Proof

Proof is based on David Auburn's play of the same name, which, since it debuted in 2000, has gone on to be the longest running Broadway production since Peter Schaeffer's Amadeus. Reuniting director John Madden with actress Gwyneth Paltrow, after their previous collaboration on Shakespeare In Love, the film offers a convincing demonstration of the theorem that good writing and sensitive performances can equal an engaging and nuanced piece of cinema.

It is her 27th birthday and Catherine (Paltrow) is a mess. Five years ago she interrupted promising if undisciplined work towards a university degree in mathematics and came home to look after her father Robert (Anthony Hopkins). A one-time mathematical genius, Robert slowly began to lose his mind when he was about Catherine's age and she now worries that his madness may be hereditary, an anxiety that is not helped by the arrival of Robert's older, interfering daughter Claire (Hope Davis), who seems even more certain of her sister's instability than Catherine herself.

"The simple fact that we can talk about this together is a good sign", Robert reassures Catherine, before offering an elegant proof of her sanity; but Catherine remains skeptical, not least because her father died about a week ago and the present conversation is all in her head.

Upstairs, young Maths geek Hal (Jake Gyllenhaal) pores over the 103 scrawled notebooks that his former teacher left behind, hoping to discover a scrap of lucidity that will write his ticket to any Maths department in the country. Yet, neither Hal nor Claire is quite prepared for the surprise that Catherine has in store for them - a surprise that is beyond all proof and will test the love and loyalty of both.

As she has already shown in Sylvia, The Royal Tenenbaums and even Sliding Doors, Paltrow has an uncanny talent for playing women who are coming apart at the seams and her Catherine veers from lovably eccentric to more disturbingly unhinged and back again with fluent ease. The scenes, which she and Hopkins share, as two difficult people bound together by affection, dependency and mutual respect, are entirely believable and all the more touching for it.

With major roles already this year in Jarhead and Brokeback Mountain, Gyllenhaal is proving to be the actor of the moment, and he plays Hal with a light touch, providing the relief of some much needed normalcy (at least of a nerdy kind) in a film whose other characters can only dream of being so grounded.

Proof unfolds in a convoluted series of flashbacks (if that is indeed what they are) that serve not so much to clarify as further to confound our understanding of Catherine's uncertain state. Is she merely paralysed with grief, or truly mad? Is it her father's, or her own genius that has been uncovered? Is she remembering, or inventing, the time that she spent alone with her father in the house?

By the end these questions can be answered only if the viewer, like Hal and even Catherine herself, is willing to make a huge leap of faith into the dark. And so Proof is cleverly structured to show that life cannot always be reduced to the neat mathematical formulae that Robert seeks so obsessively to elaborate.