15. $u \geqslant\left|z+(z-i) \omega+(z-\sqrt{3}) \omega^{2}\right|=|\omega||i+\sqrt{3} \omega|=\sqrt{7}$ (where $\omega$ $\left.=-\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)$, equality holds if and only if $z,(z-i) \omega,(z-\sqrt{3}) \omega^{2}$ are collinear in the same direction, i.e., the vectors corresponding to the complex numbers $z, z-i, z-\sqrt{3}$ form angles of $120^{\circ}$ with each other, in which case $u$ has the minimum value $\sqrt{7}$. Note: This problem can also be solved using plane geometry. By the geometric interpretation of complex numbers, we need to find the minimum sum of distances from any point $Z$ on the complex plane to the three points $O(0), A(i), B(\sqrt{3})$. It is easy to see that the minimum value $\sqrt{7}$ is achieved when point $Z$ is the Fermat point of $\triangle O A B$.
15. For a complex number $z$, find the minimum value of $u=|z|+|z-i|+|z-\sqrt{3}|$.