#### János Kollár & Claire Voisin

for their remarkable results in many central areas of algebraic geometry, which have transformed the field and led to the solution of long-standing problems that had appeared out of reach.

#### Contribution

The Shaw Prize in Mathematical Sciences 2017 is awarded in equal shares to János Kollár, Professor of Mathematics, Princeton University, USA and Claire Voisin, Professor and Chair in Algebraic Geometry, Collège de France, for their remarkable results in many central areas of algebraic geometry, which have transformed the field and led to the solution of long-standing problems that had appeared out of reach.

Since ancient times, a central theme in mathematics has been the study of polynomials and their solutions. Algebraic geometry is the study of the properties of sets of solutions to polynomial equations in several variables. A simple example of such an equation is x2 + y2 + z2 = 1, the solution set of which is the surface of a sphere of radius 1.

#### An Essay on the Prize

Since ancient times, a central theme in mathematics has been the study of polynomials and their solutions. Algebraic geometry is the study of the properties of sets of solutions to polynomial equations in several variables. A simple example of such an equation is x2+y2+z2 = 1, the solution set of which is the surface of a sphere of radius 1.

As this example demonstrates, solution sets of polynomial equations, which are known as varieties, are geometric objects. Examining the interplay between the algebra and the geometry has turned out to be remarkably fruitful, and algebraic geometry is a major branch of mathematics, the study of which has profound consequences not just for algebra and geometry but also for several other areas ranging from number theory to mathematical physics.