Young scientists in Russia are continuing the outstanding tradition of Russian mathematics in their home country, in spite of the post-Soviet diaspora. This collection, the second of two, showcases the recent achievements of young Russian mathematicians and the strong research groups they are associated with. The first collection focused on geometry and number theory; this one concentrates on combinatorial and algebraic geometry and topology. The articles are mainly surveys of the recent work of the research groups and contain a substantial number of new results. Topics covered include algebraic geometry over Lie groups, cohomological aspects of toric topology, the Borsuk partition problem, and embedding and knotting of manifolds in Euclidean spaces. The authors are A. E. Guterman, I. V. Kazachkov, A. V. Malyutin, D. V. Osipov, T. E. Panov, A. M. Raigorodskii, A. B. Skopenkov and V. V. Ten.
Author(s): Nicholas Young, Yemon Choi
Series: London Mathematical Society Lecture Note Series
Publisher: CUP
Year: 2008
Language: English
Pages: 370
Cover......Page 1
London Mathematical Society Lecture Note Series 347......Page 2
Surveys in Contemporary Mathematics......Page 4
9780521705646......Page 5
Contents......Page 6
Preface......Page 8
Introduction......Page 10
1 Preliminaries......Page 11
2 Semimodules, bases and dimension......Page 12
3 Rank functions......Page 14
4 The bideterminant function......Page 30
5 Linear transformations that preserve matrix invariants......Page 33
Bibliography......Page 39
Introduction......Page 43
1 The category of A-Lie algebras......Page 45
2 The first order language......Page 46
3 Elements of algebraic geometry......Page 47
4 The Zariski topology......Page 52
5 A-Domains......Page 54
6 The category of algebraic sets......Page 55
7 The Equivalence Theorem......Page 57
8 Prevarieties......Page 59
9 Universal classes......Page 64
10 Quasivarieties......Page 66
11 Universal closure......Page 70
12 Geometric equivalence......Page 73
13 Algebraic geometry over free metabelian Lie algebras......Page 74
14 Algebraic geometry over a free Lie algebra......Page 81
Acknowledgements......Page 88
Bibliography......Page 89
Introduction......Page 91
1 ‘Infinitely destabilizable’ braids......Page 94
2 Preliminaries: the free group, punctured disc, and hyperbolic geometry......Page 97
3 Braids and disc automorphisms......Page 114
4 The Destabilization Algorithm......Page 121
Bibliography......Page 138
1 Introduction......Page 140
2 n-dimensional local fields......Page 141
3 Adeles and adelic complexes......Page 148
4 Restricted adelic complexes......Page 158
5 Reciprocity laws......Page 168
Bibliography......Page 171
1 Introduction......Page 174
2 Simplicial complexes and face rings......Page 175
3 Toric spaces......Page 182
4 Cohomology of moment-angle complexes......Page 192
5 Applications to combinatorial commutative algebra......Page 200
Bibliography......Page 208
1 Borsuk’s conjecture: a historical overview and high-dimensional counterexamples......Page 211
2 Borsuk’s conjecture: partial solutions and other partial results......Page 228
3 Borsuk’s conjecture: the low-dimensional case......Page 241
Bibliography......Page 253
Introduction......Page 257
1 Preliminaries......Page 262
2 The simplest-to-state results on embeddings......Page 264
3 Links and knotted tori......Page 276
4 The van Kampen obstruction......Page 288
5 The Haefliger-Wu invariant......Page 296
6 The deleted product of the torus......Page 307
7 The Borromean rings and the Haefliger-Wu invariant......Page 316
8 The disjunction method......Page 328
Bibliography......Page 342
Introduction......Page 352
1 Convergence to the normal distribution......Page 353
2 Proofs of the lemmas......Page 358
3 Deviations on individual solutions......Page 362
4 Improvements of estimates......Page 365
5 Density of particles on configuration space......Page 368
Bibliography......Page 369
