I became interested in the history of the Riemann-Roch theorem, so I searched various materials.

So, I read Über die Wechselwirkungen zwischen der französischen Schule, Riemann und Weierstraß. Eine Übersicht mit zwei Quellenstudien by E. Neuenschwander. If you read pages 3 to 5 of this paper, you will see that Riemann became interested in the conformal map, and also function theory, through a conversation with his teacher Gauss. You can see that it was from. The original text was in German, so I translated the parts I found into English.

A letter from Riemann to his father from Berlin on March 30, 1849 stated the following:

“I looked for a long time in the library's data for another work by Gauss that won a prize in Copenhagen, and I finally found it, and I'm still studying it.” Gauss' paper mentioned in the last part here is about conformal mapping.

And I was intrigued, so I searched other papers to find information about the conversation between Riemann and Gauss. So I found this paper, Riemann, Betti and the birth of topology by Andre Weil. If you read this paper, you will know that after Riemann heard the concept of cuts from Gauss, he became interested in the study of Analysis Situs, now called topology, and began to study topological contents.

What gave Riemann the idea of the cuts was that Gauss defined them to him, talking about other matters, in a private conversation. In his writings one finds that analysis situs, that is, this consideration of quantities independently from their measure, is "wichtig"; in the last years of his life he has been much concerned with a problem in analysis situs, namely: given a winding thread and knowing, at every one of its selfintersections, which part is above and which below, to find whether it can be unwound without making knots; this problem he did not succeed in solving except in special cases ...

However, since the current Riemann-Roch theorem is also known as a combination of complex analysis and topology, it is possible that these contents were the basis for Riemann to think about Riemann-inequality, the first prototype of the Riemann-Roch theorem, which appeared in the above papers. I started to think that it might have been a conversation with Gauss. Is this idea right?