How does Auburn present the relationship between Catherine and Hal?

When answering this question, it's important to understand that Auburn presents the relationship between Catherine and Hal as one that evolves throughout the play. After her father passes away, Catherine maintains only two other key relationships in the story, one with Hal and the other with her sister Claire. While her connection with Claire doesn't undergo any major transformations and remains relatively stagnant, circular, and flat, her relationship with Hal dynamically changes as the story progresses. Auburn doesn't present any resolutions to the problems that exist in Catherine and Claire's relationship, which is fraught with conflict. However, the conclusion of the play suggests that Hal and Catherine resolve the issues that erupt between them, leaving us with the sense that they've found harmony and solace in each other's presence. 

With this in mind, we can then focus on the underlying dynamics that form the basis of Hal and Catherine's relationship, and the problems and resolutions that define the evolution of their connection. Auburn presents Hal as having a romantic attraction to Catherine and also a genuine desire to help her in her grief, but he is also alarmed by some of her behavior, suspecting that she may be a victim to the same illness that caused her father to lose his mind. Catherine struggles with the knowledge of her own potential. She realizes she has the potential to be a hugely influential mathematician like her father, but she also worries she may have inherited his disease. In addition she feels barred from reaching her full potential as a mathematician because of her gender, and chooses the role of a domestic caregiver to her father over pursuing a degree at a prestigious university.

In turn, the key problems that Catherine and Hal encounter in their relationship revolve around trust. Catherine doesn't trust Hal's intentions, believing that he is self-interested and would like to publish the proof he finds in her father's study room as his own in order to gain fame as a mathematician. She recognizes that Hal fundamentally misunderstands her, and does not realize how gifted she is. Meanwhile Hal is torn between is attraction to Catherine and desire to help her, and his concern that she is not mentally sound. This leads them to mistrust each other, Catherine mistrusting Hal's intentions, and Hal mistrusting Catherine's sanity. 

In the end, Hal realizes his mistake in overlooking Catherine's intelligence, and likewise Catherine recognizes that Hal would never publish a proof he hadn't written to achieve success. The two develop trust in each other again, and Hal is able to genuinely help Catherine overcome the issues she is struggling with. He encourages her to work on her proof so it an be published and she can become the mathematician that she has the potential to be. The play concludes with the implication that the relationship between the two has evolved into a long-lasting bond that provides for the needs of each character leading to greater harmony in their personal lives.