Trying to keep track of everything happening in the Langlands program area of mathematics is somewhat of a losing battle, as new ideas and results keep appearing faster than anyone could be expected to follow. Here are various items: <\/p>\n
For a popular discussion, see this article at Quanta<\/a>.<\/p>\n To put things in a wider context, one might want to take a look at the “What is not done in this paper?” section of the last paper of the five<\/a> giving the proof. It gives a list of what is still not understood:<\/p>\n Geometric Langlands with Iwahori ramification. Only the last of these touches on the original number field case of Langlands, which is a much larger subject than geometric Langlands.\n<\/li>\n As a sideline, he’s been working on the “Habiro ring” of a number field<\/a>, finding there power series that came up in the study of complex Chern-Simons theory and the volume conjecture. According to Scholze:<\/p>\n My hope was always that this q-deformation of de Rham cohomology should form a bridge between the period rings of p-adic Hodge theory and the period rings of complex Hodge theory. The power series of Garoufalidis\u2013Zagier do have miraculous properties both p-adically and over the complex numbers, seemingly related to the expected geometry in both cases (the Fargues\u2013Fontaine curve, resp. the twistor-P1<\/sup>), and one goal in this course is to understand better what\u2019s going on.<\/p><\/blockquote>\n<\/li>\n Trying to keep track of everything happening in the Langlands program area of mathematics is somewhat of a losing battle, as new ideas and results keep appearing faster than anyone could be expected to follow. Here are various items: Dennis … Continue reading
\nQuantum geometric Langlands.
\nLocal geometric Langlands with wild ramification.
\nGlobal geometric Langlands with wild ramification.
\nRestricted geometric Langlands for \u2113-adic sheaves (for curves in positive characteristic).
\nGeometric Langlands for Fargues-Fontaine curves.<\/p>\n\n<\/ul>\n","protected":false},"excerpt":{"rendered":"