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A conventional plot gets a mathematical makeover in "Proof."

At the center of it all is Paltrow, playing the daughter of a stupendously brilliant University of Chicago mathematician (Hopkins), whose career was tragically derailed when he "went bughouse" in his 20s and who has died just before the movie's action begins. Paltrow's character, a brooding and defensively petulant math whiz herself, fears she has inherited her father's illness along with his gifts. She's a mess.

One of her father's students (Jake Gyllenhaal) seems to offer love as a way out of her funk. At the same time, her big sister (Hope Davis) swoops into town, hoping to drag Paltrow away to New York, where she can drown her sorrows in exquisite coffee, jojoba oil and other symbols of inauthentic urban chic. Everything goes haywire when a brilliant, unpublished proof of uncertain authorship is discovered among her father's papers. Did Paltrow write it? In the absence of hard evidence (the other kind of proof), even the smitten Gyllenhaal has his doubts. A series of flashbacks that pop up throughout the piece chart the years Paltrow spent caring for her ailing father in their rambling Chicago house and finally reveal who done it.

Tom Stoppard's terrific play "Arcadia" also put a young female mathematical genius at the center of the action, but nothing could be further from that play's fusion of emotion and intellect than the nicely realized soap opera offered in "Proof." For one thing, if you can tell even numbers from odd, you have more than enough math to follow the action. The script would hardly need to be changed if the questionable document were a patent for a solar-powered lawn mower or a codicil to a will instead of a mathematical proof.

At the core of "Proof" is a set of reliable dramatic staples. Men are threatened by brilliant women. There is a fine line between madness and brilliance. New York careerists are contemptible. Eggheads fall in love and party hard. All this is presented in well-scrubbed, relentlessly articulate dialogue that's more at home on the stage than on the screen. Although the action takes place against the backdrop of several handsome Chicago settings, it's hard to forget that this picture started out as a single-set drawing-room drama.

Paltrow injects a measure of pathos into the barbed complaints that take up the lion's share of her part, and her scenes with Hopkins, who thunders and mutters by turns, stand at the emotional center of the movie. The other actors make the best of their two-dimensional roles. Hope Davis, in particular, tries valiantly to endow her part with some humanity, but the script stacks the cards against her character so unfairly that hardly any real warmth emerges. Even when she suggests her sister see a psychiatrist — not a bad thought when you see a family member slipping into depression — we're meant to understand her as a monster of selfishness and an enemy of genius.

Take away all the academic trappings and what remains is a fairly interesting, scrupulously produced story of young love threatened by mistrust and of a family torn up by the decline and death of a parent. The whole math angle, which is clearly supposed to be what makes this story deep and important, ends up being just a distraction. A Western set in outer space is still a Western, and a melodrama is a melodrama is a melodrama, even if the people in it occasionally announce something like, "I figured out h**1/2 S



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