For the short stories "Galatea" by Karen Brown from The Best Amaerican Short Stories, I ask that you take one of the identity markers, make a claim about it (a thesis statement), and apply this thread through two stories. You are not comparing and contrasting, but just running the thread through the story and select characters. Your claim can be one where you use ideology or a term to help you better understand how identity is being represented. Mix in some terms with identity to help you along.

The post seems to be a complete homework assignment that requires developing a thesis based on two stories. This answer addresses the identity component of "Galatea" and shows how the author develops several different components of the two main characters' identities as a thread running through it.
Karen Brown's short story is about love and the confusion it creates for an insecure young woman. The primary factors of identity that come into play are gender, sexuality, and class. William's desire for Margaret, the narrator, is indicative of the objectifying male gaze. The title derives from an allusion that he makes. In classical mythology, Galatea was the statue that Pygmalion brought to life.
Although Margaret warns the reader at the outset that William is some sort of criminal, "the Collegetown Creeper," the story develops in some ways as a traditional—albeit whirlwind—romance: they married after a very brief courtship. Margaret thinks William really needs her, but she seems so desperate to have him depend on her that she exhibits no common sense. She convinces herself that the foolish risks she takes are indicators of love.
Brown uses Sylvia Plath's poetry to indicate Margaret's emotional problems. Brown extends the idea of the male gaze into William's practice of voyeurism; although he enters Margaret's room without permission, she does not reject him. Furthermore, the author offers a distorted image of the traditional gender expectations of love and marriage; William is far from a traditional husband.
William initially is not interested in sex, although his voyeurism has a sexual component. The sexual fulfillment that Margaret expected through marriage does not materialize, but after William leaves, she becomes aggressively sexually active. In the end, William turns out to be a thief; he steals all of her earthly possessions as well as her innocence.
The class differences between them are pronounced and likely constitute part of his attraction to her. Margaret is in graduate school and has her own, tiny apartment, but William moves from one odd job to the other and lives with his sister. He is clearly educated, however, as he knows who Galatea is, and mentions that his father was formerly an attorney. He takes Margaret to a homeless encampment where his alcoholic father stays, and she imagines that he has gone to live there after cleaning out her apartment.