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SUMMARY:Georgy Sharygin
DTSTART:20241106T162000Z
DTEND:20241106T180000Z
DTSTAMP:20260423T010458Z
UID:GDEq/117
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GDEq/117/">R
 emarkable properties of the full symmetric Toda system</a>\nby Georgy Shar
 ygin as part of Geometry of differential equations seminar\n\nLecture held
  in room 303 of the Independent University of Moscow.\n\nAbstract\nA full 
 symmetric Toda system is a Hamiltonian dynamical system on the space of sy
 mmetric real matrices with zero trace\, generalizing the usual open Toda c
 hain. This system is given by the Lax equation $\\dot L=[L\,M(L)]$\, where
  $M(L)$ is the (naive) antisymmetrization of the symmetric matrix $L$: the
  difference of its super and subdiagonal parts (with zeros on the diagonal
 ). The Hamiltonianity of this system comes from the identification of the 
 space of symmetric matrices with the space dual to the algebra of upper tr
 iangular matrices\, with the Hamilton function being $1/2Tr(L^2)$. This sy
 stem can be further generalized to obtain systems on the spaces of "genera
 lized symmetric matrices"\, the symmetric components of the Cartan expansi
 on of the semi-simple real Lie algebras. In a somewhat unexpected way\, al
 l these systems turn out to be integrable (in the sense of having a suffic
 iently large commutative algebra of first integrals) and possess a number 
 of remarkable properties which I will discuss: their trajectories always c
 onnect fixed points corresponding to the elements of the Weyl group of the
  original Lie algebra\, and two such points are connected if and only if t
 he elements of the Weyl group are comparable in Bruhat order\; in the case
  of a system on spaces of generalized symmetric matrices\, this property a
 llows one to describe the intersections of the real Bruhat cells\; this sy
 stem has a large set of symmetries (sufficient for it to be Lie-Bianchi in
 tegrable)\; its additional first integrals can be obtained by a "cut" proc
 edure\, and the trajectories of the corresponding Hamiltonian fields can b
 e obtained by the QR decomposition\; if time permits\, I will describe alt
 ernative families of first integrals (commutative and non-commutative)\; f
 inally\, I will describe a way to lift the extra first integrals of the "c
 ut" into the universal enveloping algebra with commutativity preserved.\n\
 nThe talk is based on a series of works by the author jointly with Yu. Che
 rnyakov\, A. Sorin and D. Talalaev.\n
LOCATION:https://researchseminars.org/talk/GDEq/117/
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