"3-manifolds and their invariants"의 두 판 사이의 차이

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23번째 줄: 23번째 줄:
 
** E^3, E\times H^2, SL_2
 
** E^3, E\times H^2, SL_2
 
** H^3, Nil
 
** H^3, Nil
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47번째 줄: 43번째 줄:
 
*  Prove<br><math>\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt=\frac{2}{\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))=\frac{2}{\sqrt{7}}(Cl(2\pi /7})+Cl(4\pi/7})-Cl(6\pi/7}))</math><br>
 
*  Prove<br><math>\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt=\frac{2}{\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))=\frac{2}{\sqrt{7}}(Cl(2\pi /7})+Cl(4\pi/7})-Cl(6\pi/7}))</math><br>
 
* [[a log tangent integral]]<br>
 
* [[a log tangent integral]]<br>
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55번째 줄: 49번째 줄:
  
 
<h5 style="line-height: 2em; margin: 0px;">Reshetikihn, Turaev</h5>
 
<h5 style="line-height: 2em; margin: 0px;">Reshetikihn, Turaev</h5>
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84번째 줄: 76번째 줄:
 
==== 하위페이지 ====
 
==== 하위페이지 ====
  
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* [[threefolds and their invariants|hyperbolic 3-manifold]]<br>
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* [[threefolds and their invariants]]<br>
 
** [[Chern-Simons gauge theory and invariant|Chern-Simons invariant]]<br>
 
** [[Chern-Simons gauge theory and invariant|Chern-Simons invariant]]<br>
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** [[Kashaev's volume conjecture|Kashaev's volume Conjecture]]<br>
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** [[Gieseking's constant]]<br>
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** [[mathematics of x^3-x+1=0]]<br>
 
** [[triangulations and Bloch group]]<br>
 
** [[triangulations and Bloch group]]<br>
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** [[volume of hyperbolic threefolds and L-values|volume of hyperbolic 3-manifolds and L-values]]<br>
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** [[volume of hyperbolic threefolds and L-values]]<br>
  
 
 
 
 
124번째 줄: 117번째 줄:
  
 
[[4909919|]]
 
[[4909919|]]
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<h5 style="line-height: 2em; margin: 0px;">expositions</h5>
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*  Arithmetic properties of quantum invariants of manifolds http://www.mathnet.ru/php/presentation.phtml?presentid=3937&option_lang=rus<br>
  
 
 
 
 
139번째 줄: 142번째 줄:
 
** Don Zagier, Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월
 
** Don Zagier, Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월
  
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* [[2010년 books and articles|논문정리]]
 
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* http://www.ams.org/mathscinet
 
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* http://www.zentralblatt-math.org/zmath/en/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html][http://www.ams.org/mathscinet ]
 
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* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
 
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* http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
 
 
* http://dx.doi.org/10.1063/1.3085764
 
* http://dx.doi.org/10.1063/1.3085764
  

2011년 11월 4일 (금) 00:47 판

introduction
  • volume of knot complements
  • Chern-Simons invariant of manifolds
  • Turaev-Viro invariant (related to 6j symbols)
    • Kauffman and Line 'The Temperley Lie algebra recoupling theory and invariants of 3-manifolds"
    • Turaev-Viro "state sum invariants of 3-manifolds and quantum 6j-symbols)

 

 

maps between threefolds
  • maps between aspherical 3 manifolds
  • aspherical threefolds = second and higher homotopy groups vanish
  • JSJ decomposition http://en.wikipedia.org/wiki/JSJ_decomposition
    • cutting M into
      • Seifert fibered pieces ~ non hyperbolic pieces
      • atoroidal pieces ~ hyperbolic pieces
  • Thurston's geometrization
    • S^3, E\times S^2, Sol
    • E^3, E\times H^2, SL_2
    • H^3, Nil

 

 

Volume of knot complement
  1. KnotData[]
    KnotData["FigureEight", "HyperbolicVolume"]
    N[%, 20]
  • Dedekind zeta funciton evaluated at 2 gives a number related to volume of 3-manifold
  • Bloch-Wigner dilogarithm is involved

 

 

a problem
  • Prove
    \(\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt=\frac{2}{\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))=\frac{2}{\sqrt{7}}(Cl(2\pi /7})+Cl(4\pi/7})-Cl(6\pi/7}))\)
  • a log tangent integral

 

 

Reshetikihn, Turaev

 

 

software

 

 

history

 

 

 

하위페이지

 

 

related items

 

 

encyclopedia

 

 

books

 

[[4909919|]]

 

 

 

expositions

 

 

articles

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

experts on the field

 

 

links
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