We learned in class that the regular languages are closed under concatenation (e.g $L_1L_2 =\{ w_1w_2 : w_1 \in L_1,w_2 \in L_2\}$ is a regular language if $L_1$ and $L_2$ are also regular languages).
My question is: why is there a language : $L' = \{ww: w \in L\}$ which is not regular if $L$ is a regular language? Is it not just a special case where $L_1 = L_2$?
Good question. The problem is that $L'$ isn't the concatenation $L\circ L$, since it's made up to the concatenation of each word in $L$ with itself. It would be if we had defined it as $$ \{xy\mid x\in L, y\in L\} $$ but a string is in $L'$ if it's of the form $ww$: two copies of a word in $L$. That's clearly not a regular language in general, since you could take as an example $L=\{\mathtt{0}^n\mathtt{1}\mid n\ge 0\}$ and then $$ L'=\{\mathtt{0}^n\mathtt{1}\mathtt{0}^n\mathtt{1}\mid n\ge 0\} $$ which a simple Pumping Lemma proof would show that isn't regular. Intuitively, the language above would require a finite automaton that could count the number of zeros in a string, which we know is impossible.