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= Motivation = <ref name="Blo81"/> <ref name="Goe07"/> <ref name="Gon95"/> <ref name="Lew81"/> <ref name="Sus84"/> <ref name="Zic15"/> <ref name="Sus87"/> Not: <ref name="Hut13"/> but has an informative introduction... In the 1970's a beautiful story began, when mathematicians, with Spencer Bloch leading the way, began to discover a beautiful connection between algebraic K-Theory, algebraic cycles and polylogarithms. The first work was restricted to the case of codimension 2 cycles, K_2, and the dilogarithm (this next polylogarithm after the usual logarithm) and '''quite explicit'''. This story, for the next 35 years until now, triggered an enormous amount of research and was one of the motivations for many modern (and quite abstract) developments and conjectures, as for example the theory of higher algebraic K-Theory (Quillen), mixed motives (Voevodsky - the ideas going back to Grothendieck), motivic cohomology, Bloch-Kato conjectures, Beilinson conjectures ... The intention of this seminar is to study the beginnings of this subject as they evolved historically. The hope is that people might appreciate and understand better many of the recent developments in the field when they have some "easy" explicit examples in mind that go just a little bit beyound the relations between line bundles, codimension 1 cycles (divisors), invertible functions and the logarithm that everybody knows well. (Nevertheless the latter will be rediscussed in the first three talks). While the first half of the seminar focuses on the explicit relation between (algebraic) K_2, the second (algebraic) Chern class of vector bundles and codimension 2 cycles the second half is devoted to the '''dilogarithm function''' which enters these relations in various ways. The dilogarithm became quite famous in many other areas of mathematics recently. Some of those application we want to discuss as well. For instance, the relation with zeta functions, the applications in physics, and in hyperbolic geometry (all of course not completely unrelated with each other). Let me quote Don Zagier <ref name="Zag07"/>: ''Almost all of its ''[of the dilogarithm]'' appearances in mathematics, and almost all the formulas relating to it, have something of the fantastical in them, as if this function alone among all others possessed a sense of humor.'' = Schedule = {| border="1" style="text-align:left;" | 20.4. |[[#Introduction|Introduction]] |Braeunling |- | 27.4. |[[#Cech cohomology and the Hodge conjecture in codim 1|Cech cohomology and the Hodge conjecture in codim 1]] |N.N. |- | 4.5. |[[#Chow groups and Chern classes|Chow groups and Chern classes]] |N.N. |- | 11.5. |[[#Elementary algebraic K-Theory: K0 and K1|Elementary algebraic K-Theory: K0 and K1]] |N.N. |- | 18.5. |Pentecost | |- | 25.5. |[[#Elementary algebraic K-Theory: K2|Elementary algebraic K-Theory: K2]] |N.N. |- | 1.6. |[[#Cycles in codim 2|Cycles in codim 2, part I]] |N.N. |- | 8.6. |[[#Cycles in codim 2|Cycles in codim 2, part II]] |N.N. |- | 15.6. |[[#The dilogarithm|The dilogarithm]] |N.N. |- | 22.6. |[[#A generalization of the exponential sequence|A generalization of the exponential sequence]] |N.N. |- | 29.6. |[[#The dilog, scissors congruences and volumes of hyperbolic spaces|The dilog, scissors congruences and volumes of hyperbolic spaces]] |N.N. |- | 6.7. |[[#The dilog and zeta functions|The dilog and zeta functions]] |N.N. |- | 13.7. |[[#The dilog in physics|The dilog in physics]] |Wendland/Scheidegger ? |- | 20.7. |[[#Outlook: Mixed motives and periods|Outlook: Mixed motives and periods]] |Huber/Hörmann |} = Introduction = * Motivation for alg. K-Theory (with review of topological K-Theory) * Cycle classes of subvarieties in (singular) cohomology * Formulation of the Hodge conjecture * Overview on the topics of the seminar = Cech cohomology and the Hodge conjecture in codim 1 = Definition of vector bundles by Cech cocycles - Weibel, p. 54 ff. I.5.10 * Need also Lemma 2.1<ref name="Blo74"/> on a non-Abelian version of the long exact sequence. = Chow groups and Chern classes = The classical construction of Chow groups and the Chern classes using the splitting principle. = Elementary algebraic K-Theory: K0 and K1 = Also the relation of algebraic K_0 with the gamma-filtration on K_0 ? References: <ref name="Mil71"/> <ref name="Wei13"/> Weibel II, §8 = Elementary algebraic K-Theory: K2 = * Definition of <math>K_2</math> via Steinberg relations * Introduction of Milnor K-groups <math>K_n^M</math> and proof of the statement <math>K_2^M(R) \cong K_2(R)</math> for fields and local rings References: <ref name="Mil71"/> <ref name="Wei13"/> Weibel, III = Cycles in codim 2 = '''These are two talks.''' * Explain how <math>K_2</math>, and <math>St</math> (Steinberg group) can be sheafified * Prove Proposition 1.13<ref name="Blo74"/> * Remind of Lemma 2.1<ref name="Blo74"/> which has been discussed in the talk on Cech cohomology * Deduce the long exact sequence associated with the sequence of sheaves <math> 1 \rightarrow K_2 \rightarrow St \rightarrow \mathrm{SL} \rightarrow 1 </math> and obtain a map <math> CK_2: H^1(X, \mathrm{SL}(\mathcal{O}_X)) \rightarrow H^2(X, K_2(\mathcal{O}_X)) </math> 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> in such a way that <math>CK_2</math> becomes identified with the second Chern class as explained in the third talk. * Prove Proposition 2.4, Corollary 2.7, and Proposition 2.8 (maybe omitting some details, and presenting concrete examples instead) References: <ref name="Blo74"/>, (<ref name="Blo10"/>, §4) * Bloch's construction (§3<ref name="Blo10"/>) of a map <math>H^2(X, K_2) \rightarrow CH^2(X)</math> * Show for a vector bundle <math>E</math> (of trivial determinant) that this map identifies <math>SK_2(E)</math> with the second Chern class <math>SK_2(E)</math> defined in the third talk (Theorem 4.2'<ref name="Blo10"/>). * Explain that (via resolutions of the structure sheaf of codimension 2 subvarieties by vector bundles) one gets a map <math> \{\text{codim 2 subvarieties}\} \rightarrow H^2(X, K_2)</math> One would like to have that it factors through <math>CH^2(X) </math> and establishes that the morphism <math>H^2(X, K_2) \rightarrow CH^2(X)</math> is an ''isomorphism''. However, at this point, Bloch's article is a bit out-of-date. Quillen later proved by using the '''Gersten resolution''' of the sheaf <math>K_n^M(\mathcal{O}_X)</math> (Sheafified higher Milnor K-groups, cf. talk 5) that <math>H^n(X, K_n) \rightarrow CH^n(X)</math>. In the end one could say a couple of words on this (Reference: ) References: <ref name="Blo74"/>, (<ref name="Blo10"/>, §4) = The dilogarithm = * 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).) * Introduce the Bloch-Wigner dilogarithm (a single valued non-analytic variant of the dilog) * The five term relation and special values ** Present Theorem 7.4.4 from <ref name="Blo10"/> and its proof ** a good explanation for what is going on can be found in <ref name="Sus91"/> ** deduce special values as presented in <ref name="Zag07"/>, I, 1 References: <ref name="Blo00"/>, in particular 7.4 and all what is needed from before 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 If times is left some more motivations from <ref name="Zag07"/> can be presented. = A generalization of the exponential sequence = A survey of Bloch's article '78 References: <ref name="Blo78"/>, (<ref name="Blo10"/>, §6) Reference for the Dilogarithm: <ref name="Zag07"/> = The dilog, scissors congruences and volumes of hyperbolic spaces = References: <ref name="DS83"/>, <ref name="Gon99"/> = The dilog and zeta functions = References: <ref name="Blo00"/>, (<ref name="Zag07"/>, I, §5) = The dilog in physics = Starting point: The quantum dilogarithm (<ref name="Zag07"/>, I D, § 3) = Outlook: Mixed motives and periods = Historical overview about the developments in the last 35 years...: Mixed motives via their realizations and comparison encoding periods. Explain that the values of the dilog appears as periods. Relation between Extensions of Tate-motives and K-theory, Regulators. Connection with the topics discussed in the seminar. References: <ref name="Lev05"/>, <ref name="BD94"/>, <ref name="Blo91"/> = References = <references> <ref name="Blo74">Bloch, S.; ''<math>K_2</math> and algebraic cycles''. Ann. of Math. (2) 99 (1974), 349–379. </ref> <ref name="Blo00">Bloch, S.; ''Higher regulators, algebraic K-theory, and zeta functions of elliptic curves''. CRM Monograph Series, 11. American Mathematical Society, Providence, RI, 2000. x+97 pp.</ref> <ref name="Blo78">Bloch, S.; ''Applications of the dilogarithm function in algebraic K-theory and algebraic geometry''. Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), pp. 103–114, Kinokuniya Book Store, Tokyo, 1978.</ref> <ref name="Blo10">Bloch, S.; ''Lectures on algebraic cycles''. Second edition. New Mathematical Monographs, 16. Cambridge University Press, Cambridge, 2010. xxiv+130 pp.</ref> <ref name="Blo81">Bloch, S.; ''The dilogarithm and extensions of Lie algebras''. Algebraic K-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), pp. 1–23, Lecture Notes in Math., 854, Springer, Berlin-New York, 1981. </ref> <ref name="Zag07">Zagier, D.; ''The dilogarithm function''. Frontiers in number theory, physics, and geometry. II, 3–65, Springer, Berlin, 2007. </ref> <ref name="Hut13">Hutchinson, K.; ''A Bloch-Wigner complex for <math>\mathrm{SL}_2</math>''. J. K-Theory 12 (2013), no. 1, 15–68. </ref> <ref name="Goe07">Goette, S.; Zickert, Ch.; ''The extended Bloch group and the Cheeger-Chern-Simons class''. Geom. Topol. 11 (2007), 1623–1635. </ref> <ref name="Gon95">Goncharov, A. B.; ''Geometry of configurations, polylogarithms, and motivic cohomology''. Adv. Math. 114 (1995), no. 2, 197–318. </ref> <ref name="Lew81">Lewin, L.; ''Polylogarithms and associated functions''. With a foreword by A. J. Van der Poorten. North-Holland Publishing Co., New York-Amsterdam, 1981. xvii+359 pp. </ref> <ref name="Mil71">Milnor, J., ''Introduction to algebraic K-theory''. Annals of Mathematics Studies, No. 72. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. xiii+184 pp. </ref> <ref name="Wei13">Weibel, Ch. A.; ''The K-book. An introduction to algebraic K-theory''. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. xii+618 pp. </ref> <ref name="Sus84">Suslin, A. A.; ''Homology of <math>\mathrm{GL}_n</math>, characteristic classes and Milnor K-theory''. Algebraic K-theory, number theory, geometry and analysis (Bielefeld, 1982), 357–375, Lecture Notes in Math., 1046, Springer, Berlin, 1984. </ref> <ref name="Sus91">Suslin, A. A.; ''<math>K_3</math> of a field, and the Bloch group.'' (Russian) Translated in Proc. Steklov Inst. Math. 1991, no. 4, 217–239. Galois theory, rings, algebraic groups and their applications (Russian). </ref> <ref name="Sus87">Suslin, A. A.; ''Algebraic K-theory of fields''. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 222–244, Amer. Math. Soc., Providence, RI, 1987. </ref> <ref name="Zic15">Zickert, Ch. K.; ''The extended Bloch group and algebraic K-theory''. J. Reine Angew. Math. 704 (2015), 21–54. </ref> <ref name="Lev05">Levine, Marc; ''Mixed motives''. Handbook of K-theory. Vol. 1, 2, 429–521, Springer, Berlin, 2005.</ref> <ref name="Blo91"> Bloch, S.; ''Function theory of polylogarithms''. Structural properties of polylogarithms, 275–285, Math. Surveys Monogr., 37, Amer. Math. Soc., Providence, RI, 1991.</ref> <ref name="DS83">Dupont, J. L.; Sah, C. H.; ''Scissors congruences. II''. J. Pure Appl. Algebra, 25(2):159–195, 1982.</ref> <ref name="Gon99">Goncharov, A.; ''Volumes of hyperbolic manifolds and mixed Tate motives''. J. Amer. Math. Soc. 12 (1999), no. 2, 569–618. </ref> <ref name="BD94">Beĭlinson, A.; Deligne, P. ''Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs''. Motives (Seattle, WA, 1991), 97–121, Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, 1994. </ref> <ref name="Ram82">Ramakrishnan, D.; ''On the monodromy of higher logarithms''. Proc. Amer. Math. Soc. 85 (1982), no. 4, 596–599. </ref> </references>
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