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E-BookPDF1 - PDF WatermarkE-Book
792 Seiten
Englisch
Springer Berlin Heidelbergerschienen am09.04.20083rd ed. 2008
Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements.

Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)?

The third edition improves the second edition in two ways: First it removes many typos and mathematical inaccuracies that occur in the second edition (in particular in the references). Secondly, the third edition reports on five open problems (out of thirtyfour open problems of the second edition) that have been partially or fully solved since that edition appeared in 2005.



Moshe Jarden (revised and considerably enlarged the book in 2004 (2nd edition) and revised again in 2007 (the present 3rd edition).

 
Born on 23 August, 1942 in Tel Aviv, Israel.

Education:
Ph.D. 1969 at the Hebrew University of Jerusalem on
'Rational Points of Algebraic Varieties over Large Algebraic Fields'.
Thesis advisor: H. Furstenberg.
Habilitation at Heidelberg University, 1972, on
'Model Theory Methods in the Theory of Fields'.

Positions:
Dozent, Heidelberg University, 1973-1974.
Seniour Lecturer, Tel Aviv University, 1974-1978
Associate Professor, Tel Aviv University, 1978-1982
Professor, Tel Aviv University, 1982-
Incumbent of the Cissie and Aaron Beare Chair,
Tel Aviv University. 1998-

Academic and Professional Awards
Fellowship of Alexander von Humboldt-Stiftung in Heidelberg, 1971-1973.
Fellowship of Minerva Foundation, 1982.
Chairman of the Israel Mathematical Society, 1986-1988.
Member of the Institute for Advanced Study, Princeton, 1983, 1988.
Editor of the Israel Journal of Mathematics, 1992-.
Landau Prize for the book 'Field Arithmetic'. 1987.
Director of the Minkowski Center for Geometry founded by the
Minerva Foundation, 1997-.
L. Meitner-A.v.Humboldt Research Prize, 2001
Member, Max-Planck Institut für Mathematik in Bonn, 2001.


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Produkt

KlappentextField Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements.

Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)?

The third edition improves the second edition in two ways: First it removes many typos and mathematical inaccuracies that occur in the second edition (in particular in the references). Secondly, the third edition reports on five open problems (out of thirtyfour open problems of the second edition) that have been partially or fully solved since that edition appeared in 2005.



Moshe Jarden (revised and considerably enlarged the book in 2004 (2nd edition) and revised again in 2007 (the present 3rd edition).

 
Born on 23 August, 1942 in Tel Aviv, Israel.

Education:
Ph.D. 1969 at the Hebrew University of Jerusalem on
'Rational Points of Algebraic Varieties over Large Algebraic Fields'.
Thesis advisor: H. Furstenberg.
Habilitation at Heidelberg University, 1972, on
'Model Theory Methods in the Theory of Fields'.

Positions:
Dozent, Heidelberg University, 1973-1974.
Seniour Lecturer, Tel Aviv University, 1974-1978
Associate Professor, Tel Aviv University, 1978-1982
Professor, Tel Aviv University, 1982-
Incumbent of the Cissie and Aaron Beare Chair,
Tel Aviv University. 1998-

Academic and Professional Awards
Fellowship of Alexander von Humboldt-Stiftung in Heidelberg, 1971-1973.
Fellowship of Minerva Foundation, 1982.
Chairman of the Israel Mathematical Society, 1986-1988.
Member of the Institute for Advanced Study, Princeton, 1983, 1988.
Editor of the Israel Journal of Mathematics, 1992-.
Landau Prize for the book 'Field Arithmetic'. 1987.
Director of the Minkowski Center for Geometry founded by the
Minerva Foundation, 1997-.
L. Meitner-A.v.Humboldt Research Prize, 2001
Member, Max-Planck Institut für Mathematik in Bonn, 2001.


Details
Weitere ISBN/GTIN9783540772705
ProduktartE-Book
EinbandartE-Book
FormatPDF
Format Hinweis1 - PDF Watermark
FormatE107
Erscheinungsjahr2008
Erscheinungsdatum09.04.2008
Auflage3rd ed. 2008
ReiheErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
Reihen-Nr.11
Seiten792 Seiten
SpracheEnglisch
IllustrationenXXIV, 792 p.
Artikel-Nr.1425774
Rubriken
Genre9200

Inhalt/Kritik

Inhaltsverzeichnis
1;Table of Contents;6
2;Introduction to the Third Edition;15
3;Introduction to the First Edition;19
4;Notation and Convention;23
5;Chapter 1. Infinite Galois Theory and Profinite Groups;24
5.1;1.1 Inverse Limits;24
5.2;1.2 Profinite Groups;27
5.3;1.3 Infinite Galois Theory;32
5.4;1.4 The p-adic Integers and the Pr¨ufer Group;35
5.5;1.5 The Absolute Galois Group of a Finite Field;38
6;Chapter 2. Valuations and Linear Disjointness;42
6.1;2.1 Valuations, Places, and Valuation Rings;42
6.2;2.2 Discrete Valuations;44
6.3;2.3 Extensions of Valuations and Places;47
6.4;2.4 Integral Extensions and Dedekind Domains;53
6.5;2.5 Linear Disjointness of Fields;57
6.6;2.6 Separable, Regular, and Primary Extensions;61
6.7;2.7 The Imperfect Degree of a Field;67
6.8;2.8 Derivatives;71
7;Chapter 3. Algebraic Function Fields of One Variable;75
7.1;3.1 Function Fields of One Variable;75
7.2;3.2 The Riemann-Roch Theorem;77
7.3;3.3 Holomorphy Rings;79
7.4;3.4 Extensions of Function Fields;82
7.5;3.5 Completions;84
7.6;3.6 The Different;90
7.7;3.7 Hyperelliptic Fields;93
7.8;3.8 Hyperelliptic Fields with a Rational Quadratic Subfield;96
7.9;Exercises;98
7.10;Notes;99
8;Chapter 4. The Riemann Hypothesis for Function Fields;100
8.1;4.1 Class Numbers;100
8.2;4.2 Zeta Functions;102
8.3;4.3 Zeta Functions under Constant Field Extensions;104
8.4;4.4 The Functional Equation;105
8.5;4.5 The Riemann Hypothesis and Degree 1 Prime Divisors;107
8.6;4.6 Reduction Steps;109
8.7;4.7 An Upper Bound;110
8.8;4.8 A Lower Bound;112
8.9;Exercises;114
8.10;Notes;116
9;Chapter 5. Plane Curves;118
9.1;5.1 A ne and Projective Plane Curves;118
9.2;5.2 Points and Prime Divisors;120
9.3;5.3 The Genus of a Plane Curve;122
9.4;5.4 Points on a Curve over a Finite Field;127
9.5;Exercises;128
9.6;Notes;129
10;Chapter 6. The Chebotarev Density Theorem;130
10.1;6.1 Decomposition Groups;130
10.2;6.2 The Artin Symbol over Global Fields;134
10.3;6.3 Dirichlet Density;136
10.4;6.4 Function Fields;138
10.5;Exercises;152
10.6;Notes;153
11;Chapter 7. Ultraproducts;155
11.1;7.1 First Order Predicate Calculus;155
11.2;7.2 Structures;157
11.3;7.3 Models;158
11.4;7.4 Elementary Substructures;160
11.5;7.5 Ultrafilters;161
11.6;7.6 Regular Ultrafilters;162
11.7;7.7 Ultraproducts;164
11.8;7.8 Regular Ultraproducts;168
11.9;7.9 Nonprincipal Ultraproducts of Finite Fields;170
11.10;Exercises;170
11.11;Notes;171
12;Chapter 8. Decision Procedures;172
12.1;8.1 Deduction Theory;172
12.2;8.2 Gödel´s Completeness Theorem;175
12.3;8.3 Primitive Recursive Functions;177
12.4;8.4 Primitive Recursive Relations;179
12.5;8.5 Recursive Functions;180
12.6;8.6 Recursive and Primitive Recursive Procedures;182
12.7;8.7 A Reduction Step in Decidability Procedures;183
12.8;Exercises;184
12.9;Notes;185
13;Chapter 9. Algebraically Closed Fields;186
13.1;9.1 Elimination of Quantifiers;186
13.2;9.2 A Quantifiers Elimination Procedure;188
13.3;9.3 E ectiveness;191
13.4;9.4 Applications;192
13.5;Exercises;193
13.6;Notes;193
14;Chapter 10. Elements of Algebraic Geometry;195
14.1;10.1 Algebraic Sets;195
14.2;10.2 Varieties;198
14.3;10.3 Substitutions in Irreducible Polynomials;199
14.4;10.4 Rational Maps;201
14.5;10.5 Hyperplane Sections;203
14.6;10.6 Descent;205
14.7;10.7 Projective Varieties;208
14.8;10.8 About the Language of Algebraic Geometry;210
14.9;Notes;214
15;Chapter 11. Pseudo Algebraically Closed Fields;215
15.1;11.1 PAC Fields;215
15.2;11.2 Reduction to Plane Curves;216
15.3;11.3 The PAC Property is an Elementary Statement;222
15.4;11.4 PAC Fields of Positive Characteristic;224
15.5;11.5 PAC Fields with Valuations;226
15.6;11.6 The Absolute Galois Group of a PAC Field;230
15.7;11.7 A non-PAC Field K with Kins PAC;234
15.8;Exercises;240
15.9;Notes;241
16;Chapter 12. Hilbertian Fields;242
16.1;12.1 Hilbert Sets and Reduction Lemmas;242
16.2;12.2 Hilbert Sets under Separable Algebraic Extensions;246
16.3;12.3 Purely Inseparable Extensions;247
16.4;12.4 Imperfect fields;251
16.5;Exercises;252
16.6;Notes;253
17;Chapter 13. The Classical Hilbertian Fields;254
17.1;13.1 Further Reduction;254
17.2;13.2 Function Fields over Infinite Fields;259
17.3;13.3 Global Fields;260
17.4;13.4 Hilbertian Rings;264
17.5;13.5 Hilbertianity via Coverings;267
17.6;13.6 Non-Hilbertian g-Hilbertian Fields;271
17.7;13.7 Twisted Wreath Products;275
17.8;13.8 The Diamond Theorem;281
17.9;13.9 Weissauer´s Theorem;285
17.10;Exercises;287
17.11;Notes;289
18;Chapter 14. Nonstandard Structures;290
18.1;14.1 Higher Order Predicate Calculus;290
18.2;14.2 Enlargements;291
18.3;14.3 Concurrent Relations;293
18.4;14.4 The Existence of Enlargements;295
18.5;14.5 Examples;297
18.6;Exercises;298
18.7;Notes;299
19;Chapter 15. Nonstandard Approach to Hilbert´s Irreducibility Theorem;300
19.1;15.1 Criteria for Hilbertianity;300
19.2;15.2 Arithmetical Primes Versus Functional Primes;302
19.3;15.3 Fields with the Product Formula;304
19.4;15.4 Generalized Krull Domains;306
19.5;15.5 Examples;309
19.6;Exercises;312
19.7;Notes;313
20;Chapter 16. Galois Groups over Hilbertian Fields;314
20.1;16.1 Galois Groups of Polynomials;314
20.2;16.2 Stable Polynomials;317
20.3;16.3 Regular Realization of Finite Abelian Groups;321
20.4;16.4 Split Embedding Problems with Abelian Kernels;325
20.5;16.5 Embedding Quadratic Extensions in Z/2nZ-Extensions;329
20.6;16.6 Zp-Extensions of Hilbertian Fields;331
20.7;16.7 Symmetric and Alternating Groups over Hilbertian Fields;338
20.8;16.8 GAR-Realizations;344
20.9;16.9 Embedding Problems over Hilbertian Fields;348
20.10;16.10 Finitely Generated Profinite Groups;351
20.11;16.11 Abelian Extensions of Hilbertian Fields;355
20.12;16.12 Regularity of Finite Groups over Complete Discrete Valued Fields;357
20.13;Exercises;358
20.14;Notes;359
21;Chapter 17. Free Profinite Groups;361
21.1;17.1 The Rank of a Profinite Group;361
21.2;17.2 Profinite Completions of Groups;363
21.3;17.3 Formations of Finite Groups;367
21.4;17.4 Free pro-C Groups;369
21.5;17.5 Subgroups of Free Discrete Groups;373
21.6;17.6 Open Subgroups of Free Profinite Groups;381
21.7;17.7 An Embedding Property;383
21.8;Exercises;384
21.9;Notes;385
22;Chapter 18. The Haar Measure;386
22.1;18.1 The Haar Measure of a Profinite Group;386
22.2;18.2 Existence of the Haar Measure;389
22.3;18.3 Independence;393
22.4;18.4 Cartesian Product of Haar Measures;399
22.5;18.5 The Haar Measure of the Absolute Galois Group;401
22.6;18.6 The PAC Nullstellensatz;403
22.7;18.7 The Bottom Theorem;405
22.8;18.8 PAC Fields over Uncountable Hilbertian Fields;409
22.9;18.9 On the Stability of Fields;413
22.10;18.10 PAC Galois Extensions of Hilbertian Fields;417
22.11;18.11 Algebraic Groups;420
22.12;Exercises;423
22.13;Notes;424
23;Chapter 19. E ective Field Theory and Algebraic Geometry;426
23.1;19.1 Presented Rings and Fields;426
23.2;19.2 Extensions of Presented Fields;429
23.3;19.3 Galois Extensions of Presented Fields;434
23.4;19.4 The Algebraic and Separable Closures of Presented Fields;435
23.5;19.5 Constructive Algebraic Geometry;436
23.6;19.6 Presented Rings and Constructible Sets;445
23.7;19.7 Basic Normal Stratification;448
23.8;Exercises;450
23.9;Notes;451
24;Chapter 20. The Elementary Theory of e-Free PAC Fields;452
24.1;20.1 @1-Saturated PAC Fields;452
24.2;20.2 The Elementary Equivalence Theorem of @1-Saturated PAC Fields;453
24.3;20.3 Elementary Equivalence of PAC Fields;456
24.4;20.4 On e-Free PAC Fields;459
24.5;20.5 The Elementary Theory of Perfect e-Free PAC Fields;461
24.6;20.6 The Probable Truth of a Sentence;463
24.7;20.7 Change of Base Field;465
24.8;20.8 The Fields Ks( 1, . . . , e);467
24.9;20.9 The Transfer Theorem;469
24.10;20.10 The Elementary Theory of Finite Fields;471
24.11;Exercises;474
24.12;Notes;476
25;Chapter 21. Problems of Arithmetical Geometry;477
25.1;21.1 The Decomposition-Intersection Procedure;477
25.2;21.2 Ci-Fields and Weakly Ci-Fields;478
25.3;21.3 Perfect PAC Fields which are Ci;483
25.4;21.4 The Existential Theory of PAC Fields;485
25.5;21.5 Kronecker Classes of Number Fields;486
25.6;21.6 Davenport´s Problem;490
25.7;21.7 On Permutation Groups;495
25.8;21.8 Schur´s Conjecture;502
25.9;21.9 Generalized Carlitz´s Conjecture;512
25.10;Exercises;516
25.11;Notes;518
26;Chapter 22. Projective Groups and Frattini Covers;520
26.1;22.1 The Frattini Group of a Profinite Group;520
26.2;22.2 Cartesian Squares;522
26.3;22.3 On C-Projective Groups;525
26.4;22.4 Projective Groups;529
26.5;22.5 Frattini Covers;531
26.6;22.6 The Universal Frattini Cover;536
26.7;22.7 Projective Pro-p-Groups;538
26.8;22.8 Supernatural Numbers;543
26.9;22.9 The Sylow Theorems;545
26.10;22.10 On Complements of Normal Subgroups;547
26.11;22.11 The Universal Frattini p-Cover;551
26.12;22.12 Examples of Universal Frattini p-covers;555
26.13;22.13 The Special Linear Group SL(2, Zp);557
26.14;22.14 The General Linear Group GL(2,Zp);560
26.15;Exercises;562
26.16;Notes;565
27;Chapter 23. PAC Fields and Projective Absolute Galois Groups;567
27.1;23.1 Projective Groups as Absolute Galois Groups;567
27.2;23.2 Countably Generated Projective Groups;569
27.3;23.3 Perfect PAC Fields of Bounded Corank;572
27.4;23.4 Basic Elementary Statements;573
27.5;23.5 Reduction Steps;577
27.6;23.6 Application of Ultraproducts;581
27.7;Exercises;584
27.8;Notes;584
28;Chapter 24. Frobenius Fields;585
28.1;24.1 The Field Crossing Argument;585
28.2;24.2 The Beckmann-Black Problem;588
28.3;24.3 The Embedding Property and Maximal Frattini Covers;590
28.4;24.4 The Smallest Embedding Cover of a Profinite Group;592
28.5;24.5 A Decision Procedure;597
28.6;24.6 Examples;599
28.7;24.7 Non-projective Smallest Embedding Cover;602
28.8;24.8 A Theorem of Iwasawa;604
28.9;24.9 Free Profinite Groups of at most Countable Rank;606
28.10;24.10 Application of the Nielsen-Schreier Formula;609
28.11;Exercises;614
28.12;Notes;615
29;Chapter 25. Free Profinite Groups of Infinite Rank;617
29.1;25.1 Characterization of Free Profinite Groups by Embedding Problems;618
29.2;25.2 Applications of Theorem 25.1.7;624
29.3;25.3 The Pro-C Completion of a Free Discrete Group;627
29.4;25.4 The Group Theoretic Diamond Theorem;629
29.5;25.5 The Melnikov Group of a Profinite Group;636
29.6;25.6 Homogeneous Pro-C Groups;638
29.7;25.7 The S-rank of Closed Normal Subgroups;643
29.8;25.8 Closed Normal Subgroups with a Basis Element;646
29.9;25.9 Accessible Subgroups;648
29.10;Notes;656
30;Chapter 26. Random Elements in Profinite Groups;658
30.1;26.1 Random Elements in a Free Profinite Group;658
30.2;26.2 Random Elements in Free pro-p Groups;663
30.3;26.3 Random e-tuples in Z;665
30.4;26.4 On the Index of Normal Subgroups Generated by Random Elements;669
30.5;26.5 Freeness of Normal Subgroups Generated by Random Elements;674
30.6;Notes;677
31;Chapter 27. Omega-free PAC Fields;678
31.1;27.1 Model Companions;678
31.2;27.2 The Model Companion in an Augmented Theory of Fields;682
31.3;27.3 New Non-Classical Hilbertian Fields;687
31.4;27.4 An Abundance of !-Free PAC Fields;690
31.5;Notes;693
32;Chapter 28. Undecidability;694
32.1;28.1 Turing Machines;694
32.2;28.2 Computation of Functions by Turing Machines;695
32.3;28.3 Recursive Inseparability of Sets of Turing Machines;699
32.4;28.4 The Predicate Calculus;702
32.5;28.5 Undecidability in the Theory of Graphs;705
32.6;28.6 Assigning Graphs to Profinite Groups;710
32.7;28.7 The Graph Conditions;711
32.8;28.8 Assigning Profinite Groups to Graphs;713
32.9;28.9 Assigning Fields to Graphs;717
32.10;28.10 Interpretation of the Theory of Graphs in the Theory of Fields;717
32.11;Exercises;720
32.12;Notes;720
33;Chapter 29. Algebraically Closed Fields with Distinguished Automorphisms;721
33.1;29.1 The Base Field K;721
33.2;29.2 Coding in PAC Fields with Monadic Quantifiers;723
33.3;29.3 The Theory of Almost all;727
33.4;29.4 The Probability of Truth Sentences;729
34;Chapter 30. Galois Stratification;731
34.1;30.1 The Artin Symbol;731
34.2;30.2 Conjugacy Domains under Projections;733
34.3;30.3 Normal Stratification;738
34.4;30.4 Elimination of One Variable;740
34.5;30.5 The Complete Elimination Procedure;743
34.6;30.6 Model-Theoretic Applications;745
34.7;30.7 A Limit of Theories;748
34.8;Exercises;749
34.9;Notes;752
35;Chapter 31. Galois Stratification over Finite Fields;753
35.1;31.1 The Elementary Theory of Frobenius Fields;753
35.2;31.2 The Elementary Theory of Finite Fields;758
35.3;31.3 Near Rationality of the Zeta Function of a Galois Formula;762
35.4;Exercises;771
35.5;Notes;773
36;Chapter 32. Problems of Field Arithmetic;774
36.1;32.1 Open Problems of the First Edition;774
36.2;32.2 Open Problems of the Second Edition;777
36.3;32.3 Open Problems;781
37;References;784
38;Index;803mehr
Kritik
From the reviews of the second edition:
"This second and considerably enlarged edition reflects the progress made in field arithmetic during the past two decades. ??? The book also contains very useful introductions to the more general theories used later on ??? . the book contains many exercises and historical notes, as well as a comprehensive bibliography on the subject. Finally, there is an updated list of open research problems, and a discussion on the impressive progress made on the corresponding list of problems made in the first edition." (Ido Efrat, Mathematical Reviews, Issue 2005 k)
"The goal of this new edition is to enrich the book with an extensive account of the progress made in this field ??? . the book is a very rich survey of results in Field Arithmetic and could be very helpful for specialists. On the other hand, it also contains a large number of results of independent interest, and therefore it is highly recommendable to many others too." (Roberto Dvornicich, Zentralblatt MATH, Vol. 1055, 2005)mehr

Autor

Moshe Jarden (revised and considerably enlarged the book in 2004 (2nd edition) and revised again in 2007 (the present 3rd edition).

Born on 23 August, 1942 in Tel Aviv, Israel.

Education:Ph.D. 1969 at the Hebrew University of Jerusalem on"Rational Points of Algebraic Varieties over Large Algebraic Fields".Thesis advisor: H. Furstenberg.Habilitation at Heidelberg University, 1972, on"Model Theory Methods in the Theory of Fields".

Positions:Dozent, Heidelberg University, 1973-1974.Seniour Lecturer, Tel Aviv University, 1974-1978Associate Professor, Tel Aviv University, 1978-1982Professor, Tel Aviv University, 1982-Incumbent of the Cissie and Aaron Beare Chair,Tel Aviv University. 1998-

Academic and Professional AwardsFellowship of Alexander von Humboldt-Stiftung in Heidelberg, 1971-1973.Fellowship of Minerva Foundation, 1982.Chairman of the Israel Mathematical Society, 1986-1988.Member of the Institute for Advanced Study, Princeton, 1983, 1988.Editor of the Israel Journal of Mathematics, 1992-.Landau Prize for the book "Field Arithmetic". 1987.Director of the Minkowski Center for Geometry founded by theMinerva Foundation, 1997-.L. Meitner-A.v.Humboldt Research Prize, 2001Member, Max-Planck Institut f\"ur Mathematik in Bonn, 2001.