Periodic Reporting for period 5 - HAS (Harmonic Analysis and l-adic sheaves)
Okres sprawozdawczy: 2021-10-01 do 2022-09-30
The conjecture of Langlands is one of the main conjectures in mathematics. Even now, when this conjecture is only known partially, it led to important results in Number Theory and Algebraic Geometry including proofs of the Ferma and the Sato-Tate conjectures.
There exists also a (conjectural) Geometric Langlands correspondence related to dualities in the 4-dimensional quantum field theory (see Gauge Theory and the Analytic Form of the Geometric Langlands Program by D.Gaiotto E.Witten).
There exists also a (conjectural) Geometric Langlands correspondence related to dualities in the 4-dimensional quantum field theory (see Gauge Theory and the Analytic Form of the Geometric Langlands Program by D.Gaiotto E.Witten).
I worked (with a number of other mathematicians) on different aspects of the Langlands conjecture. There are three main problems I considered.
1) The stability of characters of representations of reductive groups over local non-archimedian fields and the related question about the presentation of characters as linear combinations of character sheaves.
2) The geometric construction of the L-functions of representations of reductive groups over local non-archimedian fields and
3) The extension of the Langlands conjecture to the new area of representations of Kac-Moody groups.
1) The stability of characters of representations of reductive groups over local non-archimedian fields and the related question about the presentation of characters as linear combinations of character sheaves.
2) The geometric construction of the L-functions of representations of reductive groups over local non-archimedian fields and
3) The extension of the Langlands conjecture to the new area of representations of Kac-Moody groups.
I expect that these achievements will lead to new and important results in Algebraic Geometry and Number Theory.
The main objective is to advance the area of the representation theory through the application of concepts and results from Algebraic Geometry and Category Theory.
In all of these three areas we obtain new and important results. In the part 1) we defined the notion of character sheaves for groups over local fields. In the part 2) we provided a geometric construction of the L-functions in a number of new and important cases and In the part 3) we developed a completely new theory of the Langlands correspondence for Kac-Moody groups.
The main objective is to advance the area of the representation theory through the application of concepts and results from Algebraic Geometry and Category Theory.
In all of these three areas we obtain new and important results. In the part 1) we defined the notion of character sheaves for groups over local fields. In the part 2) we provided a geometric construction of the L-functions in a number of new and important cases and In the part 3) we developed a completely new theory of the Langlands correspondence for Kac-Moody groups.