EVENT: Colluqium

There are four talks this week, reporting on applications of differential geometry to nonlinear problems in dynamics, interpolation, and problems of interpolating rigid body motion. Following each talk there will be a short discussion on topics of general interest. The speakers are (note that Shreya's time has been switched with Phil's)

Shreya Bhattarai, UWA Maths Maths LR 2 Wednesday February 23, 3:00pm

Prof. Wayne Lawton, NUS Maths and UWA Adjunct Maths LR2 Wednesday February 23, 4:00pm

Philip Schrader, UWA Maths Maths LR2 Thursday February 24, 3:00pm

Michael Pauley, UWA Maths Maths LR2 Thursday February 24, 4:00pm.

Abstracts for the four talks are given below.

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Introduction to Conditional Extremals -----------------------------

Shreya Bhattarai (UWA)

Maths LR2 Wednesday February 23, 3:00pm

Conditional extremals arise as solutions to a certain variational problem on Riemannian manifolds for which there is an underlying "prior vector field" A. When A is the zero field, conditional extremals are simply geodesics. For geodesics, we can talk about Jacobi fields and conjugate points and we can extend these ideas to conditional extremals as well. In this talk, I will be giving a brief introduction to conditional extremals and then discussing results and observations about their conjugate points -------------------------------------------------------------------------

Periodicity and Recurrence --------------------

Wayne Lawton (NUS and UWA)

Maths LR2 Wednesday February 23, 4:00pm

Abstract: Fourier's observation that periodic functions admit approximation by trigonometric series led him to construct foundations for the digital age. This approximation occurs because trigonometric series admit arbitrary localisation within their period interval. What happens when the set of frequencies are restricted? We show how this question is related to recurrence phenomena in dynamical systems studied by Poincare, Birkhoff, Morse, and others. ------------------------------------------

Conditional Extremals and Global Analysis ---------------------------------

Philip Schrader (UWA)

Maths LR2 Thursday February 24, 3:00pm

We define conditional extremal curves on a Riemannian manifold $M$ as the critical points of the the $L^{2}$ distance between the tangent vector field and a `prior' vector field. The natural setting for proving necessary and sufficient conditions for existence of these curves is critical point theory on Hilbert manifolds of curves. I will briefly describe these manifolds and show how the geometry developed by Eliasson for such manifolds can be used to calculate the Euler-Lagrange equation and Hessian of this $L^{2}$ distance. I will also show that this distance satisfies a substitute for compactness known as the Palais-Smale condition, thus establishing the existence of critical points. If time permits, I will present a Morse-index theorem and some results on multiplicity of critical points. -----------------------------------------------------------------

Cubics and Negative Curvature ------------------------

Michael Pauley (UWA)

Maths LR2 Thursday February 24, 4:00pm

Abstract: Riemannian cubics are curves that generalise cubic polynomials to arbitrary Riemannian manifolds, in the same way that geodesics generalise straight lines. In any complete Riemannian manifold, geodesics can be extended indefinitely. In this talk I will discuss the question of whether Riemannian cubics can be extended indefinitely. The sectional curvature of the manifold plays a role. Assumed knowledge: definition of smooth manifolds, smooth maps and vector fields.

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Convener,

Lyle Noakes 3358