In the first act, there are two important conflicts. One of the main conflicts are that Cassius got Brutus to join his side for his own gain. Brutus is struggling with himself because he can’t persuade himself that he acted for the good of rome. The second main conflict is that there's currently no future leadership of Rome. Cassius wants all of the power, so he is trying to convince Brutus not to let Mark Antony speak at the funeral. He doesn’t want Antony speaking at the funeral because he would end up being a better leader. Cassius is power hungry.
