Riemann’s Doctoral Dissertation

Chapters 5 to 9 take up several of Riemann's most important works, which are discussed from the vantage point of prior developments and contemporary events. Gauss oversaw Riemann's doctoral work, but not in the manner of a modern-day mentor (for his broader intellectual influence on Riemann, see Chapter 10). Gauss perhaps had an idea of the topic Riemann chose for his dissertation, but he had little idea of its content before the day this work was submitted and forwarded to him by the Dean for his assessment. Gauss praised the originality of Riemann's work, but offered not even a hint as to its subject matter, probably because he assumed that his colleagues, being specialists in other fields, knew far too little higher mathematics for him to bother describing what this candidate had achieved. Riemann's general theory of complex functions was, in fact, rooted in ideas found in various works by Gauss, who introduced imaginaries in number theory in 1831 and highlighted their importance again in 1849. Riemann also cited Gauss's results on conformal mappings, a central aspect in Riemann's conception of analytic functions, which possess the property of preserving similarity locally. Although he had also studied the work of Cauchy, he nowhere cited his name in the text. He employed an early form of Dirichlet's principle to establish the existence of harmonic functions that take on given values on the boundary of a region. Riemann's dissertation is famous today, but it remained almost unknown until after 1857, the year he published his paper on Abelian functions in Crelle's Journal. There he made direct reference to his dissertation, alongside other remarks that spelled out when he first found various results. This chapter closes with a section that cites the opinions of eminent mathematicians who have commented on the difficulties involved in reading Riemann's works. As with much else, opinions about that topic tend to vary.