy theories is wrong

Friday, May 07th, 2010 | Author:

I guess most people who believe in conspiracy theories either have some benefit in pretending to believe or they really think the theories are likely to be true. Those who think conspiracy theories are likely to be true, are victims of some kind of "Bayesian fallacy":

Bayes (English mathematician, 1702-1761) proved a theorem about conditional probabilities, nowadays called "Bayes' theorem". Suppose there are two statements A and B, which might overlap (e.g. A="it's raining today" and B="it's raining the whole week"¹, where the truth of B implies the truth of A). Now imagine these statements are more or less likely, so you attach some probability to these statements, p(A) and p(B), with values in 0-100% (or, for the mathematically oriented readers: let p be a probability measure on some discrete $\sigma$-algebra containing A and B). It's not only the probability of A and B we might be interested in, but also the conditional probability "How likely is A when B is true?", which we write p(A|B). Bayes' theorem now reads:
$P(A|B)\cdot P(B) = P(B | A)\cdot P(A)$, and this means in words, that the probability of A under the condition that B is true, multiplied by the probability of B, is the same as the probability of B under the condition that A is true, multiplied by the probability of A.

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Beautiful new geometry videos

Thursday, February 04th, 2010 | Author:

On dimensions-math.org you can see a whole bunch of before unseen geometry videos introducing 2-, 3- and 4-dimensional space, complex numbers and even more to the non-mathematician (somewhat similar to the well-known Not-Knot-videos and the Moebius transformations on YouTube, but with lots of explanations). The computer animations are available on DVD and online, for free. The explanations are in many different languages.

This is something not to miss if you're interested in mathematics, and it might also be valuable if you're taking a first course in complex analysis. Even after you've taken a course on complex analysis, you might enjoy the animation of the Hopf fibration (which I liked most).

Go straight to watching the videos in English.

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