It’s entirely imaginable that Mary Callahan Erdoes recited Euler’s Formula, Fermat’s Theorem, and Li Shan-Lan’s Chinese Hypothesis, not to mention Einstein’s Equation, before she uttered a declarative sentence.

“I’ve always been good at math,” the CEO of JPMorgan Private Bank said, alluding to the formulae that have intrigued mathematicians for decades.

Ms. Callahan Erdoes has never been a mathematician, but her felicity with figures helped her sparkle at high school in Chicago; during her undergraduate years at Georgetown University, where she was the only female math major in her class, and at Harvard, where she earned her MBA and also met her future husband.

“My parents urged me to drive myself, to work hard,” Ms. Callahan Erdoes said. “My father would take me to his office on Saturdays when I was a little girl. So I suppose I absorbed certain skills early in life.”

Her father, Patrick Callahan, was a partner at the venerable investment banking firm of Lazard Freres. His strategy of exposing his young daughter to his professional life may explain why Ms. Callahan Erdoes brings her daughters, Mia, 3, and Morgan, 1, to her Park Avenue office every Sunday, a working day for her.

Yesterday, for example, while her venture capitalist husband, Philip Erdoes, was working downtown in his SoHo office, Ms. Callahan Erdoes – her daughters in tow – was tending to business uptown. She is responsible for JPMorgan Private Bank’s ultra-highnet-worth clients around the world.

Those clients account for more than $300 billion that Ms. Callahan Erdoes manages. Only UBS and Credit Suisse have larger private banking portfolios.

To obtain the services of her bank, Ms. Callahan Erdoes said, clients need to have a minimum of $25 million.

“Our responsibility is to keep them wealthy – not just for their own generation but also generations to come,” she said.

Ms. Callahan Erdoes operates in that rarified zone of finance where a distinction is made between the rich and the wealthy.

“I believe rich people are no different from wealthy people, and wealthy people are no different from poor people,” she said. “But the complexities and focus in their lives are very different. So in terms of wealth management, as you move up the food chain, it correspondingly becomes more complicated.”

What does that mean?

“You [the wealthy client] worry less about managing to meet spending targets and more to meet giving targets,” Ms. Callahan Erdoes said. “You worry less about making enough for the next generation and more about what effects it will have on the next generation.

“And because you don’t worry about possibly spending it all, your investment decisions can have much longer time horizons, exposing portfolios to longer-term investment opportunities such as timber, private equity, venture capital, and real estate,” she said.

Private banks such as JPMorgan are vying to offer those investment opportunities for very wealthy clients – not least because more wealth is being generated now than ever before.

A Merrill Lynch/Capgemini study predicts good times for the wealthy around the globe in coming years, powered by rising stock markets, growth in emerging markets like China and India, and the broader economic recovery, according to published reports.

The study also says that the number of ultra-high-net-worth individuals is expected to grow 7% a year during the next few years. The total global wealth of financial millionaires is expected to hit $40.7 trillion by 2008, up from $28.8 trillion in 2003.

What Ms. Callahan Erdoes notes especially is that this surge in wealth is also fueling growing interest in inheritance, especially among the baby boomers. Various studies estimate between $41 trillion and $136 trillion will be transferred across generations in the next 50 years in America alone. Already, high-net-worth individuals in America and Canada hold nearly $10 trillion in liquid and financial assets, up from $7.4 trillion in 2002.

These are intimidating numbers to encounter, let alone manage. But Ms. Callahan Erdoes talks about them with sangfroid. After all, she’s been crunching mega-numbers since graduation, when she started at Stein Roe & Farnham, a well-established investment company in Chicago. While her formal job concerned computers, she made it a point to seek counsel from senior staff members and gained valuable mentors along the way.

“It’s so important for a young person to find and keep a mentor,” she said. “And once you find a mentor, you must treasure them and never take them for granted. It is hard to find someone successful who’s willing to help and pull you up. I know I owe a large debt to the cumulative number of people who helped me.”

Those people include colleagues at Bankers Trust, where Ms. Callahan Erdoes worked in corporate finance, merchant banking, and high-yield debt underwriting. She then moved to Meredith, Martin & Kaye, a fixed-income specialty advisory firm, where she was responsible for credit research, trading, and individual portfolio management.

It was barely a decade ago that Ms. Callahan Erdoes came to JPMorgan as head of fixed income for high-net-worth individuals, foundations, and endowments. Following the JPMorgan/Chase merger in 2000,she became responsible for investment solutions and investment strategy for private banking clients worldwide. And in March 2005, she was named CEO.

The day she got her job, Ms. Callahan Erdoes was riding an elevator at 345 Park Ave. to her fifth-floor office.

“Nobody spoke to me in the elevator,” she said. “I thought about it and then figured out that there’s this aura around a top job that others create. But I’m the same person. If you let the public persona of your job get to you, you won’t be very good.”

That persona is understandably discreet. While that discretion is no doubt de rigueur for someone in her position, she attributes her sense of style to the influence of her mother, Patricia, who was prominent in fashion and philanthropy circles in Chicago.

It is a style that combines reserve with cordiality. It mixes firmness in management with accessibility for her global staff of 3,500. It bows to JPMorgan’s 150-year history, but also is innovative. It demands passion and commitment from her colleagues and also requires that she lead by example.

“It’s also a style that recognizes that the only stupid question is the one that you don’t ask,” Ms.Callahan Erdoes said.

It is a style, finally, that continually emphasizes integrity.

“When someone gives you their money to manage – isn’t that one of the greatest expressions of trust?” she said.