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AMCS Seminar : Prof. Patrick Fitzpatrick, Univ. of Maryland, USA

1/31/2017

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â€‹Many physical problems can be modeled by the description of the solutions of a one-parameter family of nonlinear operator equations. Examples go back as far as the analysis by Cauchy in 1744 of the bending of a rod under the application of an external load. As a general set-up, consider a nonlinear operator F : R X ! X, where X is a normed linear space, and solutions of the path of nonlinear equations F(; x) = 0; under the assumption that x = 0 is a solution for every parameter : At certain parameters there is bifurcation of nonzero solutions of the above path of equations. For each ; suppose the linear operator L : X ! Y is the linearization at x = 0 of the nonlinear mapping x 7! F(; x): Variations of the Implicit Function Theorem have long been used to deduce conclusions regarding bifurcation from properties of the path 7! L of linearizations. In contrast, the aim of this talk is to describe criteria related to topological properties of paths of mappings and paths of linear operators which force bifurcation of nonzero solutions. There are criteria which are quite di erent from those arising using more classical techniques. In the special case of bifurcation of critical points which occurs when F is a gradient (so F(; x) = rx(; x)) each L is symmetric, in which case for the path ! L there is dened a concept of spectral ow. Properties of spectral ow and its relevance to bifurcation will be described. Examples of bifurcation for paths of di erential equations will be discussed. Technical details will be passed over and no specialized background in topology will be assumed. Much of all this is related to work of the speaker with Jacobo Pejsachowicz. Some pertinent references may be found in a recent paper of the speaker and James Alexander (Spectral Flow is a complete invariant for detecting bifurcation of critical points, Trans. Amer. Math Soc. 368 (2016), 4439-4459.)

Date: Tuesday 31st Jan 2017

Time: 02:30 PM - 03:30 PM

Location: Building 1, Level 4, Room 4214

Refreshments will be Provided at 2:30pm