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# Flat manifolds isospectral on $p$-forms

### Juan Pablo Rossetti

Universidad Nacional de Cordoba

###
Thursday, September 28, 2000

102 Bradley Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: ** The *spectrum* of a differential manifold
$M$ is the collection of the eigenvalues ---counted with
multiplicities--- of the Laplacian $\Delta$ acting on smooth functions
on $M$. Two Riemannian manifolds are called *isospectral* when
their spectra coincide. Since $\Delta$ acts naturally on smooth
$p$-forms on $M$, one can similarly define $p$-spectrum and
$p$-isospectrality. The first example of manifolds isospectral but
not $1$-isospectral was given by C.\ Gordon in 1986. Other examples
have been given by A.\ Ikeda, R.\ Gornet and D.\ Schueth.

In this
talk we will describe the Bieberbach groups and the compact flat
Riemannian manifolds (c.f.m.) which provide unexpected tools for
obtaining simple examples of $p$-isospectral but not isospectral
manifolds. We will see an explicit formula for the multiplicities of
the eigenvalues of $\Delta$ acting on $p$-forms on a c.f.m. When one
considers what we call a c.f.m. of *diagonal type* this formula is
entirely combinatorial and it involves the integral values of the *
Krawtchouk polynomials*. As in Coding theory, the integral zeros of
these polynomials will play an important role here.

This talk will be accessible to graduate students.