## Columbia Mathematics Department Colloquium

*"Geometric-analytic dualities on arithmetic surfaces"*

### by

### Ivan Fesenko

### University
of Nottingham

Abstract:

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**