##### Department of Mathematics,

University of California San Diego

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### Advancement to Candidacy

## Christopher S. Nelson

#### UCSD

## Noncommutative Partial Differential Equations

##### Abstract:

This talk classifies all harmonic noncommutative polynomials, as well as all polynomial solutions to other selected noncommutative partial differential equations. The directional derivative of a noncommutative polynomial in the direction $h$ is defined as $D[p(x_1,\ldots,x_g),x_i,h]:= \frac{d}{dt}[p(x_1, \ldots, (x_i+th), \ldots, x_g)]_{|_{t=0}}$. From this noncommutative derivative, one may define differential equations which take as solutions polynomials in free variables. A noncommutative harmonic polynomial is a polynomial such that its noncommutative Laplacian is zero.

Advisor: Bill Helton

### May 2, 2010

### 11:00 AM

### AP&M 5829

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