15. A mathematics interest group at a school consists of 14 students, who form $n$ different project teams. Each project team has 6 students, each student participates in at least 2 project teams, and any two project teams have at most 2 students in common. Find the maximum value of $n$.
Let 14 students be denoted as $s_{1}, s_{2}, \cdots, s_{14}$, and the research group sets as $G_{1}, G_{2}, \cdots, G_{14}$. Then $\left|G_{i}\right|=6$, $i=1,2, \cdots, n$, and $\left|G_{i} \bigcap G_{j}\right| \leqslant 2,1 \leqslant i<j \leqslant n$.
Let $s_{k}(k=1,2, \cdots, 14)$ belong to $r_{k}$ sets among $G_{1}, G_{2}, \cdots, G_{14}$, then $r_{k} \geqslant 2$, and $r_{1}+r_{2}+\cdots+r_{14}=6 n$. Consider the number of triples $\left(s_{k}, G_{i}, G_{j}\right)$, where $s_{k} \in G_{i} \cap G_{j}$.
On one hand, fixing $G_{i}, G_{j}(1 \leqslant i<j \leqslant n)$, since at most 2 $s_{k}$ can belong to the triple $\left(s_{k}, G_{i}, G_{j}\right)$, thus $S \leqslant 2 C_{n}^{2}=n(n-1)$;
On the other hand, fixing $s_{k}$, since $s_{k}$ belongs to $r_{k}$ sets among $G_{1}, G_{2}, \cdots, G_{14}$, the number of triples $\left(s_{k}, G_{i}, G_{j}\right)$ is $C_{r_{k}}^{2}$. Therefore, $S=\sum_{k=1}^{14} C_{r_{k}}^{2}=\sum_{k=1}^{14} \frac{r_{k}\left(r_{k}-1\right)}{2}$
$$
\begin{array}{l}
=\frac{1}{2}\left(r_{1}^{2}+r_{2}^{2}+\cdots+r_{14}^{2}\right)-\frac{1}{2}\left(r_{1}+r_{2}+\cdots+r_{14}\right) \\
\geqslant \frac{1}{2} \cdot \frac{1}{14}\left(r_{1}+r_{2}+\cdots+r_{14}\right)^{2}-\frac{1}{2}\left(r_{1}+r_{2}+\cdots+r_{14}\right) \\
=\frac{1}{28} \cdot 36 n^{2}-\frac{1}{2} \cdot 6 n=\frac{9}{7} n^{2}-3 n, \text { thus } \frac{9}{7} n^{2}-3 n \leqslant n(n-1) \Rightarrow n \leqslant 7 .
\end{array}
$$

Furthermore, 14 students can be grouped into 7 research groups in the following way to meet the requirements:
$$
\begin{array}{l}
G_{1}=\left\{s_{1}, s_{5}, s_{7}, s_{8}, s_{12}, s_{14}\right\} ; G_{2}=\left\{s_{1}, s_{2}, s_{6}, s_{8}, s_{9}, s_{13}\right\} ; \\
G_{3}=\left\{s_{2}, s_{3}, s_{7}, s_{9}, s_{10}, s_{14}\right\} ; G_{4}=\left\{s_{1}, s_{3}, s_{4}, s_{8}, s_{10}, s_{11}\right\} ; \\
G_{5}=\left\{s_{2}, s_{4}, s_{5}, s_{9}, s_{11}, s_{12}\right\} ; G_{6}=\left\{s_{3}, s_{5}, s_{6}, s_{10}, s_{12}, s_{13}\right\} ; \\
G_{7}=\left\{s_{4}, s_{6}, s_{7}, s_{11}, s_{13}, s_{14}\right\} . \text { In conclusion, the maximum value of } n \text { is } 7 .
\end{array}
$$