2pm - 3pm
Wednesday 26 June 2019

The Maslov index and the spectrum of differential operators

Speaker: Yuri Latushkin (Missouri)


We will review some recent results on connections between the Maslov and the Morse indices
for differential operators. The Morse index is a spectral quantity defined as the number of
negative eigenvalues counting multiplicities while the Maslov index is a geometric
characteristic defined as the signed number of intersections of a path in the space of
Lagrangian planes with the train of a given plane. The problem of relating these two quantities
is rooted in Sturm's Theory and has a long history going back to the classical work by Arnold,
Bott and Smale, and has attracted recent attention of several groups of mathematicians.

We will briefly mention how the relation between the two indices helps to prove the fact
that a pulse in a gradient system of reaction diffusion equations is unstable.
We will also discuss a fairly general theorem relating the indices for a broad class
of multidimensional elliptic selfadjoint operators. Connections of the Maslov index and
Hadamard-type formulas for the derivative of eigenvalues will be also discussed.

This talk is based on a joint work with M. Beck, G. Cox, C. Jones, P. Howard, R. Marangell,
K. McQuighan, A. Sukhtayev, and S. Sukhtaiev.


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