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Theory of symmetric polynomials and applications
01HVXUR
A.A. 2023/24
Course Language
Inglese
Degree programme(s)
Doctorate Research in Scienze Matematiche - Torino
Course structure
| Teaching | Hours |
|---|---|
| Lezioni | 30 |
Lecturers
| Teacher | Status | SSD | h.Les | h.Ex | h.Lab | h.Tut | Years teaching |
|---|---|---|---|---|---|---|---|
| Gatto Letterio | Professore Associato | MATH-02/B | 30 | 0 | 0 | 0 | 1 |
Co-lectures
Context
| SSD | CFU | Activities | Area context |
|---|---|---|---|
| *** N/A *** |
The aim of the course is to offer a presentation of the theory of symmetric polynomials in the light of new
methodologies that allow a more flexible use of the formalism and with applications to multilinear algebra,
combinatorics, algebra, representation theory, mathematical physics (Bose-Fermi correspondence and integrable
systems), to algebraic geometry (classical, quantum and equivariant cohomology of Grassmannians and, possibly, to the cohomoloy of zero-dimensional Hilbert schemes on algebraic surfaces)
Linear Algebra and Geometry, Analisys 1, Analysis 2. The needing Multilinear algebra will be developped along the
course
Actions of groups on sets. Actions of the symmetric group on polynomial rings and fixed points of the action.
Introduction to the theory of partitions, Young's diagrams and Young's Tableaux. Complete, elementary and Schur
complete polynomials, power sums and their generating functions. The exterior algebra as a (non-irreducible)
representation of the algebra of symmetric polynomials. Jacobi-Trudi formula, Pieri formulas. Introduction to the theory
of symmetric polynomials by Jack and Macdonald. Application to universal linear ordinary differential equations. The cayley-Hamilton theorem for infinite dimensional vector spacs. The
Bose-Fermi correspondence and top operators. Discussion of open problems and proposing of research topics.
On site
Written report presentation - Oral presentation
P.D.2-2 - March
The course will have a strong interdisciplinary connotation and therefore of interest to students oriented both to algebra and algebraic geometry and to mathematical physics and analysis.