Codimension 2

From Mathematics
Jump to: navigation, search
(Schedule)
 
Line 26: Line 26:
 
| 27.4.
 
| 27.4.
 
|[[#Cech cohomology and the Hodge conjecture in codim 1|Cech cohomology and the Hodge conjecture in codim 1]]
 
|[[#Cech cohomology and the Hodge conjecture in codim 1|Cech cohomology and the Hodge conjecture in codim 1]]
|N.N.
+
|Vetere
 
|-
 
|-
 
| 4.5.
 
| 4.5.
 
|[[#Chow groups and Chern classes|Chow groups and Chern classes]]
 
|[[#Chow groups and Chern classes|Chow groups and Chern classes]]
|N.N.
+
|Bertini
 
|-
 
|-
 
| 11.5.
 
| 11.5.
 
|[[#Elementary algebraic K-Theory: K0 and K1|Elementary algebraic K-Theory: K0 and K1]]
 
|[[#Elementary algebraic K-Theory: K0 and K1|Elementary algebraic K-Theory: K0 and K1]]
|N.N.
+
|Schwald
 
|-
 
|-
 
| 18.5.
 
| 18.5.
Line 42: Line 42:
 
| 25.5.
 
| 25.5.
 
|[[#Elementary algebraic K-Theory: K2|Elementary algebraic K-Theory: K2]]
 
|[[#Elementary algebraic K-Theory: K2|Elementary algebraic K-Theory: K2]]
|N.N.
+
|Zaccanelli
 
|-
 
|-
 
| 1.6.
 
| 1.6.
 
|[[#Cycles in codim 2|Cycles in codim 2, part I]]
 
|[[#Cycles in codim 2|Cycles in codim 2, part I]]
|N.N.
+
|Voelkel
 
|-
 
|-
 
| 8.6.
 
| 8.6.
 
|[[#Cycles in codim 2|Cycles in codim 2, part II]]
 
|[[#Cycles in codim 2|Cycles in codim 2, part II]]
|N.N.
+
|Kelly
 
|-
 
|-
 
| 15.6.
 
| 15.6.
 
|[[#The dilogarithm|The dilogarithm]]
 
|[[#The dilogarithm|The dilogarithm]]
|N.N.
+
|Bergner, Sartori
 
|-
 
|-
 
| 22.6.
 
| 22.6.
 
|[[#A generalization of the exponential sequence|A generalization of the exponential sequence]]
 
|[[#A generalization of the exponential sequence|A generalization of the exponential sequence]]
|N.N.
+
|Soergel
 
|-
 
|-
 
| 29.6.
 
| 29.6.
 
|[[#The dilog, scissors congruences and volumes of hyperbolic spaces|The dilog, scissors congruences and volumes of hyperbolic spaces]]
 
|[[#The dilog, scissors congruences and volumes of hyperbolic spaces|The dilog, scissors congruences and volumes of hyperbolic spaces]]
|N.N.
+
|Lye
 
|-
 
|-
 
| 6.7.
 
| 6.7.
 
|[[#The dilog and zeta functions|The dilog and zeta functions]]
 
|[[#The dilog and zeta functions|The dilog and zeta functions]]
|N.N.
+
|Eberhardt
 
|-
 
|-
 
| 13.7.
 
| 13.7.
 
|[[#The dilog in physics|The dilog in physics]]
 
|[[#The dilog in physics|The dilog in physics]]
|Wendland/Scheidegger ?
+
|Scheidegger
 
|-
 
|-
 
| 20.7.
 
| 20.7.
 
|[[#Outlook: Mixed motives and periods|Outlook: Mixed motives and periods]]
 
|[[#Outlook: Mixed motives and periods|Outlook: Mixed motives and periods]]
|Huber/Hörmann
+
|Hörmann
 
|}
 
|}
  
All talks have 60 min + 30 min time for discussions.
 
  
 
== General comments ==
 
== General comments ==
  
The following talk descriptions are merely '''suggestions'''. People should feel free to change the content of their talk as they like and concentrate rather on things they found interesting during the preparation. This should not be a problem as the talks are relatively independent. There are some restrictings, however:  
+
All talks have 60 min + 30 min time for discussions.
 +
 
 +
The following outlines of the talks are merely '''suggestions'''. People should feel free to change the content of their talk as they like and concentrate rather on things they found interesting during the preparation. This should not be a problem as the talks are relatively independent. There are some restrictions, however:  
  
 
* The topics mentioned in the description of talks 2-5 (though largely independent from each other) are needed for talks 6, 7 and 9. In case of doubt ask me or the speakers of these talks whether they need something.  
 
* The topics mentioned in the description of talks 2-5 (though largely independent from each other) are needed for talks 6, 7 and 9. In case of doubt ask me or the speakers of these talks whether they need something.  
* Talk 8 (introduction to the dilog) is needed for the remaining talks on the dilog.
+
* Talk 8 (The dilogarithm) is needed for the remaining talks on the dilog.
  
 
= Introduction =
 
= Introduction =
Line 101: Line 102:
 
<math> 0 \rightarrow 2\pi i \mathbb{Z}_X \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X^* \rightarrow 1</math>
 
<math> 0 \rightarrow 2\pi i \mathbb{Z}_X \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X^* \rightarrow 1</math>
 
* Proof of the Hodge conjecture for divisors using the long exact sequence associated with the exponential sequence
 
* Proof of the Hodge conjecture for divisors using the long exact sequence associated with the exponential sequence
* Need also Lemma 2.1<ref name="Blo74"/> on a non-Abelian version of the long exact sequence.
+
* Explain also Lemma 2.1<ref name="Blo74"/> on a non-Abelian version of the long exact sequence. (This will be needed later)
  
 
= Chow groups and Chern classes =
 
= Chow groups and Chern classes =
Line 109: Line 110:
 
* Definition of the Chow groups of a smooth variety X
 
* Definition of the Chow groups of a smooth variety X
 
* A survey over its properties, including the intersection pairing
 
* A survey over its properties, including the intersection pairing
* Cycle class map to Cohomology for varieties over <math>\mathbb{C}</math>.  
+
* Cycle class map to singular cohomology for varieties over <math>\mathbb{C}</math>.  
 
* Splitting principle: Theorem 5.19<ref name="Wei13"/>  
 
* Splitting principle: Theorem 5.19<ref name="Wei13"/>  
* Definition of Chern classes using this (Reference?)
+
* Definition of algebraic Chern classes using the splitting prinicple <ref name="Gro58"/>, axiomatic characterization
 +
* Mention also the compatibility (via the cycle class map to singular cohomology) with the complex analytic construction of Chern classes using metrics and connections
 
* The isomorphism:
 
* The isomorphism:
  
Line 120: Line 122:
 
= Elementary algebraic K-Theory: <math>K_0</math> and <math>K_1</math> =
 
= Elementary algebraic K-Theory: <math>K_0</math> and <math>K_1</math> =
  
Also the relation of algebraic K_0 with the gamma-filtration on K_0 ?
 
 
 
* The definition of <math>K_0(R)</math> for a ring <math>R</math> and <math>K_0(X)</math> for a variety <math>X</math>.
 
* The definition of <math>K_0(R)</math> for a ring <math>R</math> and <math>K_0(X)</math> for a variety <math>X</math>.
 +
* Chern classes are maps from <math>K_0(X)</math>
 +
* Also mention that there is a map
 +
 +
<math>\{\text{codim } n \text{ cycles}\} \rightarrow K_0(X) </math>
 +
 +
which is right inverse to the <math>n</math>-th Chern class up to a rational factor.
 
* The Whitehead Lemma, 3.1<ref name="Mil71"/> and the definition of <math>K_1(R)</math>
 
* The Whitehead Lemma, 3.1<ref name="Mil71"/> and the definition of <math>K_1(R)</math>
 
* The localization exact sequence, if <math>R</math> is a Dedekind ring, <math>K</math> its quotient field:
 
* The localization exact sequence, if <math>R</math> is a Dedekind ring, <math>K</math> its quotient field:
Line 129: Line 135:
  
 
References: §1-§5<ref name="Mil71"/>,
 
References: §1-§5<ref name="Mil71"/>,
II, §8<ref name="Wei13"/>, Chapter I<ref name="Sri91"/>.
+
II, §8<ref name="Wei13"/>, Chapter I<ref name="Sri91"/>, 1.1-1.2 and 2.1-2.2<ref name="Ros94"/>
  
 
= Elementary algebraic K-Theory: <math>K_2</math> =
 
= Elementary algebraic K-Theory: <math>K_2</math> =
Line 151: Line 157:
 
<math>K_2^M(R) \cong K_2(R)</math>
 
<math>K_2^M(R) \cong K_2(R)</math>
  
for fields (cf. §17, §18<ref name="Mil71"/>).
+
for fields (cf. §11, §12<ref name="Mil71"/>).
  
References: §5-§12<ref name="Mil71"/>, Chapter III<ref name="Wei13"/>, Chapter I<ref name="Sri91"/>.
+
* extension of the localization squence to <math>K_2</math> (the tame symbol).
 +
 
 +
References: §5-§12<ref name="Mil71"/>, Chapter III<ref name="Wei13"/>, Chapter I<ref name="Sri91"/>, Chapter 4<ref name="Ros94"/>.
 
A nice collection of facts about <math>K_2</math> can be found here: <ref name="Dal06"/>.
 
A nice collection of facts about <math>K_2</math> can be found here: <ref name="Dal06"/>.
  
Line 180: Line 188:
 
of Cech cohomology groups. The purpose of the rest of the two talks is to see that  
 
of Cech cohomology groups. The purpose of the rest of the two talks is to see that  
  
<math> H^2(X, K_2(\mathcal{O}_X)) \cong CH_2(X)</math>
+
<math> H^2(X, K_2(\mathcal{O}_X)) \cong CH^2(X)</math>
  
 
in such a way that <math>CK_2</math> becomes identified with the second Chern class as explained in the third talk.  
 
in such a way that <math>CK_2</math> becomes identified with the second Chern class as explained in the third talk.  
Line 207: Line 215:
 
* Introduction of the dilogarithm via integral formula and power series <ref name="Zag07"/>
 
* Introduction of the dilogarithm via integral formula and power series <ref name="Zag07"/>
 
* Multivalence (dependence of paths of integration) and how this problem is solved by considering the values as a cosets of certain unipotent matrices. Present the ideas from <ref name="Ram82"/> but restricted to the dilogarithm. (A theoretical, more modern explanation in terms of mixed Hodge structures can be found in <ref name="Blo91"/> but this should not be presented this way (yet).)  
 
* Multivalence (dependence of paths of integration) and how this problem is solved by considering the values as a cosets of certain unipotent matrices. Present the ideas from <ref name="Ram82"/> but restricted to the dilogarithm. (A theoretical, more modern explanation in terms of mixed Hodge structures can be found in <ref name="Blo91"/> but this should not be presented this way (yet).)  
* Introduce the Bloch-Wigner dilogarithm (a single valued non-analytic variant of the dilog)
+
* Introduce the Bloch-Wigner dilogarithm (a single valued real-analytic variant of the dilog)
 
* The five term relation and special values
 
* The five term relation and special values
 
** Present Theorem 7.4.4 from <ref name="Blo10"/> and its proof
 
** Present Theorem 7.4.4 from <ref name="Blo10"/> and its proof
Line 213: Line 221:
 
** deduce special values as presented in <ref name="Zag07"/>, I, 1
 
** deduce special values as presented in <ref name="Zag07"/>, I, 1
  
References: <ref name="Blo10"/>, <ref name="Ram82"/>, <ref name="Blo00"/>, in particular 7.4 and all what is needed from before, <ref name="Zag07"/>
+
References: <ref name="Blo10"/>, <ref name="Ram82"/>, <ref name="Blo00"/>, in particular 7.4 and all what is needed from before, <ref name="Zag07"/>, <ref name="Hai94"/>
  
 
There should/could be some overlap with the following two talks; speakers should coordinate; some material can be shifted to the talk on the relation with hyperbolic geometry
 
There should/could be some overlap with the following two talks; speakers should coordinate; some material can be shifted to the talk on the relation with hyperbolic geometry
Line 229: Line 237:
 
References: <ref name="Blo78"/>, (<ref name="Blo10"/>, §6)
 
References: <ref name="Blo78"/>, (<ref name="Blo10"/>, §6)
  
Reference for the Dilogarithm: <ref name="Zag07"/>
+
Reference for the Dilogarithm: <ref name="Zag07"/>, <ref name="Hai94"/>
  
 
= The dilog, scissors congruences and volumes of hyperbolic spaces =
 
= The dilog, scissors congruences and volumes of hyperbolic spaces =
  
References: <ref name="DS83"/>, <ref name="Gon99"/>
+
References: <ref name="DS83"/>, <ref name="Gon99"/>, <ref name="Hai94"/>
  
 
More advanced reference: <ref name="Gon95"/>.
 
More advanced reference: <ref name="Gon95"/>.
Line 241: Line 249:
 
= The dilog and zeta functions =
 
= The dilog and zeta functions =
  
References: <ref name="Blo00"/>, (<ref name="Zag07"/>, I, §5)
+
References: <ref name="Blo00"/>, (<ref name="Zag07"/>, I, §5), <ref name="Hai94"/>
  
 
More advanced reference: <ref name="Gon95"/>.
 
More advanced reference: <ref name="Gon95"/>.
Line 256: Line 264:
 
Connection with the topics discussed in the seminar.
 
Connection with the topics discussed in the seminar.
  
References: <ref name="Lev05"/>, <ref name="BD94"/>, <ref name="Blo91"/>, <ref name="Gon95"/>
+
References: <ref name="Lev05"/>, <ref name="BD94"/>, <ref name="Blo91"/>, <ref name="Gon95"/>, <ref name="Hai94"/>
  
 
= References =
 
= References =
Line 319: Line 327:
 
''Algebraic K-theory''.
 
''Algebraic K-theory''.
 
Progress in Mathematics, 90. Birkhäuser Boston, Inc., Boston, MA, 1991. xvi+314 pp.</ref>
 
Progress in Mathematics, 90. Birkhäuser Boston, Inc., Boston, MA, 1991. xvi+314 pp.</ref>
 +
 +
<ref name="Ros94">Rosenberg, J.;
 +
''Algebraic K-theory and its applications''.
 +
Graduate Texts in Mathematics, 147. Springer-Verlag, New York, 1994. x+392 pp.</ref>
 +
 +
<ref name="Gro58">Grothendieck, A.
 +
''La théorie des classes de Chern''. (French)
 +
Bull. Soc. Math. France 86 1958 137–154. </ref>
 +
 +
<ref name="Hai94">Hain,R.M.; ''Classical Polylogarithms'', Motives, in Motives (Seattle, WA, 1991), manual, vol. 55 (1994), pp. 3-42, American Mathematical Society</ref> 
  
 
  </references>
 
  </references>

Latest revision as of 17:58, 25 July 2016

Personal tools
Namespaces

Variants
Actions
Navigation
Toolbox