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Box splines, vector partition functions and Riemann-Roch theorem


Speaker: Michele Vergne, Institut de Mathématiques de Jussieu
Location: 2112 MSB
Start time: Tue, Jan 19 2010, 3:10PM

Let $\R^r$ be the real vector space of dimension $r$. Consider a sequence $\Delta^+$ of vectors $[a_1,a_1,...,a_n]$ in $\Z_+^r$. For $\lambda$ in $\Z_+^r$, the value of the vector partition function $K(\lambda)$ is the number of ways an element $\lambda$ can be expressed as an integral combination of the elements $a_i$. For $\lambda$ in $\R^r$, the value of the Box spline $B(\lambda)$ is the volume of the polytope $\{0 \le t_i \le 1; \sum_i t_i a_i = \lambda \}$, while the value of the spline function $T(\lambda)$ is the volume of the polytope $P := \{ 0 \le t_i \le \infty; \sum_i t_i a_i = \lambda \}$.

We will recall the Dahmen-Micchelli relations between $B,K,T$. We will show that this relation is an instance of the Riemann-Roch theorem for indices of elliptic operators.