Stanford University’s Maryam Mirzakhani was the first and only woman to win the prestigious Fields Medal in mathematics.

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PALO ALTO, Calif. — Maryam Mirzakhani, 40, a renowned mathematician and Stanford University professor best known for being the first woman to receive the prestigious Fields Medal for mathematics, died Saturday, the university announced.

She had breast cancer.

The Fields Medal, often described as the mathematician’s Nobel Prize, is given every four years to no more than four mathematicians, all of whom are 40 or younger. She was named for her work on complex geometry and dynamic systems.

Ms. Mirzakhani, of Iran, received the award in Seoul, South Korea, in 2014. “This is a great honor. I will be happy if it encourages young female scientists and mathematicians,” she said at the time. “I am sure there will be many more women winning this kind of award in coming years.”

Ms. Mirzakhani — known for taking the difficult, complicated path to solve mathematical problems — studied the symmetry of curved surfaces and other theoretical concepts known as “pure mathematics.” She joined Stanford in 2008, where she was a mathematics professor until her death.

“Mirzakhani was fascinated by the geometric and dynamic complexities of curved surfaces — spheres, doughnut shapes and even amoebas,” the university said in a news release. “Despite the highly theoretical nature of her work, it has implications in physics, quantum mechanics and other disciplines outside of math. She was ambitious, resolute and fearless in the face of problems others would not, or could not, tackle.”

She attended an all-girls high school in her native Tehran, where she competed for Iran’s International Mathematical Olympiad Team. She gained international recognition in 1994 as a gold-medal recipient and again in 1995, after achieving a perfect score and two gold medals.

She went on to college at Sharif University in Tehran, and then graduate school at Harvard University, where she was guided by Fields Medal winner Curtis McMullen, who once said she was filled with “fearless ambition.”

At Harvard, she was known for her persistence and intense questioning, despite a language barrier, according to Stanford. “Her questions came in English. Her notes were jotted in Farsi.”

Her 2004 dissertation, in which she solved two long-standing problems and connected the two into a masterful thesis, catapulted her into even greater fame. It yielded papers in each of the top three mathematics journals, Stanford said.

“The majority of mathematicians will never produce something as good,” said Benson Farb, a mathematician at the University of Chicago. “And that’s what she did in her thesis.”

Iran’s President Hassan Rouhani issued a statement Saturday praising Mirzakhani. “The grievous passing of Maryam Mirzakhani, the eminent Iranian and world-renowned mathematician, is very much heart-rending,” Rouhani said in a message that was reported by the Tehran Times.

Ms. Mirzakhani was a professor at Princeton University and a research fellow at the Clay Mathematics Institute before joining the Stanford faculty.

When she was working, Ms. Mirzakhani would doodle on sheets of paper and scribble formulas on the edges of her drawings, leading her daughter to describe the work as painting, according to Stanford’s statement.

Ms. Mirzakhani once described her work as “like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out.”

Stanford President Marc Tessier-Lavigne called Mirzakhani a brilliant theorist who made enduring contributions and inspired thousands of women to pursue math and science.

Ms. Mirzakhani is survived by her husband, Jan Vondrák, and her daughter, Anahita.