Skip to content →

neverendingbooks Posts

wankers

Published April 10, 2008 by lievenlb

You may not have noticed, but I’m in a foul mood for weeks now because of comments and reactions to the last line of the post on Finding Moonshine. I wrote

Du Sautoy is a softy! I’d throw such students out of the window…

and got everyone against me for this (first floor) defenestration threat…

That’s OK! I sometimes post what’s on my mind and if you don’t like it you are free to leave a comment, and, usually I won’t even bother to reply to it. But occasionally, stuff is bottling up un-healthily.

So, I thought it was a good idea to have a prolonged easter-vacation, somewhere in the south of France. The weather, food, rest, drinks, company and all that were just gorgeous

but …

A quick recap. Here’s the relevant section in duSautoy’s book again :

One of my graduate students has just left my office. He’s done some great work over the past three years and is starting to write up his doctorate, but he’s just confessed that he’s not sure that he wants to be a mathematician. I’m feeling quite sobered by this news. My graduate students are like my children. They are the future of the subject. Who’s going to read all the details of my papers if not my mathematical offspring? The subject feels so tribal that anyone who says they want out is almost a threat to everything the tribe stands for.
Anton has been working on a project very close to my current problem. There’s no denying that one can feel quite disillusioned by not finding a way into a problem. Last year one of my post-docs left for the City after attempting to scale this mountain with me. I’d already rescued him from being dragged off to the City once before. But after battling with our problem and seeing it become more and more complex, he felt that he wasn’t really cut out for it.
What is unsettling for me is that they both questioned the importance of what we are doing. They’ve asked that ‘What’s it all for?’ question, and think they’ve seen the Emperor without any clothes.
Anton has questioned whether the problems we are working on are really important. I’ve explained why I think these are fundamental questions about basic objects in nature, but I can see that he isn’t convinced. I feel I am having to defend my whole existence. I’ve arranged for him to join me at a conference in Israel later this month, and I hope that seeing the rest of the tribe enthused and excited about these problems will re-inspire him. It will also show him that people are interested in what he is dedicating his time to.

For starters, I’m getting old so I’m from those long-forgotten days when you had to do a Ph.D. to prove that you could conduct research independently.

A fortiori this meant that the topic of your thesis was your own choice and interest. The role of your Ph.D. advisor was to get you going and, occasionally, to warn you when you were straying too far afield but that was it.

You, and only you, were responsible to get the thesis finished and defended.

Today, the Ph.D. is just another item on the market to be consumed.

Graduate students shop around looking for the advisor having the best sales-pitch, offers the best deal and, if possibly, the best after-phd service aka the promise of an academic career.

Topic and main outline of the proofs are provided by the advisor and an exceptionally good student today means that (s)he proved a few results along the way on her/his own.

University policy and the promotion-rat-race appear to make the Ph.D. more important to the advisor than to the defendent.

Independence of research today means that after your PhD is obtained, you ditch your advisor and try to get into the slipstream of another more powerful guru, having better after-phd service prospects…

For those who stick with their old advisor, the moment of truth comes when they fail to get a renewal of their grant or a permanent position.

At that time one can hear complaints such as : “That persons’ student got ranked ahead of me and I always thought you were better than that person?” or “The better ranked people for the position are all doing that topic instead of ‘ours’, so I guess your topic isn’t so important after all!”. duSautoy’s captures it all in this one key sentence :

They’ve asked that ‘What’s it all for?’ question, and think they’ve seen the Emperor without any clothes.

As if, failing to get a permanent position is the advisors fault, more than yours…

Just for once, try to be honest to yourself : count the number of hours a day your brain-power gets you over 120 IQ. Substract from this the number of hours a day lost surfing the web idly, trying to read unreadable hep-th papers, socializing, kissing asses, socializing, doing fun things with you fellow graduate students, socializing, working on a relation, chatting, texing, emailing insults but softening it all with a closing smily 🙂 , socializing, etc… (you know the daily-drill of a 20-30-something phd-student a lot better than I do)

I’ll be damned if you get a positive outcome. But if you do, I’ll be happy to take you on as a PhD student…

Well, it’s no threat, it’s a promise : the first ex-student who gets me into a ‘why was it all good for?’ discussion will experience first floor defenestration! (provided I’ll get my window open in time)

And, to soften it all, I’ll add the obligatory 🙂

4 Comments

Surreal numbers & chess

Published April 8, 2008 by lievenlb

Most chess programs are able to give a numerical evaluation of a position. For example, the position below is considered to be worth +8.7 with white to move, and, -0.7 with black to move (by a certain program). But, if one applies combinatorial game theory as in John Conway’s ONAG and the Berlekamp-Conway-Guy masterpiece Winning Ways for your Mathematical Plays it will turn out that the position can be proved to have an infinitesimal advantage for white…

So, what do we mean by this? First some basic rules of combinatorial game theory. To start, we evaluate a position without knowing which player has the move. A zero-game is by definition a position in which neither player has a good move, that is, any move by either player quickly leads to losing the game. Hence, a zero-game is a position in which the second player to move wins.

What is the chess-equivalent of a zero-position game? A position in which neither player has a good move is called a Mutual Zugzwang in chess literature. An example is given by the above position, if we restrict attention only to the 4 pieces in the upper right-hand corner and forget the rest. We don’t know who has the move, but, White cannot move at all and Black cannot move the King or Bishop without losing the Bishop and allowing White to promote the pawn and win quickly. In CGT-parlance, the upper-right position has value $\{ \emptyset | \emptyset \} = 0 $ where the left options denote the White moves and the right options the Black moves.

All other values are determined by recursion. For example, consider a position in which White has just one move left before the sitution is again a Mutual Zugzwang, and, Black has no good move whatsoever. After white’s move, the position will again be a zero-position and Black has no options, so the value of this position would be denoted by $\{ 0 | \emptyset \} $ and we call the value of this position to be $+1 $. Similarly, if white has no options and black has one final move to make, the position would be considered to have value $\{ \emptyset | 0 \}= -1 $.

Clearly, these are just the three easiest game-values to have and the real kick comes further down the road when one can prove by recursion that some games have non-integer values (such as $\{ 0 | 1 \} = \frac{1}{2} $ for a position in which white has one move to get to a mutual zugzwang and black has a move leading to a position of value $+1 $ (defined as before)), or non-number values such as $\ast = \{ 0 | 0 \} $ where both white and black’s best move is to get to a mutal zugzwang. Game-values such as $\ast $ are called fuzzy (or confused with zero) and are defined by the property that the first player to move wins.

Similarly, positive game-values are those positions where White wins, independent of who has the move and negatives are those that Black wins. There is a whole menagery of game-values and the WinningWays-booklets give an example based introduction to this fascinating theory.

Brief as this introduction was, it will allow us to determine the exact value of the position in the above diagram. We know already that we can forget about the right-hand upper corner (as this is a zero-position) and concentrate attention to the left-hand side of the board.

It is easy to see that neither Knight can move without loosing quickly, nor can the pawns on a5 and b7. That is, white has just 2 options : either c3-c4 (quickly loosing after d5xc4 2. d3xc4,d4-d3 3. Nc1xd3,Na1-b3) or, and this is the only valid option c3xd4 leading to the position on the left below. Black has only one valid move : d4xc3 leading to the position on the right below.

Clearly, the left-diagram has value 0 as it is a mutual Zugzwang. The position on the right takes a moment’s thought : White has one move left d3-d4 leading to a 0-position, whereas black has one move d5-d4 leading to a position of value -1 (as black still has one move left d6-d5, whereas white has none). That is, the CGT-value of the right-hand position is $\{ 0 | -1 \} $ and therefore, the value of the starting position is precisely equal to

\[ \{ 0 | \{ 0 | -1 \} \} = +_{1} \]

(called tiny-one among ONAGers)

It can be shown that $+_1 $ has a positive value (that is, White wins independently of who has the first move) but smaller than any positive number-valued games!

Noam Elkies has written a beautiful paper On numbers and endgames: Combinatorial game theory in chess endgames containing many interesting examples (the example above is an adaptation of his diagram9).

2 Comments

New world record obscurification

Published April 7, 2008 by lievenlb

I’ve always thought of Alain Connes as the unchallengeable world-champion opaque mathematical writing, but then again, I was proven wrong.

Alain’s writings are crystal clear compared to the monstrosity the AMS released to the world : In search of the Riemann zeros – Strings, fractal membranes and noncommutative spacetimes by Michel L. Lapidus.

Here’s a generic half-page from a total of 558 pages (or rather 314, as the remainder consists of appendices, bibliography and indices…). I couldn’t find a single precise, well-defined and proven statement in the entire book.

4.2. Fractal Membranes and the Second Quantization of Fractal Strings
“The first quantization is a mystery while the second quantization is a functor” Edward Nelson (quoted in [Con6,p.515])

We briefly discuss here joint work in preparation with Ryszard Nest [LapNe1]. This work was referred to several times in Chapter 3, and, as we pointed out there, it provides mathematically rigorous construction of fractal membranes (as well as of self-similar membranes), in the spirit of noncommutative geometry and quantum field theory (as well as of string theory). It also enables us to show that the expected properties of fractal (or self-similar) membranes, derived in our semi-heuristic model presented in Sections 3.2 and 3.2. are actually satisfied by the rigorous model in [LapNe1]. In particular, there is a surprisingly good agreement between the author’s original intuition on fractal (or self-similar) membrane, conceived as an (adelic) Riemann surface with infinite genus or as an (adelic) infinite dimensional torus, and properties of the noncommutative geometric model in [LapNe1]. In future joint work, we hope to go beyond [LapNe1] and to give even more (noncommutative) geometric content to this analogy, possibly along the lines suggested in the next section (4.3).
We will merely outline some aspects of the construction, without supplying any technical details, instead referring the interested reader to the forthcoming paper [LapNe1] for a complete exposition of the construction and precise statements of results.

Can the AMS please explain to the interested person buying this book why (s)he will have to await a (possible) forthcoming paper to (hopefully) make some sense of this apparent nonsense?

3 Comments