Non-linear PDEs and Secondary Calculus via Derived Algebraic Geometry
Organizers
Speaker
Time
Thursday, December 12, 2024 3:00 PM - 4:00 PM
Venue
A6-101
Online
Zoom 638 227 8222
(BIMSA)
Abstract
Secondary calculus, due to Alexander Vinogradov, is a formal replacement for the differential calculus on the typically infinite dimensional space of solutions to a non-linear partial differential equation.
On the one hand, one may associate a Variational Bicomplex to any PDE, which roughly speaking, plays the same role as the de Rham complex of an ordinary manifold or scheme. It is a central object of study in the geometric theory of PDEs, the theory of Integrable systems, Variational Calculus as well as Gauge Field Theory, and the objects of Secondary calculus are obtained from its cohomology groups, possibly with coefficients. By studying these groups, we may extract information about the original PDE, such as existence of symmetries, recursion operators, conservation laws etc.
In my talk I will discuss a vast generalization of this object to the derived and stacky setting using the language of (relative) homotopical algebraic geometry.
The refined object we obtain- the so called “Derived Variational Tricomplex” then provides a powerful tool for equipping moduli spaces of non-linear PDEs with shifted symplectic structures, as well as approaching the problem of formulating a global, hence non-perturbative, approach to the BV-BRST formalism.
Time permitting I will discuss applications to these topics.
This is based on a joint work with A. Sheshmani and S-T Yau and a work in progress.
On the one hand, one may associate a Variational Bicomplex to any PDE, which roughly speaking, plays the same role as the de Rham complex of an ordinary manifold or scheme. It is a central object of study in the geometric theory of PDEs, the theory of Integrable systems, Variational Calculus as well as Gauge Field Theory, and the objects of Secondary calculus are obtained from its cohomology groups, possibly with coefficients. By studying these groups, we may extract information about the original PDE, such as existence of symmetries, recursion operators, conservation laws etc.
In my talk I will discuss a vast generalization of this object to the derived and stacky setting using the language of (relative) homotopical algebraic geometry.
The refined object we obtain- the so called “Derived Variational Tricomplex” then provides a powerful tool for equipping moduli spaces of non-linear PDEs with shifted symplectic structures, as well as approaching the problem of formulating a global, hence non-perturbative, approach to the BV-BRST formalism.
Time permitting I will discuss applications to these topics.
This is based on a joint work with A. Sheshmani and S-T Yau and a work in progress.