The Story of Reimann’s Moduli Space

Abstract

Compact Riemann surfaces and projective algebraic curves over C are two realizations of the same class of objects. Their moduli space was first introduced by Riemann and is one of the central objects of the contemporary mathematics and mathematical physics. In this note, we explain the birth and evolution of Riemann's moduli space by emphasizing several points which might not be so well-known to the general mathematicians and possibly even to some mathematicians who are interested in either complex analytic or algebraic geometric theories of moduli spaces. For example, who was the first to formulate the moduli problem precisely? What is the meaning of moduli spaces? Who was the first to give complex analytic and algebraic variety structures to Riemann's moduli space? Why did they study these problems? How was Riemann's moduli space first used? And why did Riemann choose and use the word moduli? Why did Teichmüller study Teichmüller space? What were Teichmüller's contributions to Riemann's moduli space? What were Teichmüller's contributions to algebraic geometry? Contents