Columbia Mathematics Department Colloquium


"Geometric-analytic dualities on arithmetic surfaces"


 Ivan Fesenko  

University of Nottingham


 Higher local and global fields are associated to flags of subvarieties on an arithmetic variety, similar to how usual local fields are associated

to points of arithmetic curves. Two-dimensional local fields include formal power series over R, C and p-adic numbers.  Such formal

 loop objects play an important role and are intensively studied in several parts of mathematics, mostly related to quantum physics.


Several years ago the speaker introduced new translation invariant measure, integration and harmonic analysis on higher local and adelic objects,

which take values in power series over C, appropriately topologized. These integrals correspond to higher rank integral structures,

which are finer than infinite dimensional vector space structures or other classical structures in common use. Remarkably, some of integrals

in this higher theory  are quite similar to the (still mathematically nonrigorous) Feynman path integrals in physics. The higher analysis on

arithmetic schemes  uncovers new deep geometric-analytic dualities which underly and explain several fundamental conjectural properties

of zeta functions of arithmetic schemes. This development uses and has connections to analysis, functional analysis, algebraic topology, algebraic

K-theory, geometric representation theory, Langlands and geometric Langlands correspondences.


  April 1st, Wednesday, 5:00-6:00 pm

Mathematics 312

Tea will be served at 4:30pm