Algebraic Introduction to Kadomtsev-Petviashvili Hierarchy

2024/2025

Academic Year

ENG

Instruction in English

6

ECTS credits

Delivered by:

Faculty of Mathematics

Type:

Optional course (faculty)

When:

1, 2 module

Instructors

Bychkov, Boris

Dunin-Barkowski, Petr

Course Syllabus

Abstract

Kadomtsev--Petviashvili hierarchy is an infinite system of pairwise commuting PDEs. It has a proper description in terms of the Lax operators and commuting flows, but in this course we will work with the KP hierarchy from the point of view of its solutions and will give a description of the formal solutions of the KP hierarchy through the points of the semi infinite Grassmannian. We start with the bosonic and fermionic Fock spaces and the isomorphism between them, then describe a symmetry group which maps one solution to the different one. Then we describe an orbit of this action as an infinite dimensional Grassmannian and rewrite the conditions on tau functions as Hirota bilinear equations. This point of view on KP hierarchy turns out to be very fruitful in applications. We will presents such example as Konstevich--Witten tau function, Orlov--Scherbin tau function and others.

Learning Objectives

• -

Expected Learning Outcomes

• ---

Course Contents

• Fock space
• Boson – Fermion correspondence
• KP hierarchy
• tau functions and algebra gl(∞)
• Infinite dimensional Grassmaninans
• Hirota bilinear equations
• Examples of tau functions from enumerative geometry and enumerative combinatorics

Assessment Elements

• HW
• Exam

Interim Assessment

• 2024/2025 2nd module
  4 ∗ 0.1 ∗ 𝐻𝑊 + 0.6 ∗ 𝐸, where HW is a grade for the homework (4 during the semester), E is a final exam grade.

Bibliography

Recommended Core Bibliography

• Солитоны: дифференциальные уравнения, симметрии и бесконечномерные алгебры, Мива, Т., 2005

Recommended Additional Bibliography

• Lie algebras, Jacobson, N., 1979

Authors

• Dunin-Barkovskii PETR IGOREVICH
• BYCHKOV BORIS SERGEEVICH
• Иконописцева Юлия Вахтаногвна