The book begins with preliminary explorations of mathematical and geometric principles, before going into the

geometry of numbers, Arakelov Geometry on arithmetic curves, Arakelov Geometry on arithmetic surfaces, Arakelov Geometry on general arithmetic varieties, arithmetic volume function and its continuity, Nakai-Moishezon criterion on an arithmetic variety, arithmetic Bogolov inequality, and Lang-Bogomolov conjecture.

The Fields winners were selected for their contributions to topics ranging from dynamical systems to the

geometry of numbers and the solution of equations of the type that describe many physical phenomena.

Bhargava, born in 1974 in Canada, was awarded the Fields Medal for developing powerful new methods in the

geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves.

Their topics include divisibility, polynomial congruences, quadratic reciprocity, the

geometry of numbers, and algebraic integers.

The book is suitable for researchers and graduate students interested in combinatorial aspects of commutative algebra, optimization, discrete geometry, statistics, mirror symmetry, and

geometry of numbers.

The

geometry of numbers is illustrated by the following hands-on task using 7 matchsticks or straight rods of equal length.