When John ends up in hot water in grad school for not going to class or having made any progress on his research, his professor shows him a ceremony among Princeton faculty as a kind of "kick in the pants" to his young mentee. In the ritual, the professors show up at one particular colleague's table and present him with their pens.

The point/symbolism? To recognize some massive achievement that the faculty member has just racked up:

HELINGER: Do you see what they're doing in there? It's the pens. Reserved for a member of the department that makes the achievement of a lifetime. Now, what do you see, John?

JOHN: Recognition.

[...]

HELINGER: Well, try seeing accomplishment.

Years later, when John is being considered for the Nobel Prize, some Princeton professors perform this ceremony with John. Even though his career got, er, slightly derailed by his illness, the presentation of the pens really drives home just how much he's managed to achieve nonetheless.