# Gerd Faltings

# Gerd Faltings

**1954-**

**German Mathematician**

Mathematician Gerd Faltings was born in Gelsenkirchen-Buer, Germany. Faltings undertook his graduate work at the University of Münster and completed his doctoral work in 1978. He then went on to do postdoctoral research at Harvard, taught at the University of Wuppertal, and eventually accepted a professorial appointment at Princeton University. In 1996 Faltings was invited to study at the Max Planck Institute for Mathematics in Bonn.

Faltings's proof of Mordell's conjecture earned him a 1986 Fields Medal and provided a major stepping stone toward the elusive proof of Fermat's Last Theorem, advanced in 1995 by fellow Princeton mathematics professor Andrew Wiles (1953- ).

Fermat's Last Theorem is a variation on the famous Pythagorean theorem stating that for right triangles the square of the hypotenuse is equal to the sum of the squares of the sides (*x*^{2} + *y*^{2} = *z*^{2}). Both are based on Diophantine equations of the form *x*^{n} + *y*^{n} = *z*^{n} where *x, y, z* and *n* are integers—equations first explored by the Greek mathematician Diophantus (circa 250 a.d.). Based on the observation that it was impossible for a cube to be the sum of two cubes and a fourth power to be the sum of two fourth powers, in 1637 French mathematician Pierre de Fermat (1601-1665) offered the theorem that, in general, it was impossible for any number that is a power greater than the second to be the sum of two like powers (that is, there were no non-zero solutions for equation *x*^{n} + *y*^{n} = *z*^{n} where *n* is greater than 2). In his notes, Fermat claimed to have found proof for his theorem, but none was discovered prior to Wiles's proof. The problem vexed mathematicians for centuries.

Faltings's contribution to proof of Fermat's Last Theorem came in 1983 when Faltings's proved Louis Mordell's 1922 conjecture regarding systems of polynomial equations that define curves as having a finite number of solutions (that is, a finite number of prime integers *x, y, z *where *x*^{n} + *y*^{n} = *z*^{n}). Mordell's conjecture stipulated that within three-dimensional space, two-dimensional surfaces are grouped according to their genus (the number of holes in the surface). A ring, for example, has one hole and therefore its genus is considered one. If the surface of solutions for equations contained two or more holes, then the underlying equations must have a finite number of integer solutions.

By proving Mordell's conjecture, Faltings showed (using methodology based in arithmetic algebraic geometry) that the Diophantine equations *x*^{n} + *y*^{n} = *z*^{n} underlying Fermat's Last Theorem could only contain a finite number of integer
solutions for each exponent of Fermat's Last Theorem (that is, for *n* > 2).

Faltings's proof stopped short of proof that the finite number was, in accordance with Fermat's Last Theorem, zero—meaning that there were no solutions. Although proof of Fermat's theorem was not Faltings's goal (a proof of Fermat's Last Theorem would have required a proof encompassing all exponents *n* > 2), his proof was a major step along the road to a solution.

Faltings's work was of such siginificance that when Wiles sought verification of his proof of Fermat's Last Theorem he turned to Faltings for critical review. Among other works, Faltings published commentary and analysis of Wiles's proof for the American Mathematical Society.

The proof of Fermat's Last Theorem captured popular attention and propelled mathematicians, including Faltings, from scholarly renown to wider celebrity.

Much of Faltings's work has dealt with the arithmetic of elliptic curves, the determination of algebraic structure, and geometric combination of the solution. Faltings's well-regarded work ranges over, and mixes, algebra, arithmetic, and analysis. His published works include studies of the Shafarevich and Tate conjecture (the proof of which is often credited to Faltings), Arakelov theory, degeneration of Abelian varieties, vector bundles on curves, and the arithmetic Riemann-Roch theorem.

**ADRIENNE WILMOTH LERNER**

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**Gerd Faltings**