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Category: stories

mathematics for 2008 (and beyond)

Published December 30, 2007 by lievenlb

Via the n-category cafe (and just now also the Arcadian functor ) I learned that Benjamin Mann of DARPA has constructed a list of 23 challenges for mathematics for this century.

DARPA is the “Defense Advanced Research Projects Agency” and is an agency of the United States Department of Defense ‘responsible for the development of new technology for use by the military’.

Bejamin Mann is someone in their subdivision DSO, that is, the “Defense Sciences Office” that ‘vigorously pursues the most promising technologies within a broad spectrum of the science and engineering research communities and develops those technologies into important, radically new military capabilities’.

I’m not the greatest fan of the US military, but the proposed list of 23 mathematical challenges is actually quite original and interesting.

What follows is my personal selection of what I consider the top 5 challenges from the list (please disagree) :

1. The Mathematics of Quantum Computing, Algorithms, and Entanglement (DARPA 15) : “In the last century we learned how quantum phenomena shape
our world. In the coming century we need to develop the
mathematics required to control the quantum world.”

2. Settle the Riemann Hypothesis (DARPA 19) : “The Holy Grail of number theory.”

3. Geometric Langlands and Quantum Physics (DARPA 17) : “How does the Langlands program, which originated in number
theory and representation theory, explain the fundamental
symmetries of physics? And vice versa?”

4. The Geometry of Genome Space (DARPA 15) : “What notion of distance is needed to incorporate biological utility?”

5. Algorithmic Origami and Biology (DARPA 10) : “Build a stronger mathematical theory for isometric and rigid
embedding that can give insight into protein folding.”

All of this will have to wait a bit, for now

HAPPY & HEALTHY 2008

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SMS-Math Meme (SMM)

Published December 21, 2007 by lievenlb

Hey, here’s an idea : The Text-Math Book! Trying to promote mathematics while at the same time acknowledging the fairly limited attention-span of the intended generation, let’s try to write a book on serious maths following just one rule

EVERY DEFINITION, THEOREM AND PROOF IN THE BOOK SHOULD NOT BE LONGER THAN A TEXT-MESSAGE (ie. 160 chars)

I don’t even own a cell phone (( waiting for the iPhone to arrive in Belgium )), so PLEASE educate me youngsters! SMS your contribution, either as a comment left here or hosted at your own blog (please link, so that I can learn…, a full text explanation of abbreviations used will be applauded.)

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daddy wasn’t impressed

Published December 19, 2007 by lievenlb

A first year-first semester course on group theory has its hilarious moments. Whereas they can relate the two other pure math courses (linear algebra and analysis) _somewhat_ to what they’ve learned before, with group theory they appear to enter an entirely new and strange world. So, it is best to give them concrete examples : symmetry groups of regular polygons and Platonic solids, the symmetric group etc. One of the lesser traditional examples I like to give is Nim addition and its relation to combinatorial games.

For their first test they had (among other things) to find a winning move for the position below in the Lenstra’s turtle turning game. At each move a player must put one turtle on its back and may also turn over any single turtle to the left of it. This second turtle, unlike the first, may be turned either onto its feet or onto its back. The player wins who turns the last turtle upside-down.

So, all they needed to see was that one turtle on its feet at place n is equivalent to a Nim-heap of height n and use the fact that all elements have order two to show that any zero-move in the sum game can indeed be played by using the second-turtle alternative. (( for the curious : the answer is turning both 9 and 4 on their back ))

A week later, one of the girls asked at the start of the lecture :

Are there real-life applications of group-theory? I mean, my father asked me what I was learning at school and I told him we were playing games turning turtles. I have to say that he was not impressed at all!.

She may have had an hidden agenda to slow me down because I spend an hour talking about a lot of things ranging from codes to cryptography and from representations to elementary particles…

For test three (on group-actions) I asked them to prove (among other things) Wilson’s theorem that is

$~(p-1)! \equiv -1~\text{mod}~p $

for any prime number $p $. The hint being : take the trivial action of $S_p $ on a one-element set and use the orbit theorem. (they know the number of elements in an $S_n $-conjugacy class)

Fingers crossed, hopefully daddy approved…

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