Why is the maximum likelihood estimator $y_{\operatorname{max}}$ instead of $y_{\operatorname{min}}$ Use the method of maximum likelihood to estimate the parameter $\theta$ in the uniform pdf,
$f_y(y ;\theta)=\frac{1}\theta$ where $0\leq y\leq\theta$
According to the solution manual $\theta_e=y_{\operatorname{max}}$, however the likelihood function $L(\theta)=(\frac{1}\theta)^n,$
For me, what would maximize $L(\theta)$ would be $\theta_e=y_{\operatorname{min}}$

Well if you think about it your answer does not make much sense!
Maybe better to see it as follows:
Let $\hat \theta$ be the MLE, then the likelihood of all points $y_i$ that would be greater than $\hat\theta$ would be zero according to that model, and the entire likelihood would be zero. Therefore, in order to maximize the likelihood, $\hat\theta = y_{\text{max}}$.
