New Tools in the Local and Global Langlands Programs (25w5400)

The Institute for Advanced Study in Mathematics will host the "New Tools in the Local and Global Langlands Programs" workshop in Hangzhou, China from August 10 to August 15, 2025.

The Langlands program is one of the pillars of modern mathematics. It relates the following two very different types of mathematical structures: The first is the type of continuous and smooth structures, equipped with symmetries. The second is the type of discrete structures, having to do with integers and rational numbers; these include the most mysterious properties of integers that have baffled mathematicians since the beginning of the subject, for instance properties of primes, or solving equations in integers like Fermat's Last Theorem. Thus the Langlands program is sometimes referred to as a "grand unification of mathematics''.

There are several sub-areas in the Langlands program, working under different mathematical settings. Each sub-area has developed its own set of insights and techniques, and these are often difficult to be applied to the other sub-areas because of the significant difference in the natures of the objects being considered. In the past five to ten years, sufficiently many new tools were introduced into the Langlands program, and an increasing amount of meaningful communication between the sub-areas became possible and fruitful. Most remarkably, information was flowing from the more geometric theories to the more discrete (that is, arithmetic) theories. One of the new tools which made this possible is modern $p$-adic or perfectoid geometry. At this moment, collaboration between different fields is predictably the most fruitful in advancing the Langlands program. It is therefore timely to hold this workshop in order to facilitate communication between researchers in different aspects of the Langlands program, and to create an inclusive atmosphere to encourage collaboration between different subject areas, career stages, and backgrounds.

The Institute for Advanced Study in Mathematics (IASM) in Hangzhou, China, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, are collaborative Canada-US-Mexico ventures that provide an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station in Banff is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).