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mzveng [2016/03/06 22:42]
Уланский Евгений Алесандрович 		mzveng [2016/05/08 23:00]
Уланский Евгений Алесандрович [Course programm]
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 ======Course programm====== ======Course programm======
  
-1) Zeta values. ​Basel problem and its solution by Leonard Euler.+1) Zeta values. ​Multiple polylogarithms. Differential equations. 
  
-2) Roger Apery’s theorem on irrationality ​of ζ(3).+2)      ​Integral representation and analitic continuation. General integral ​of hypergeometric type.
  
-3) Tanguy Rivoal and Keith Ball’s theorem on irrationality of  ​ζ(2n+1) for infinitely many n.+3) Formulae for ζ(2n).
  
-4) Wadim Zudilin’s theorem on irrationality ​of at least one of four numbers ​ ζ(5),​ζ(7),​ζ(9),​ζ(11).+4) Closed formulae for values ​of dilogarithm and trilogarithm.
  
-5) Closed formulae for zeta values.+5) Periodic ​zeta values.
  
-6) Multiple zeta values (MZV) and generalized polylogarithms. Weight and length. Classical polylogarithms. Euler formulae ​for MZV including ​ζ(2,​1)=ζ(3).+6) Duality for MZVs. Euler formula ​for MZVs. Special case ζ(2,​1)=ζ(3).
  
-7) Closed formulae ​for MZV and some values ​of generalized polylogarithms.+7) Standard relations ​for MZVs. Proof of shuffle relations.
  
-8) Standard ​relations ​for MZV.+8)      Proof of stuffle ​relations.
  
-9) Michael Hoffman relations and their connection with standard relations.+9) Michael Hoffman relations and their connection with standard relations. Proof.
  
-10) Integral representations ​for MZV and generalized polylogarithms.+10) Sum formula ​for MZVs. Proof of Okuda and Ueno.
  
-11) Sum relation for MZV. Duality for MZV.+11)     Igarashi'​s proof of sum formula.
  
-12) Yasuo Ohno relations for MZV and their connection with sum formula and duality.+12) Yasuo Ohno relations for MZVs and their connection with sum formula and duality. Scetch of proof à la Igarashi. 
  
-13) Transformations ​-z/1-z and 1-z for generalized polylogarithms.+13) Transformation ​-z/1-z for generalized polylogarithms.
  
-14) Linear independence of generalized polylogarithms. Algebraic independence of classical polylogarithms. +14) Transformation 1-z for generalized polylogarithms.
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-15) Colored generalized polylogarithms. Connection between different integral representations of generalized polylogarithms. +
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-16) Identities ​for integrals of hypergeometric type. Consequences for generalized polylogarithms. +
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-17) Linear spaces generated by values of generalized polylogarithms ​of fixed weight.+