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The conflict in the story centres around the husband misunderstanding that a 'rest cure' is ultimately destructive for an intelligent woman such as the narrator. She can see the destructive effects from their beginning, but he dismisses her warnings to his peril - and hers.
The resolution to the conflict is noted at the end of the story. He, after breaking down the door, lies protstrate - presumably in a faint after seeing the extent to which his wife has deteriorated mentally. She continues to 'creep' around the room, stepping over him in the process. Her madness has defeated him, and his dismissal of her mental state has reduced him to apoplexy and her to insanity.
Besides the patriarchal (male) dominance of women in the 19th century, there are other representations of the conflict. Consider themes like the room as a symbol of her marriage as a prison. The woman in the wallpaper is a manifestation of her mental illness. There is also the dichotomy of the rational (her husband) and the creative (the narrator). The ending of the story has the narrator ripping off the wallpaper and creeping around the house. The pressure of all the conflicts has made her insane.