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Abstracts and Talk Materials
Geometric and Topological Methods in Variational Calculus
February 20 - May 8, 2014


Scot Adams (University of Minnesota, Twin Cities)

Frame Bundle Freeness Results

In smooth dynamics of Lie groups, it's a consequence of the Frobenius Theorem that locally free actions have a foliated orbit structure. This allows for the formation of local transversals which reduces the classification of the orbits down to a problem in countable equivalence relations. For actions that are not locally free, this fails, but it's natural to ask if one can work around this difficulty by finding a related action that is locally free, and, with luck, even free. Passage to higher order frame bundles (and various associated bundles) is one technique for implementing this work-around. We will discuss situations where freeness can be found in frame bundles, and situations where it cannot.

Joseph Benson (University of Minnesota Twin Cities)
http://math.umn.edu/~benso700/

Invariant Curve and Surface Flows and Evolution of their Associated Differential Invariants

In this talk, I will describe some of the uses of the invariant variational bicomplex structure in studying the evolution of differential invariants such as curvature, torsion, mean curvature, etc. under a group invariant flow. This talk will also discuss integrability of the resulting equations describing these evolutions, and in the case of curves in Euclidean 3-space, the connection to the non-linear Schrödinger equation via the Hasimoto transform.

Joseph Benson (University of Minnesota Twin Cities)
http://math.umn.edu/~benso700/

The Invariant Variational Bicomplex

In this talk I will use the method of moving frames to construct a group invariant version of the variational bicomplex. Explicit formulae for the group invariant versions of the Euler-Lagrange equations associated to an invariant variational problem arise in this context. If time permits, I will give further applications to invariant curve and surface flows in low dimensions. The talk is based on papers by Kogan and Olver and leads into the problems considered in my dissertation.

Irina Kogan (North Carolina State University)
http://www4.ncsu.edu/~iakogan/

Inverse Problem of Calculus of Variations

This week seminar will be devoted to the inverse problem of calculus of variations: 1) given a system of equations, decide whether or not it comes from a variational principle; 2) if the answer is positive, find a corresponding Lagrangian. I will present Vainberg, Volterra solution to this problem, as well as a solution in terms of the variational bicomplex. I will not prove the results but illustrate them by examples.

David Paul Morrissey (University of Minnesota, Twin Cities)
www.math.umn.edu/~morri495

Loss of a Free Energy Functional Through Boundary Conditions

Pattern forming PDE often rely on a Lyapunov functional for uniqueness and existence, in turn the existence of such functionals often rely on infinitely extended domains. Exotic boundary conditions break this variational structure. I will present an overview of the Cahn-Hilliard equation and Swift-Hohenberg equation as examples of such pattern forming differential equations. For small bifurcation parameter values in the Swift-Hohenberg equation it is well known that there exists a family of solutions parameterized by the wavenumber. For the Swift-Hohenberg equation I will show a Numerical Homotopy of this family of solutions from Neumann to Transparent boundary conditions on a simulated half line which exhibit a transition from phase selection to wavenumber selection.

Peter J. Olver (University of Minnesota, Twin Cities)
http://www.math.umn.edu/~olver

Introduction to the Variational Bicomplex

I will present the basics of jet space, contact forms, and the variational bicomplex. Applications to the calculus of variations, the inverse problem, conservation laws, and symmetries, possibly getting to Noether's Theorem(s), will be discussed. No prior knowledge beyond multivariable calculus and a little familiarity with differential forms will be assumed.

Peter J. Olver (University of Minnesota, Twin Cities)
http://www.math.umn.edu/~olver

Introduction to the variational bicomplex (part II)

I will present the basics of jet space, contact forms, and the variational bicomplex. Applications to the calculus of variations, the inverse problem, conservation laws, and symmetries, possibly getting to Noether's Theorem(s), will be discussed. No prior knowledge beyond multivariable calculus and a little familiarity with differential forms will be assumed

Alexander A. (Sasha) Voronov (University of Minnesota, Twin Cities)

A Batalin-Vilkovisky extension of the variational bicomplex

I will review a physically-oriented approach to constraints and symmetries in the variational bicomplex developed by Barnich, Fulp, Henneaux, Lada, McCloud, and Stasheff.

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