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In an undergraduate linear algebra course, one learns about inner
products on $\Bbb R^n$. Since the dot product is essentially the only
one, they are not very interesting. People actually study a broader
class of objects --- {\it symmetric bilinear forms} --- which include
what physicists call Minkowski space.

The desire to classify symmetric bilinear forms leads to investigating
their invariants. This talk will discuss symmetric bilinear forms as a
case study to motivate the general theory of cohomological invariants,
as presented in a forthcoming book by Serre, et al. A good portion of
this talk will be classical in nature, and so accessible to a more
general audience.

We present a new version of the path model for representations of a semisimple group G. In the new setting the LS-paths are replaced by certain galleries in the affine Coxeter complex. Using the corresponding affine Tits building, we associate in a canonical way to the galleries some finite dimensional projective varieties in the affine grassmannian associated to G. It turns out that these are precisely the intersection homology cycles investigated by Mirkovic and Vilonen.

A long time ago, astronomer Johann Bode noticed a remarkable regularity
in the interplanetary spacings in our solar system. It is not just the
regularity in the spacings, but the fact that a few planets previously
unknown were discovered by using the Bode law.

Is it a coincidence ? Is it a semi-physical law ? Or, could it just be
that the mathematical formula characterzing the interplanetary spacings
is embeddable in a parametric class of sequences that will give great fit
to a lot of empirical sequences of small length ? (The suggestion has been
made that it COULD be a sign from a creator too)

We will present the story of Bode\'s law, the story of discovery of the
planets, and the resultant mathematical questions, with some calculations,
and many examples.