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Mirror Symmetry

February 21, 2008 at 1:22 am | In Mathematics | Leave a Comment

So today, I accidentally saw two talks on mirror symmetry and Calabi-Yau manifolds. I didn’t understand much but I have to write reports on some of these things anyway so some quick notes may as well go on here.

The first talk (link to 1.7mb talk) (in the grad student colloquium):
After a large number of slides about vector bundles we were able to define Chern classes in two (or was it three?) ways. The first was axiomatic (satisfied reasonable sounding things and played nicely with the Whitney sum) and the second was by this formula (thanks Wikipedia):

$\det \left(\frac {it\Omega}{2\pi} +I\right) = \sum_k c_k(V) t^k$

Calabi-Yau manifolds are manifolds with Kahler complex structure (these were handwaved under the carpet) whose first Chern class is zero. There was a really convincing reason why these are useful in string theory – some equation needed to be true for surfaces of all genus. For it not to depend on genus needed the first Chern class to be zero and the complex dimension of the manifold to be 3 (real dimension 6). So that’s where the 6 extra dimensions of (some kind of) string theory come from?

An example of a Calabi-Yau manifold is a hypersurface in $\mathbb{CP}^4$ given by an equation like $x_0^2+\ldots+x_4^2+\lambda(x_0 \ldots x_4) = 0$ .

The second talk (by Yi Hu; in the series of talks introducing department research to the first year grad students):
Physics happens in some spacetime, a Calabi Yau manifold $X$ . Then strings ( $S^1$ ) move through spacetime tracing out a (complex) curve i.e. a Riemann surface. What kind of things can there be in this surface? Surfaces of various genus, g, that are immersed in the spacetime by a map of degree d.

So let $N_{g,d}$ be the number of these surfaces. Then you go and talk to some theoretical physicists. With their physics voodoo they can use “mirror symmetry” to relate these numbers, the “A model”, on $X$ to facts about different complex Kahler structures, the “B model”, on a different space $X'$ . This symmetry isn’t rigorous but they can use it to come up with equations like (from memory, this is for the g=0 case)

$\frac{5}{6} T^3 + \sum_{d=1}^{\infty} N_{0,d} e^{dT} = (J_1(t)J_2(t) - J_3(t))$

where the $J_i$ ’s depended on some fairly complicated polynomial $I_i$ ’s and t was related to T somehow.

Mathematicians can compute both sides and see that they are equal in low genus cases (for g=0 in 1997 and g=1 in 2007) but they don’t really have a good reason why the equations should be true.

Then we talked about Riemann surfaces and drew some pictures. Next week there is more algebraic geometry which should be interesting.

IPL lineups

February 20, 2008 at 5:52 pm | In Cricket | 3 Comments

Possible IPL batting orders (once they get rid of that pesky test match cricket) and some comments. A line (———) indicates they’ll have to fill in with local players or buy someone not in the auction.

Teams from wikipedia.

Edit: Apparently they are only allowed to field 4 international players at a time. So some of these teams won’t ever hit the park…

Bangalore

Wasim Jaffer
Shivnarine Chanderpaul
Rahul Dravid
Jacques Kallis
Cameron White
———–
Mark Boucher (wk)
Anil Kumble
Zaheer Khan
Dale Steyn
Nathan Bracken

Snoozefest. Dravid and Kallis: we are playing Twenty20 here. At least Shiv can go nuts and White is always good for some runs.

Chennai

Matt Hayden
Stephen Fleming
Michael Hussey
Jacob Oram
MS Dhoni (wk)
Suresh Raina
Albie Morkel
Joginder Sharma
Makhaya Ntini
Muttiah Muralitharan
—————–

Strong top order although if Hayden and Hussey are playing for Australia they lose something. There are two categories of people I know nothing about: Indian and South African Twenty20 specialists. But I’m sure Albie Morkel and Joginder Sharma will be handy….

Delhi

Virender Sehwag
AB de Villiers
Tillakaratne Dilshan
Gautam Gambhir
Manoj Tiwary
Shoaib Malik
Dinesh Karthik (wk)
Daniel Vettori
Farveez Maharoof
Mohammad Asif
Glenn McGrath

Meh. Old man McGrath was cheap.

Hyderabad

Adam Gilchrist (wk)
Herschelle Gibbs
Shahid Afridi
VVS Laxman
Andrew Symonds
Rohit Sharma
Chamara Silva
Scott Styris
Chaminda Vaas
Nuwan Zoysa
RP Singh

This is what this competition is about! Gilly, Gibbs, Afridi and Symonds in the one team.

Jaipur

Graeme Smith
Justin Langer
Younis Khan
Mohammad Kaif
Yusuf Pathan
————–
Kamran Akmal (wk)
Shane Warne
Munaf Patel
————–
————-

I don’t know how Warnie’s going to feel about this crappy team.

Kolkata

Sourav Ganguly
Chris Gayle
Ricky Ponting
David Hussey
—————
Brendan McCullum (wk)
Ajit Agarkar
Umar Gul
Murali Kartik
Shoaib Akhtar
Ishant Sharma

Nice team. Classy batsmen and subcontinental bowlers.

Mohali

Simon Katich
————–
Mahela Jayawardene
Kumar Sangakkara (wk)
Ramnaresh Sarwan
Yuvraj Singh
Ramesh Powar
Piyush Chawla
Irfan Pathan
Brett Lee
Sreesanth

They forgot to buy opening batsmen?

Mumbai

Sanath Jayasuriya
Sachin Tendulkar
Loots Bosman
Robin Uthappa
—————
—————- (wk)
—————-
Shaun Pollock
Harbhajan Singh
Lasith Malinga
Dilhara Fernando

They spent 5 million on only 8 players?

Stuff white people like / I need a project

February 20, 2008 at 4:49 am | In Internet, Life in America, Mathematics | 6 Comments

It’s

all

true

Also, I need to think of a topic to write a paper on for my topology/geometry class. Anything from differential geometry or algebraic topology that will give me something to talk about for 5 pages. Any ideas?

Etymology of Algebraic Structures

February 16, 2008 at 5:39 pm | In Mathematics | 8 Comments

Standard names for algebraic structures come from German/French words for objects from number theory or Galois theory. The reasons for the names in the modern English are opaque to say the least.

Magma – the thing with least structure. From Bourbaki.

Ring – Short for Zahlring (German for number ring). Think of Z[2^(1/3)]. Here the generating element loops around like a ring.

Group – Galois used the phrase ‘groupe de l’équation’ for subgroups of permutations of roots. Why did he use the word group?

Field – Dedekind used Zahlenkorper (body/corpus of numbers). This is where Korper (German) and Corps (French) come from. E.H. Moore ‘translated’ Korper to ‘Field’?

(Integral/UF/PI) Domain – The domain of numbers that you are working in.

Module – Dedekind came up with this one as well, spelling it Modul and using it for a subgroup of the additive group of a ring. Originally from Latin modulus meaning ’small measure’.

Monoid – From the Greek, mon(o) + oid. Fulfills only one of the axioms for a group (associative but no identity or inverses).

So we’re still left with Group, Field and Module invented by a Frenchman, an American and a German with apparently no reason for the names. Suggestions/references for why these names were picked would be appreciated.

References:

The history of ring theory

Earliest known uses of some of the words of mathematics

The Words of Mathematics

A History of Abstract Algebra (this one your uni probably needs to have a springerlink subscription)

Law 32

February 15, 2008 at 10:19 am | In Cricket | 9 Comments

Part of Law 32:

3. A fair catch
A catch shall be considered to have been fairly made if
(a) throughout the act of making the catch
(i) any fielder in contact with the ball is within the field of play. See 4 below.
(ii) the ball is at no time in contact with any object grounded beyond the boundary.

The act of making the catch shall start from the time when a fielder first handles the ball and shall end when a fielder obtains complete control both over the ball and over his own movement.

(b) the ball is hugged to the body of the catcher or accidentally lodges in his clothing or, in the case of the wicket-keeper, in his pads. However, it is not a fair catch if the ball lodges in a protective helmet worn by a fielder. See Law 23 (Dead ball).

(c) the ball does not touch the ground, even though the hand holding it does so in effecting the catch.

I didn’t know that the law was this stringent – if you dive for the ball, take a clean catch then land on the ground with the ball touching the ground it’s not out? Even though you clearly caught the ball and it never bobbled.

Now I see what the controversy about Ponting’s catch was…

The fine art of cricket claymation

February 11, 2008 at 1:54 pm | In Cricket, Internet | 2 Comments

Thanks to Are you a left arm chinaman we can now see a liquorice allsort version of Jesse Ryder.

Errata

February 8, 2008 at 1:44 pm | In Mathematics, sport | Leave a Comment

Martin would like to release the following errata for conversations he has recently had:

Australia did have conscription during Vietnam. (I for some reason thought we didn’t).
cf is Latin and means confer. (I thought it was French).
The morphisms in the category of metric spaces are short maps. (I said I thought they were probably just continuous maps).
The AFL has average attendances of about 38000 and this places it second in the list of attendance figures for domestic professional sports leagues. (I said the attendance was higher but that it ranked third).

Thank you for reading and I shall try to be more accurate in the future.

The Victorian Solution

February 5, 2008 at 2:21 pm | In Cricket | 11 Comments

So what are the problems with the current Australian lineup?

- There isn’t a good spin bowler in the country.
- It isn’t worth playing 4 quicks because Ponting will just hurry up the fourth one and cause him to retire.
- We lack quality slips fielders.

The solution to all these problems: Cameron White! A part time legspinner who can field at first slip and bolster the batting.

The team:

Hayden
Jaques
Ponting
Hussey
Clarke
Symonds
White
Haddin
Lee
Johnson
Clark

It really hurts me to advocate this but if MacGill is injured and they don’t want to consider McGain then this isn’t a bad lineup. You may as well go with your batting strength and rely on having three different styles of part time spin.

Acknowledgements: Thanks to Banh for the idea.

Sabermetrics in Baseball

February 5, 2008 at 4:59 am | In Baseball | 2 Comments

For those who don’t know, there has been a revolution in baseball statistics in the past 25 years (we discuss this with plans of writing about whether a similar idea is necessary in cricket). Baseball statistics were first formulated by Henry Chadwick who brought a cricket background and a kind of Victorian moralism to the numbers he decided to emphasise.

Traditional baseball statistics include:
- batting average: ratio of hits (getting on to some base by hitting the ball) to at bats (times you are at the plate but not including walks or hit by pitch (or sacrifice bunt/flys)).
- Runs Batted In (RBI): for example if a player hits a home run with 2 runners on base then he is credited with 3 RBIs.
- Runs : the number of times a batter gets over home plate.
- Earned Run Average (ERA): number of earned runs (runs not caused by fielding errors or an earlier pitcher) given up by a pitcher per nine innings pitched.
- Fielding errors
- Wins: A win is credited to the pitcher who was pitching when his team took the lead for the final time.

It should be fairly clear that these numbers don’t tell the whole story: there is no measurement of how often a batsman gets to base (walks are important and disciplined batters draw a lot more of them). They also make no distinction between getting to first base and hitting home runs (except runs and RBIs which are very much dependent on your teammates). Fielding errors are about not making mistakes rather than whether you get to the ball.

Bill James and his disciples in the The Society for American Baseball Research (SABR – where the name comes from) developed new statistics, the main concept of which is to better predict run scoring and wins created:

- Defence Independent Pitching Statistics (DIPS)
- On Base Percentage
- On Base plus Slugging (OPS)
- Value over replacement player (VORP)
- Walks plus hits per innings pitched (WHIP)

Some of these are fairly arbitrary numbers: OPS is an oft-used statistic that has no reasonable interpretation or units. It is the percentage of times you get on base + the number of bases that you get per at bat. This seems to be a reasonable measure of batting ability but really should be modified because these two numbers are generally of slightly different magnitudes and of differing importance.

But these numbers are much more effective at predicting future performance. For example DIPS takes out a lot of the reliance on other players and also some of the luck from pitchers performance.

Another idea from sabermetrics is putting a run value on each play based on the vast number of times the situation has played out in the past (162 games per season for 31 teams). A failed attempt at stealing a base when you are on first and there are no outs is worth -0.58, a double from the leadoff hitter is worth 0.74 and so on… Fielding statistics are less well understood but there is at least a pretty good idea of how much a hitter is worth.

The well known book ‘Moneyball’ describes the way the Oakland Athletics used the tools of Sabermetrics to exploit inefficiencies in the baseball player market to remain competitive even with a much smaller payroll than other teams.

Some concepts from Moneyball:
- Drafting: Other teams are basing their picks on scouts who are looking for players with potential or who look good in a uniform. Major League baseball teams aren’t very good at teaching people how to play so you’re better off picking college batters who already have discipline. Their numbers have high predictive value whereas high school pitchers for example are very hard to predict.
- Trading: Closers who can throw hard are over rated. You can get some fireball thrower from the minor leagues and turn him into a successful closer then sell him off for large amounts of money to the Yankees.
- Getting on base is the most important thing about batting.
- Outs are the most important resource when you are batting: stealing bases/sacrifice flys/sacrifice bunts and other ’small ball’ strategies are a bad idea.
- ‘Trick’ pitchers (submarine etc.) are undervalued.

The main idea, the definition of sabermetrics, is the search for objective knowledge about baseball. In the period that Moneyball was written this objective knowledge of the baseball market suggested that batters who were capable of getting on base and hitting home runs were undervalued and base running speed/fielding ability was overrated. The manager of the A’s, Billy Beane, used this to put together a ragtag team of outcasts and ugly players who were competitive with teams with much higher payrolls.

So that’s a whirlwind rundown of some of the ideas of sabermetrics. There is still much debate between the old scouts and the new technocrats about how baseball should be run both off and on the field. But we’re more interested in whether some of these ideas apply to cricket also.

To see some ideas of sabermetrics applied to all kinds of sports (especially in the baseball off season) have a look at Sabermetric Research. One of the more amusing formulae is this one for wine quality prediction:

Wine Quality = 12.145 + .00117 * winter rainfall + .0614 average growing season – .00386 harvest rainfall

Back to cricket

February 4, 2008 at 4:55 am | In Cricket | 1 Comment

So after my last post I feel that I should reaffirm my allegiance to cricket as the one true sport. (Apologies to any readers of this blog that don’t care about sport whatever country it is from).

To this end, an idea inspired by Dave alerting me to the existence of the outstandingly mediocre Roger Kynaston (averaged 9.66 in 166 first class matches, never bowled and didn’t keep wicket). He played some quixotic matches including ‘MCC v The Bs’, ‘A to K’ v ‘L to Z’ and ‘Married v Single’.

But the one that I would like to bring back would be ‘Fast Bowlers v Slow Bowlers’ or as it was also known ‘Eight Gentlemen and Players with Three Slow Bowlers v Eight Gentlemen and Players with Three Fast Bowlers’.

So here are my teams picked from players currently available for test match cricket. The qualification is 5 test or 20 first class wickets and me knowing what they bowl. One FC wicket in the appropriate style for wicketkeepers.

Fast Bowlers

Michael Hussey
Collingwood
Jayawardene
Ponting
Kallis
Flintoff
Boucher (wk)
Lee (c)
Asif
Zaheer Khan
Steyn

Slow Bowlers

Gayle
Sehwag
Vaughan
Tendulkar
Pietersen
Symonds
Sangakkara (wk)
Vettori
Kumble (c)
Macgill
Murali

I would seriously like to see this game played (on a a fairly dry SCG, say). With Kumble and Symonds to open the bowling the slow bowlers could take the shine off the ball and then bring on some magic.

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