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SUMMARY:Georgy Sharygin
DTSTART:20261028T162000Z
DTEND:20261028T180000Z
DTSTAMP:20261025T080008Z
UID:GDEq/163
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GDEq/163/">T
 he full symmetric Toda system\, Lie-Bianchi integrability and the "stochas
 tic" group</a>\nby Georgy Sharygin as part of Geometry of differential equ
 ations seminar\n\nLecture held in room 303 of the Independent University o
 f Moscow.\n\nAbstract\nIn my talk I will continue to tell the story of the
  full symmetric Toda system. It is a rather straightforward generalisation
  of the usual "tridiagonal" open Toda chain: the corresponding equation ha
 s the form $\\dot L= [M\,L]$\, where $L$ is a real symmetric matrix and $M
  = M(L)$ is the "naive" antisymmetrisation of $L$. This system is Hamilton
 ian and Liouville integrable. In a recent paper with D.Talalaev and Yu.Che
 rnyakov we proved that this system is also integrable in the sense of the 
 Lie–Bianchi criterion\, its symmetries forming the Lie algebra of the so
 -called "stochastic" group: the group of invertible matrices that preserve
  the vector (or convector) (1\,1\,...\,1). In this talk (also based on a j
 oint work with D.Talalaev and Yu.Chernyakov) we describe how the action of
  the stochastic Lie algebra integrates to an action of the corresponding g
 roup itself\, an action that preserves the trajectories of the Toda system
 . It turns out that every orbit of this action contains a tridiagonal symm
 etric matrix\, so that solving the full symmetric Toda system reduces to s
 olving the open Toda lattice. Moreover\, the passage from a full symmetric
  matrix to a tridiagonal one can be carried out by means of the Lanczos-Kr
 ylov transformation\, a rather explicit and simple numerical procedure.\n
LOCATION:https://researchseminars.org/talk/GDEq/163/
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