Selberg trace formula

This lecture by Richard Borcherds provides an overview of the Selberg trace formula, a powerful tool in number theory and representation theory. The speaker begins with a simple finite group example to illustrate the underlying idea: computing the character of the induced representation by counting fixed points. He then generalizes to the case of SL2(R) acting on functions on SL2(R)/SL2(Z), highlighting the complications that arise when the group is infinite and non-compact. Key issues include the need to replace sums by integrals, the distributional nature of characters, the complexity of conjugacy classes (especially over adeles), and the necessity of using Eisenstein series to handle the continuous spectrum. The lecture concludes by summarizing the modifications to the Frobenius formula and mentioning the comprehensive treatment in Hejhal’s two-volume book.