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SUMMARY:Alexander Zheglov
DTSTART:20241113T162000Z
DTEND:20241113T180000Z
DTSTAMP:20260423T023008Z
UID:GDEq/118
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GDEq/118/">N
 ormal forms for differential operators</a>\nby Alexander Zheglov as part o
 f Geometry of differential equations seminar\n\nLecture held in room 303 o
 f the Independent University of Moscow.\n\nAbstract\nIn my talk I'll give 
 an overview of the results obtained by me\, as well as jointly with co-aut
 hors\,  related to the problem of classifying commuting (scalar) differen
 tial\, or more generally\, differential-difference or  integral-different
 ial operators in several variables.\n\nConsidering such rings as subrings 
 of a certain complete non-commutative ring $\\hat{D}_n^{sym}$ (not the kno
 wn  ring of formal pseudo-differential operators!)\, the normal forms of 
 differential operators mentioned in the title are obtained after conjugati
 on by some invertible operator ("Schur operator")\, calculated with the he
 lp of one of the operators in a ring. Normal forms of  <i>commuting</i> op
 erators  are polynomials with constant coefficients in the differentiatio
 n\, integration and shift operators\, which have a finite order in each va
 riable\, and can be effectively calculated for any given commuting operato
 rs.\n\nI'll talk about some recent applications of the theory of normal fo
 rms:  an effective parametrisation of torsion free sheaves with vanishing
  cohomologies on a projective curve\, and a correspondence between solutio
 ns to the string equation and pairs of commuting ordinary differential op
 erators  of rank one.\n
LOCATION:https://researchseminars.org/talk/GDEq/118/
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