Motion of zeros of polynomial solutions of the one-dimensional heat equation: A first-order Calogero-Moser system

Abstract: I study the motion of zeros of polynomial solutions $\phi(x, t)=\prod_{i=1}^n[x-x_{i}(t)]$ of the one-dimensional heat equation $\displaystyle\frac{\partial \phi}{\partial t}=\kappa\frac{\partial^2\phi}{\partial x^2}$; they satisfy the first-order Calogero–Moser system \[ \frac{{\rm d}x_i}{{\rm d}t}=\sum_{j\ne i}\frac{-2\kappa}{x_i-x_j}. \] I am interested in the behavior at complex time $t$ (usually with real initial conditions). My goals are to

(a) Determine the complex times t at which collisions can or cannot occur; and

(b) Control the location of $x_1(t),\ldots, x_n(t)$ in the complex plane. I have no nontrivial theorems, but many interesting conjectures.

complex variablesdynamical systems

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

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