Lecture notes on the Carlsson-Mellit proof of the shuffle conjecture

This note is based on the original proof of the shuffle conjecture by Carlsson and Mellit (arXiv:1508.06239, version 2), which seems to be too concise for the combinatorial community. James Haglund spent a semester to check through the proof line by line and took explicit note. Guoce Xin was asked by Adriano Garsia to elaborate the proof. The whole paper is thoroughly checked and filled in with details. Section 1 and the abstract of the original paper is included. Section 2 is rewritten for clarity. Other sections are filled with detailed proof and computation. We hope this note is accessible to graduate students.