Mikhael Gromov (mathematician) facts for kids

Quick facts for kids Mikhael Gromov
Михаил Громов
Gromov in 2014
Born	(1943-12-23) 23 December 1943 (age 81) Boksitogorsk, Russian SFSR, Soviet Union
Nationality	Russian and French
Education	Leningrad State University (PhD)
Known for	List Geometric group theory Symplectic geometry Systolic geometry Gromov boundary Gromov's compactness theorem (geometry) Gromov's compactness theorem (topology) Gromov's theorem on groups of polynomial growth Gromov–Hausdorff convergence Gromov–Ruh theorem Gromov–Witten invariant Gromov hyperbolic group Gromov δ-hyperbolic space Gromov norm Gromov product Gromov topology Gromov's inequality for complex projective space Gromov's systolic inequality Bishop–Gromov inequality Asymptotic dimension Essential manifold Filling area conjecture Filling radius Mean dimension Minimal volume Non-squeezing theorem Pseudoholomorphic curve Random group Sofic group Systolic freedom 2π theorem
Awards	Oswald Veblen Prize in Geometry (1981) Wolf Prize (1993) Balzan Prize (1999) Kyoto Prize (2002) Nemmers Prize in Mathematics (2004) Bolyai Prize (2005) Abel Prize (2009)
Scientific career
Fields	Mathematics
Institutions	Institut des Hautes Études Scientifiques New York University
Doctoral advisor	Vladimir Rokhlin
Doctoral students	Denis Auroux François Labourie< Pierre Pansu Mikhail Katz

Mikhael Leonidovich Gromov is a famous Russian-French mathematician. He is known for his important work in geometry, analysis, and group theory. He is a permanent member of the Institut des Hautes Études Scientifiques in France. He is also a professor of mathematics at New York University.

Gromov has received many awards for his work. One of the most important is the Abel Prize, which he won in 2009. He received this prize "for his revolutionary contributions to geometry."

Early Life and Education

Mikhael Gromov was born on December 23, 1943, in Boksitogorsk, which was part of the Soviet Union at the time. His father, Leonid Gromov, was Russian-Slavic, and his mother, Lea, was of Jewish heritage. Both of his parents were pathologists, which means they studied diseases.

Gromov was born during World War II. His mother, who was a medical doctor in the Soviet Army, had to leave the front lines to give birth to him. When he was nine years old, his mother gave him a book called The Enjoyment of Mathematics. This book sparked his interest in math and had a big impact on him.

Gromov studied mathematics at Leningrad State University. He earned his master's degree in 1965 and his doctorate in 1969. He completed his postdoctoral thesis in 1973. His main teacher and advisor was Vladimir Rokhlin.

Career and Moving to the West

Gromov got married in 1967. In 1970, he was invited to give a presentation at a big math conference. This was the International Congress of Mathematicians in Nice, France. However, the Soviet government did not allow him to leave the country. Even so, his lecture was still published in the conference's official papers.

Gromov disagreed with the Soviet system. He had thought about leaving the country since he was 14. In the early 1970s, he stopped publishing his work. He hoped this would help him get permission to move to Israel. He even changed his last name to his mother's. In 1974, his request was finally approved. He moved directly to New York and started working at Stony Brook University.

In 1981, he left Stony Brook University. He joined the faculty at the University of Paris VI in France. In 1982, he became a permanent professor at the Institut des Hautes Études Scientifiques. He still works there today. He also held professorships at the University of Maryland, College Park from 1991 to 1996. Since 1996, he has been a professor at the Courant Institute of Mathematical Sciences in New York. He became a French citizen in 1992.

His Mathematical Work

Gromov's approach to geometry is often described as "coarse" or "soft." This means he looks at the overall shape and large-scale properties of things. He is also interested in mathematical biology, how the brain works, and how scientific ideas develop.

Flexible Solutions in Geometry

Gromov introduced something called the h-principle. This is a way to find "flexible" solutions to certain math problems. It often shows that there are more ways to solve a problem than mathematicians first thought. For example, he showed that you can create curved surfaces in many unexpected ways. This was different from what was known about surfaces with very specific curvatures. His work helped to restart the study of similar ideas from the 1950s.

Understanding Large Data Sets

Gromov, along with Vitali Milman, helped explain a concept called "concentration of measure." Imagine you have a huge amount of data. This idea says that most of the data points will be very close to the average. They showed how this happens in many different mathematical spaces. This is similar to the law of large numbers, which says that if you repeat an experiment many times, the average result will get closer to the expected value.

Geometry of Spaces and Groups

Gromov made big contributions to understanding the geometry of spaces. He introduced the Gromov–Hausdorff metric in 1981. This is a way to measure the "distance" between two different geometric spaces. It helps mathematicians compare and classify different shapes and structures.

He also developed a key idea called Gromov's compactness theorem. This theorem helps mathematicians understand when a sequence of geometric shapes will "settle down" and get closer to a specific shape. This has been very important in a field called geometric group theory. This field studies groups (like sets of numbers with certain operations) by looking at their geometric properties.

Gromov also helped introduce the idea of hyperbolic groups with Eliyahu Rips. These are groups that behave in a way similar to hyperbolic geometry, which is a type of geometry where parallel lines can diverge.

Symplectic Geometry and Curves

Gromov's work on pseudoholomorphic curves is a key part of modern symplectic geometry. Symplectic geometry is a branch of math that deals with spaces where you can measure areas and volumes in a special way. He discovered a "bubbling" effect in these curves. This means that sometimes, when you try to make these curves "smooth," small bubbles can form.

One of his most famous results is the "non-squeezing theorem." This theorem shows a surprising property of symplectic spaces. It's like trying to squeeze a watermelon into a smaller pipe. The theorem says you can't squeeze a larger "ball" into a smaller one in a symplectic way, even if it seems like you could. This result showed how unique and rigid symplectic geometry is. His work also influenced Gromov-Witten theory, which connects to string theory and algebraic geometry.

Awards and Recognition

Mikhael Gromov has received many prestigious awards and honors for his groundbreaking work in mathematics:

Prizes

• Prize of the Mathematical Society of Moscow (1971)
• Oswald Veblen Prize in Geometry (1981)
• Prix Elie Cartan de l'Académie des Sciences de Paris (1984)
• Prix de l'Union des Assurances de Paris (1989)
• Wolf Prize in Mathematics (1993)
• Leroy P. Steele Prize for Seminal Contribution to Research (1997)
• Lobachevsky Medal (1997)
• Balzan Prize for Mathematics (1999)
• Kyoto Prize in Mathematical Sciences (2002)
• Nemmers Prize in Mathematics (2004)
• Bolyai Prize (2005)
• Abel Prize (2009)

Honors

• He was an invited speaker at the International Congress of Mathematicians multiple times: in 1970, 1978, 1983, and 1986.
• He is a foreign member of several important academies, including the National Academy of Sciences (1989), the American Academy of Arts and Sciences (1989), the Norwegian Academy of Science and Letters, the Royal Society (2011), and the National Academy of Sciences of Ukraine (2023).
• He became a member of the French Academy of Sciences in 1997.

See also

In Spanish: Mijaíl Grómov para niños

• Cartan–Hadamard theorem
• Non-squeezing theorem
• Systoles of surfaces