The rose in the concrete is basically how someone can fulfill greatness despite their environment or background.
Bernard ignores the conversation because he knows his fate would be sealed even more if he spoke up.
Explanation:
In the conversation between Mond, John and Helmholtz he knows that Mond will have the upper hand and anything that Bernard might say about what he was willing to do against the new society by reading the texts and organizing people who fail even further.
So he stays silent and only talks up to beg Mond to reduce his sentence for his crime of doing it but his friends had until them pretty much spilled the beans and had engaged in a discussion about the things they were doing.
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Keating is all about 'carpe diem', which means 'seize the day, go for it'. He encourages his students to live their lives to the fullest, the way they want to, and not to be restrained by anything. Having this in mind, Knox gathers the courage to go after the girlfriend of a football player, who was way out of his league.
