Write with genuine curiosity—mention one specific paper and a small doubt you had, not just praise. They look for humility and hunger, not perfect grades. It is too late only after their PhD students are full; email even in December if you must.
My cousin from a tier-3 college did this, attaching a simple analysis of a graph from the professor's own work. He got a reply when others with better marks did not.
#work#career
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