# The Variational Bicomplex

Wednesday, January 29, 2014 - 1:30pm - 2:30pm

Lind 305

Irina Kogan (North Carolina State University)

Introduction of the variational bicomplex can be motivated by drawing

an analogy with vector calculus. It is well known that one can

reformulate vector calculus in terms of differential forms, and use

the exterior derivative to represent gradient, divergence, and curl

operators. Thus vector calculus can be efficiently expressed by the de

Rham complex. Similarly to vector calculus, many important aspects of

variational calculus, such as Euler-Lagrange operator, Helmholtz

operator, or Noether correspondence, can be formulated in terms of

complexes of differential forms. This leads to notion of the

variational bicomplex (and related notions of the variational complex,

the variational spectral sequence). These constructions originated

with the work of Dedecker (1957) and a large body of literature has

appeared afterwards. The purpose of my talk is to define the

variational bicomplex, explain how it encodes various aspects of

variational calculus and what role is played by its cohomology.

an analogy with vector calculus. It is well known that one can

reformulate vector calculus in terms of differential forms, and use

the exterior derivative to represent gradient, divergence, and curl

operators. Thus vector calculus can be efficiently expressed by the de

Rham complex. Similarly to vector calculus, many important aspects of

variational calculus, such as Euler-Lagrange operator, Helmholtz

operator, or Noether correspondence, can be formulated in terms of

complexes of differential forms. This leads to notion of the

variational bicomplex (and related notions of the variational complex,

the variational spectral sequence). These constructions originated

with the work of Dedecker (1957) and a large body of literature has

appeared afterwards. The purpose of my talk is to define the

variational bicomplex, explain how it encodes various aspects of

variational calculus and what role is played by its cohomology.