2024/2025
Algebraic Introduction to Kadomtsev-Petviashvili Hierarchy
Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
1, 2 module
Open to:
students of all HSE University campuses
Language:
English
ECTS credits:
6
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.
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
Interim Assessment
- 2024/2025 2nd module4 ∗ 0.1 ∗ 𝐻𝑊 + 0.6 ∗ 𝐸, where HW is a grade for the homework (4 during the semester), E is a final exam grade.
