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SUMMARY:Georgy Sharygin
DTSTART:20250319T162000Z
DTEND:20250319T180000Z
DTSTAMP:20260423T010459Z
UID:GDEq/129
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GDEq/129/">G
 eometry of the full symmetric Toda system</a>\nby Georgy Sharygin as part 
 of Geometry of differential equations seminar\n\nLecture held in room 303 
 of the Independent University of Moscow.\n\nAbstract\nFull symmetric Toda 
 system is the Lax-type system $\\dot L=[M(L)\,L]$\, where the variable $L$
  is a real symmetric $n\\times n$ matrix and $M(L)=L_+-L_-$ denotes its "
 naive" anti-symmetrisation\, i.e. the matrix constructed by taking the dif
 ference of strictly upper- and lower-triangular parts $L_+$ and $L_-$ of $
 L$. This system has lots of interesting properties: it is a Liouville-in
 tegrable Hamiltonian system (with rational first integrals)\, it is also 
 super-integrable (in the sense of Nekhoroshev)\, its singular points and t
 rajectories represent the Hasse diagram of Bruhat order on permutations gr
 oup. Its generalizations to other semisimple real Lie algebras have simila
 r properties. In my talk I will sketch the proof of some of these properti
 es and will describe a construction of infinitesimal symmetries of the Tod
 a system. It turns out that there are many such symmetries\, their constru
 ction depends on representations of $\\mathfrak{sl}_n$. As a byproduct w
 e prove that the full symmetric Toda system is integrable in the sense of 
 Lie-Bianchi criterion.\n\nThe talk is based on a series of papers joint wi
 th Yu.Chernyakov\, D.Talalaev and A.Sorin.\n
LOCATION:https://researchseminars.org/talk/GDEq/129/
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