The Minkowski Connexion

When Albert Einstein was going to school, his math teacher called him "a slow dog". That teacher was named Hermann Minkowski.

Minkowski was an expert at number theory :
"Minkowski explored the arithmetic of quadratic forms, especially
concerning n variables, and his research into that topic led him to
consider certain geometric properties in a space of n dimensions. In
1896, he presented his geometry of numbers, a geometrical method that
solved problems in number theory."

The geometrical method can be used to create weak crypto. Imagine the visual possibilities from geometry. Later, examples of “isospectral” lattices of dimensions 12 and 8 will be given by Hellmuth Kneser and Yoshiyuki Kitaoka, respectively. The latter considered theta series and used the theory of modular forms. In 1988 the dimension of such examples was pushed down to 4 by Alexander Schiemann and independently by Ken-Ichi Shiota. Schiemann made exhaustive computer tabulations to approach systematically the smallest determinant 1729 for which such an example exists!

Since some results in adelic representation theory on theta liftings imply the representability of modular forms as linear combinations of theta series of not fixed type, it is then desirable to understand whether it is possible to turn linear combinations of “generic” theta series into combinations of the convenient (i.e. relevant to the same congruence subgroup) theta series.

The great Riemann said in 1854, "It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it."

Minkowski and Einstein lit a candle in that darkness, but they still needed to invoke Newton to pretend to fill in the remaining unknowables, which still remain void in 2011.