KTH/SU Mathematics
Colloquium
den 5 oktober
05
Gert Almkvist, Lund
University
Calabi-Yau differential
equations
ABSTRACT
When in 1978
Ap\'{e}ry proved the irrationality of $\zeta (3)$ there
appeared a
3rd order differential equation with polynomial coefficients.
25
years later Zudilin working on the irrationality of $\zeta (4)$
and some di-
and trilogarithms found similar differential equations
of order 4 and 5.
These equations were similar to the differential
equations satisfied by the
periods of Calabi-Yau manifolds. In
particular the ''Yukawa coupling''
(counting rational curves?) had
integer coefficients. This led to a whole
industry of finding
''Calabi-Yau differential equations'' using computers
and various
tricks like Hadamard products of lower order equations. As
byproducts
we found numerous identities for sums of binomial coefficients
and
harmonic numbers, $H_n=\sum_{j=1}^n\frac 1j.$ (Based on joint works
with
Zudilin, van Straten, van Enckevort and
Krattenthaler.)