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edited Jul 15 2010 at 8:29

Bo Peng
925●3●13

The number $12$ (or, probably we shall say Bernoulli numbers in general) appears in so many places in Mathematicsthat it always fascinates me whenever I see one, sometimes leading to unexpected connections between different topics.

For instance, some time ago there was a very interesting explanation for

1) its occurrence in the Todd class

and

2) its occurrence in the Euler-Maclaurin formula

in terms of Riemann-Roch for toric varieties, as explained in:

http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch

===========================================================

My questions question is, will there be some relation between 1) and

3) its occurrence in the Baker-Campbell-Hausdorff formula.

I guess this might be related to some explicit local expressions in some method of proving the index theorem on Lie groups, or even the Duflo map (which I don't really understand).

Thank you very much.

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edited Jul 15 2010 at 8:20

Bo Peng
925●3●13

The number $12$ (or, probably we shall say Bernoulli numbers in general) appears in so many places in Mathematics that it always fascinates me whenever I see one.

For instance, some time ago there was a very interesting explanation for

1) its occurrence in the Todd class

and

2) its occurrence in the Euler-Maclaurin formula

in terms of Riemann-Roch for toric varieties, as explained in:

http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch

===========================================================

My questions is, will there be some relation between 1) and

3) its occurrence in the Baker-Campbell-Hausdorff formula.

I guess this might be related to some explicit local expressions in some method of proving the index theorem on Lie groups, or even the Duflo map (which I don't really understand).

Thank you very much.

===========================================================

EDITED: why I think this is probably not numerology... Consider

http://upload.wikimedia.org/math/9/e/1/9e139d2231ccc0080d4835284ca11aa2.png

and

$td(E) = 1 + c_{1}/2 + (c_{1}^{2}+c_{2})/12 + c_{1} c_{2}/24 + (−c_{1}^{4} + 4c_1^2c_2 + c_1c_3 + 3c_2^2 − c_4)/720 + ...$

We have 1, 2, 12 and 24, and even the number $720$ matches as well :-)

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edited Jul 15 2010 at 7:37

Bo Peng
925●3●13

The number $12$ (or, probably we shall say Bernoulli numbers in general) appears in so many places in Mathematics that it always fascinates me whenever I see one.

For instance, some time ago there was a very interesting explanation for

1) its occurrence in the Todd class

and

2) its occurrence in the Euler-Maclaurin formula

in terms of Riemann-Roch for toric varieties, as explained in:

http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch

===========================================================

My questions is, will there be some relation between 1) and

3) its occurrence in the Baker-Campbell-Hausdorff formula.

I guess this might be related to some explicit local expressions in some method of proving the index theorem on Lie groups, or even the Duflo map (which I don't really understand).