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## Numerical cubature from geometry and coding theory

### Greg Kuperberg

U.C. Davis and Cornell University

###
Thursday, April 7, 2005

L02 Carson Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: ** The numerical cubature problem is the
generalization to higher dimensions of integration methods such as
Simpson's rule. Given a measure *\mu* on *R^n*,
a *t*-cubature formula
is a finite set *C* such that the integral of any polynomial *P* of
degree *t* with respect to *\mu* equals a weighted sum over values on
*C*. The main interest is in cubature formulas with few points, with
positive weights, and without points outside of the domain of
*\mu*. Gaussian quadrature satisfies all three conditions in one
dimension, but the problem is already open-ended in two dimensions and
increasingly non-trivial in higher dimensions.

I will discuss new
methods for the cubature problem coming from error-correcting codes,
symplectic moment maps, and lattice packings of discretized convex
bodies. The methods yield many new explicit, efficient, positive,
interior, cubature formulas for the most standard choices of *\mu*. In
one context, they also lead to an interesting local lower bound on the
number of points needed for cubature.

This talk will be accessible to graduate students.