*By Prof. PATRICK M. FITZPATRICK
(Department of Mathematics, University of Maryland, USA*)

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.)

### More Information:

**For more info contact: **Prof. Athanasios Tzavaras: email: athanasios.tzavaras@kaust.edu.sa

Date: Tuesday 31

^{st} Jan 2017

Time: 02:30 PM - 03:30 PM

Location: Building 1, Level 4, Room 4214

Refreshments will be Provided at 2:30pm