Which woman controls the most funds in the world? Not Nicola Horlick, who runs SocGen Asset Management. She makes a lot of noise but looks after only £2 billion sterling (€3.03 billion). Not Carol Galley, co-head of Merrill Lynch Mercury, Britain's largest pension fund manager. She makes rather less noise and controls £153 billion.

The biggest by far is Pattie Dunn, who makes no noise at all but is in charge of £370 billion. She is chairman of Barclays Global Investors, the world's largest institutional fund manager, based in San Francisco.

But have you heard of her? I certainly hadn't until her office contacted me.

They sent me one of the shortest CVs I have ever seen. It said she had read journalism at the University of California at Berkeley, that she had been at BGI for 23 years, that she was on the board of Hewlett-Packard and on the executive committee of Barclays Bank.

The woman who comes swinging into the Coq d'Argent, the swanky new Conran restaurant in the City of London, looks as immaculate as her picture. She flashes the same white-toothed smile at the waiter.

"Is this place called the Chicken of Money?" she asks him. He gives her a cold stare. "D'argent means silver. It means the silver cock."

"Qu'est-ce que c'est `darne' ?" she persists, unabashed, glancing at the menu.

Why the low profile? I ask.

She seems taken aback and explains that in the US no one outside her industry finds her even slightly interesting.

"What I do doesn't merit news. The American business press tends to pay attention to people who create wealth. That is just as it should be. Fund managers do not create wealth. We trade it around.

"I would have to jump off a building naked to get any coverage in the San Francisco Chronicle. And it wouldn't be because I am a fund manager. It would be: Stupid Lady Jumps Off Building."

This is hard to understand in Britain, where Ms Horlick gets column inches for getting out of bed in the morning.

That very morning, Ms Horlick, after getting out of bed, was being quoted on the BBC's Today programme attacking her fellow fund managers. In particular she had laid into unimaginative tracker funds, which happen to be the bedrock of BGI's business.

I mention this to Ms Dunn, who listens politely. "I am very aware of who she is," she says, just about managing not to look dismissive.

"There is a lot of show biz in active fund management," she says, hitting back. "Investors pay a lot for a promise that can only be delivered by chance. It signifies the greed of people - they are time and again willing to suspend their rational judgment in the hope that someone has that magic touch."

By contrast, BGI, with its index funds and scientific approach to investment, is boring and proud of it. "We are like the designated driver at a big party. We are not the fun people. People are glad there is someone who is OK behind the wheel."

Ms Dunn herself appears to have all the traits that the softer female manager is supposed to have. She claims never to have felt competitive towards people inside the company. She talks about the importance of persuasion over coercion. She says - several times - what a great bunch her colleagues are.

"I would rather that people came to conclusions on their own than push them into certain courses of action. Managing in this industry is like herding cats. People will do amazing things for the right reasons and very little for the wrong reasons."

I ask what she was like as a journalist.

"I never succeeded in making any money at journalism at all. I used to joke with Martin Taylor (ex-chief executive of Barclays and former Financial Times journalist) that he wasn't trained as a journalist but could make a living of it, while I was trained as a journalist but couldn't get a job."

Instead, she answered an advertisement for a temporary secretary at BGI. It was just starting to sell index funds, and the young Ms Dunn saw that its marketing material was terrible.

"That's how I got started. Trying to understand it myself so that I could explain it to other people."

I say something about her having come a long way.

She laughs. "I agree, it's improbable. It helps to be in the right place at the right time. I have worked very hard. And I have been very lucky."

These days she works harder than ever. Travels the whole time, and says that the responsibility of the job rests heavily on her shoulders. If she looks well rested, she says, it is thanks to wonder travel drug, Melatonin.

Last year, she was appointed to the executive committee of Barclays - and most weeks she is beamed by satellite to be with the suits in London.

I try to get her to talk about the succession problems at Barclays, but nothing doing. Instead she offers me a talk on succession problems in general.

I ask what she does when she is not working.

There is a long pause.

"Well," she says. "Ummm. I have a family."

It turns out that at the age of 26 she got married and took on four young step-children. She tells me about this as if it were the easiest thing in the world, explaining that in those days she didn't travel so much, and that she had some help with the cleaning - as if that made all the difference.

"I enjoy time with the family. We remind each other who we are. Other than that, I hit the trail every Saturday or Sunday and hike for a couple of hours and chill out. That's pretty much my life. Not very exciting."

Maybe not. Does she ever wonder why she works so hard, why she spends much of her life on an aircraft and reflect on whether it is worth it?

She is so puzzled by the question, I have to repeat it. "No," she says, "No."