"Proof": Darkness Visible

In mathematics, a proof is a closely reasoned and airtight theoretical argument that establishes the truth of a proposition beyond question. I'm pretty sure that's what it is. In any event, it's all you need to know, math-wise, to be caught up in this elegant film version of David Auburn's Pulitzer Prize-winning play about genius and madness and the fathomless complexity of real life, in which there are no proofs, only crumbs of inconclusive evidence. Sounds grim, I know; but the movie is gripping, and studded with moments of electrifying dramatic invention. It's also quite funny.

The pivotal character in "Proof" is a dead man, a mathematical genius named Robert (Anthony Hopkins). Robert's groundbreaking theoretical work at the University of Chicago, we are told, "revolutionized his field" before he turned 22. (His brilliance is a given that we gratefully accept without much in the way of actual demonstration, which would surely leave us baffled and grumpy.) Following his early accomplishments, Robert sank into a gathering darkness of schizophrenia. One of his two daughters, the troubled and possibly gifted Catherine (Gwyneth Paltrow), gave up her own academic ambitions to live with and care for her deteriorating father, rather than allow him to be institutionalized. Now, at the age of 27, she's convinced that her deadline for intellectual flowering has passed; and although she feels she may have inherited some part of her father's genius, she also fears that she may have inherited his madness, too. Since the movie opens shortly after Robert's death from a brain aneurysm, and yet we see Catherine engaged in quiet conversation with him about his mental breakdown and when it started, we understand her anxiety in this regard with startling suddenness.

To her initial irritation, Catherine has become entangled with a university brainiac named Hal (Jake Gyllenhaal), a bestubbled young Ph.D. who was once one of Robert's students. She is allowing him to go through her father's raft of private notebooks in search of some unpublished brilliance with which Hal -- who has ruefully concluded that he is no genius himself -- might burnish his scholarly reputation. He also has eyes for Catherine, who is clearly a stranger to romance. (They do tumble into bed at one point, and the next day Hal asks her, "How embarrassing would it be if I said last night was wonderful?" Says Catherine, "It's only embarrassing if I don't agree.")

Swanning into this situation comes Catherine's annoying older sister, Claire (Hope Davis), who escaped the family fevers years ago by moving to New York. Claire's weightiest personal concerns seem to be gourmet coffee and high-end hair conditioners, and Catherine has never liked her. Claire has returned to Chicago to sell the family house out from under Catherine, whom she regards as increasingly unstable, and take her sister back to New York, where she intends to put her in the care of a psychiatrist. As intended, perhaps a little too schematically, we don't like Claire, either.

Poring over Robert's notebooks, Hal eventually realizes that they're filled with useless psychotic ravings. But Catherine knows of another notebook, which is locked away in a drawer. She gives this to Robert, and after examining it, he realizes that it's the real thing: a breakthrough proof, so complex he himself can barely follow it. He asks Catherine where she found it. "I didn't find it," she says. "I wrote it."

Did she? Catherine never even finished college; and as Claire points out, the handwriting in this new notebook seems almost certainly to be her father's. Can its authorship be proved, beyond argument? It's an engrossing question, and it keeps us wondering -- and hoping -- right to the end of the picture.

Gwyneth Paltrow, with her hair pulled back in a ratty ponytail, her thoroughbred figure obscured in drab jeans and sweaters, and flickers of pain, terror and sardonic disregard playing across her face in seamless succession, inhabits the character of Catherine with formidable control and an exquisite attention to emotional detail. I can't recall when she's ever been better. (It helps that she knows the role inside-out: She starred in an acclaimed London production of the play, which, like the movie, was directed by John Madden, who also presided over Paltrow's Oscar-winning performance in the 1998 "Shakespeare in Love.") The once-again-entrancing Hope Davis ("American Splendor") manages to humanize her tightly-wrapped character in small ways not always provided by the script; and Anthony Hopkins brings a burly, irrational cheer to the part of a man whose desperate hope for a return to sanity is, to everyone but himself, clearly in vain. Jake Gyllenhaal seems a little too hunky for a math geek -- but he is, after all, Jake Gyllenhaal; what're you gonna do?

"Proof" was a masterpiece before it reached the screen. Given the blunt compromises so often imposed in the course of transforming a play into a movie, maybe the highest praise that can be offered in celebration of this film version is that it remains one.

"Just Like Heaven": She's Not There

"Just Like Heaven" is a cute romantic comedy that doesn't really aspire to be much else. The clever lines come zinging in on cue and the plot complications are almost comforting in their predictability. But while the picture adheres to romantic-comedy formula, it's not entirely formulaic (especially toward the end); and the actors, who appear to be having fun, give it a bright and undeniable appeal.

Reese Witherspoon, in familiar Reese Witherspoon mode (pert, but borderline-snippy), plays Elizabeth, a workaholic hospital doctor who has no time for a love life. ("I'm completely capable of meeting men on my own," she tells her ever-inquiring sister, Abby -- (Dina Waters). "I'd just like you to meet one who's not bleeding," Abby says.)

Driving down a highway one night, Elizabeth's car gets hit by a truck. The screen goes black. The next thing we see is a rumpled shlub named David (Mark Ruffalo) being shown around Elizabeth's suddenly-available apartment by a rental agent. The place comes furnished, and David really likes the couch; he's still a shambles after the death of his wife, and he plans to spend most of his time there, slobbing out with pizza and beer in front of the television. When Elizabeth suddenly appears and wants to know what David is doing in her apartment ... well, that's the first plot complication, and you know what's gonna happen, right?

David is of course the only one who can see Elizabeth. He tries to convey the strangeness of this situation to his psychiatrist, Jack (Donal Logue), and is naturally met with incomprehension. ("I'm seeing someone who's not really there," he whines. Says Jack: "You mean she's emotionally unavailable?") A clerk at an occult bookstore, Darryl (Jon Heder, of "Napoleon Dynamite"), does buy his yarn, though -- not only can he sense Elizabeth's presence, he can even feel her anger at David's appropriation of her home. ("There's a cancer-causing ray of spirit-hate aiming at you," Daryl warns.)

David and Elizabeth naturally warm up to each other -- if they were both alive (or, presumably, both dead), they might have hooked up. She decides to help him shed his sorrow, and he agrees to help her find out who she was (she's forgotten) and what happened to her. David begins feeling such fondness for Elizabeth that he loyally turns away the advances of an alarmingly nubile neighbor (Ivana Milicevic) and scurries back to her. ("I told her I was seeing someone," he says. "I didn't tell her I was the only one who could.")

"Just Like Heaven" is deftly made, and it comes to an agreeable conclusion. The many fans of Mark Ruffalo will want to see it for another display of his shuffling comic charm. (The scene in which Elizabeth takes possession of his body and flings it around a crowded barroom is a delirious slapstick eruption.) Most people, however, in a crowded fall movie season, may have to pass this one up. But it'll be on cable soon enough. And there it may shine brighter, like the small, unassuming jewel that it is.

--Kurt Loder

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