Surveys in Contemporary Mathematics

370 Pages·2007·3.143 MB·English

by  Nicholas Young, Yemon Choi

LONDONMATHEMATICALSOCIETYLECTURENOTESERIES
ManagingEditor:ProfessorN.J.Hitchin,MathematicalInstitute,
UniversityofOxford,24–29StGiles,OxfordOX13LB,UnitedKingdom
Thetitlesbelowareavailablefrombooksellers,or,fromCambridgeUniversityPressatwww.cambridge.org/mathematics
191 Finitegeometryandcombinatorics, F.DECLERCKetal
192 Symplecticgeometry, D.SALAMON(ed)
194 Independentrandomvariablesandrearrangementinvariantspaces, M.BRAVERMAN
195 Arithmeticofblowupalgebras, W.VASCONCELOS
196 Microlocalanalysisfordifferentialoperators, A.GRIGIS&J.SJO¨STRAND
197 Two-dimensionalhomotopyandcombinatorialgrouptheory, C.HOG-ANGELONIetal
198 Thealgebraiccharacterizationofgeometric4-manifolds, J.A.HILLMAN
199 InvariantpotentialtheoryintheunitballofC(cid:2)(cid:2), M.STOLL
200 TheGrothendiecktheoryofdessinsd’enfant, L.SCHNEPS(ed)
201 Singularities, J.-P.BRASSELET(ed)
202 Thetechniqueofpseudodifferentialoperators, H.O.CORDES
203 HochschildcohomologyofvonNeumannalgebras, A.SINCLAIR&R.SMITH
204 Combinatorialandgeometricgrouptheory, A.J.DUNCAN,N.D.GILBERT&J.HOWIE(eds)
205 Ergodictheoryanditsconnectionswithharmonicanalysis, K.PETERSEN&I.SALAMA(eds)
207 GroupsofLietypeandtheirgeometries, W.M.KANTOR&L.DIMARTINO(eds)
208 Vectorbundlesinalgebraicgeometry, N.J.HITCHIN,P.NEWSTEAD&W.M.OXBURY(eds)
209 Arithmeticofdiagonalhypersurfacesoverinfitefields, F.Q.GOUVE´A&N.YUI
210 HilbertC∗-modules, E.C.LANCE
211 Groups93Galway/StAndrewsI, C.M.CAMPBELLetal(eds)
212 Groups93Galway/StAndrewsII, C.M.CAMPBELLetal(eds)
214 GeneralisedEuler-Jacobiinversionformulaandasymptoticsbeyondallorders, V.KOWALENKOetal
215 Numbertheory1992–93, S.DAVID(ed)
216 Stochasticpartialdifferentialequations, A.ETHERIDGE(ed)
217 Quadraticformswithapplicationstoalgebraicgeometryandtopology, A.PFISTER
218 Surveysincombinatorics,1995, P.ROWLINSON(ed)
220 Algebraicsettheory, A.JOYAL&I.MOERDIJK
221 Harmonicapproximation, S.J.GARDINER
222 Advancesinlinearlogic, J.-Y.GIRARD,Y.LAFONT&L.REGNIER(eds)
223 Analyticsemigroupsandsemilinearinitialboundaryvalueproblems, KAZUAKITAIRA
224 Computability,enumerability,unsolvability, S.B.COOPER,T.A.SLAMAN&S.S.WAINER(eds)
225 Amathematicalintroductiontostringtheory, S.ALBEVERIOetal
226 Novikovconjectures,indextheoremsandrigidityI, S.FERRY,A.RANICKI&J.ROSENBERG(eds)
227 Novikovconjectures,indextheoremsandrigidityII, S.FERRY,A.RANICKI&J.ROSENBERG(eds)
228 ErgodictheoryofZdactions, M.POLLICOTT&K.SCHMIDT(eds)
229 Ergodicityforinfinitedimensionalsystems, G.DAPRATO&J.ZABCZYK
230 Prolegomenatoamiddlebrowarithmeticofcurvesofgenus2, J.W.S.CASSELS&E.V.FLYNN
231 Semigrouptheoryanditsapplications, K.H.HOFMANN&M.W.MISLOVE(eds)
232 ThedescriptivesettheoryofPolishgroupactions, H.BECKER&A.S.KECHRIS
233 Finitefieldsandapplications, S.COHEN&H.NIEDERREITER(eds)
234 Introductiontosubfactors, V.JONES&V.S.SUNDER
235 Numbertheory1993–94, S.DAVID(ed)
236 TheJamesforest, H.FETTER&B.G.DEBUEN
237 Sievemethods,exponentialsums,andtheirapplicationsinnumbertheory, G.R.H.GREAVESetal
238 Representationtheoryandalgebraicgeometry, A.MARTSINKOVSKY&G.TODOROV(eds)
240 Stablegroups, F.O.WAGNER
241 Surveysincombinatorics,1997, R.A.BAILEY(ed)
242 GeometricGaloisactionsI, L.SCHNEPS&P.LOCHAK(eds)
243 GeometricGaloisactionsII, L.SCHNEPS&P.LOCHAK(eds)
244 Modeltheoryofgroupsandautomorphismgroups, D.EVANS(ed)
245 Geometry,combinatorialdesignsandrelatedstructures, J.W.P.HIRSCHFELDetal
246 p-Automorphismsoffinitep-groups, E.I.KHUKHRO
247 Analyticnumbertheory, Y.MOTOHASHI(ed)
248 Tametopologyando-minimalstructures, L.VANDENDRIES
249 Theatlasoffinitegroups:tenyearson, R.CURTIS&R.WILSON(eds)
250 Charactersandblocksoffinitegroups, G.NAVARRO
251 Gro¨bnerbasesandapplications, B.BUCHBERGER&F.WINKLER(eds)
252 Geometryandcohomologyingrouptheory, P.KROPHOLLER,G.NIBLO,R.STO¨HR(eds)
253 Theq-Schuralgebra, S.DONKIN
254 Galoisrepresentationsinarithmeticalgebraicgeometry, A.J.SCHOLL&R.L.TAYLOR(eds)
255 Symmetriesandintegrabilityofdifferenceequations, P.A.CLARKSON&F.W.NIJHOFF(eds)
256 AspectsofGaloistheory, H.VO¨LKLEINetal
257 Anintroductiontononcommutativedifferentialgeometryanditsphysicalapplications2ed, J.MADORE
258 Setsandproofs, S.B.COOPER&J.TRUSS(eds)
259 Modelsandcomputability, S.B.COOPER&J.TRUSS(eds)
260 GroupsStAndrews1997inBath,I, C.M.CAMPBELLetal
261 GroupsStAndrews1997inBath,II, C.M.CAMPBELLetal
262 Analysisandlogic, C.W.HENSON,J.IOVINO,A.S.KECHRIS&E.ODELL
263 Singularitytheory, B.BRUCE&D.MOND(eds)
264 Newtrendsinalgebraicgeometry, K.HULEK,F.CATANESE,C.PETERS&M.REID(eds)
265 Ellipticcurvesincryptography, I.BLAKE,G.SEROUSSI&N.SMART
267 Surveysincombinatorics,1999, J.D.LAMB&D.A.PREECE(eds)
268 Spectralasymptoticsinthesemi-classicallimit, M.DIMASSI&J.SJO¨STRAND
269 Ergodictheoryandtopologicaldynamics, M.B.BEKKA&M.MAYER
270 AnalysisonLieGroups, N.T.VAROPOULOS&S.MUSTAPHA
271 Singularperturbationsofdifferentialoperators, S.ALBEVERIO&P.KURASOV
272 Charactertheoryfortheoddorderfunction, T.PETERFALVI
273 Spectraltheoryandgeometry, E.B.DAVIES&Y.SAFAROV(eds)
274 TheMandelbrotset,themeandvariations, TANLEI(ed)
275 Descriptivesettheoryanddynamicalsystems, M.FOREMANetal
276 Singularitiesofplanecurves, E.CASAS-ALVERO
277 Computationalandgeometricaspectsofmodernalgebra, M.D.ATKINSONetal
278 Globalattractorsinabstractparabolicproblems, J.W.CHOLEWA&T.DLOTKO
279 Topicsinsymbolicdynamicsandapplications, F.BLANCHARD,A.MAASS&A.NOGUEIRA(eds)
280 CharactersandAutomorphismGroupsofCompactRiemannSurfaces, T.BREUER
281 Explicitbirationalgeometryof3-folds, A.CORTI&M.REID(eds)
282 Auslander-Buchweitzapproximationsofequivariantmodules, M.HASHIMOTO
283 Nonlinearelasticity, Y.FU&R.W.OGDEN(eds)
284 Foundationsofcomputationalmathematics, R.DEVORE,A.ISERLES&E.SU¨LI(eds)
285 Rationalpointsoncurvesoverfinitefield, H.NIEDERREITER&C.XING
286 Cliffordalgebrasandspinors2ed, P.LOUNESTO
287 TopicsonRiemannsurfacesandFuchsiangroups, E.BUJALANCEetal
288 Surveysincombinatorics,2001, J.HIRSCHFELD(ed)
289 AspectsofSobolev-typeinequalities, L.SALOFF-COSTE
290 QuantumgroupsandLietheory, A.PRESSLEY(ed)
291 Titsbuildingsandthemodeltheoryofgroups, K.TENT(ed)
292 Aquantumgroupsprimer, S.MAJID
293 SecondorderpartialdifferentialequationsinHilbertspaces, G.DAPRATO&J.ZABCZYK
294 Introductiontothetheoryofoperatorspace, G.PISIER
295 Geometryandintegrability, L.MASON&YAVUZNUTKU(eds)
296 Lecturesoninvarianttheory, I.DOLGACHEV
297 Thehomotopycategoryofsimplyconnected4-manifolds, H.-J.BAUES
298 Higheroperads,highercategories, T.LEINSTER
299 Kleiniangroupsandhyperbolic3-manifolds, Y.KOMORI,V.MARKOVIC&C.SERIES(eds)
300 IntroductiontoMo¨biusdifferentialgeometry, U.HERTRICH-JEROMIN
301 StablemodulesandtheD(2)-problem, F.E.A.JOHNSON
302 DiscreteandcontinuousnonlinearSchro¨dingersystems, M.J.ABLOWITZ,B.PRINARI&A.D.TRUBATCH
303 NumberTheoryandAlgebraicGeometry, M.REID&A.SKOROBOGATOV(eds)
304 GroupsStAndrews2001inOxfordVol.1, C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds)
305 GroupsStAndrews2001inOxfordVol.2, C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds)
306 Peyresqlecturesongeometricmechanicsandsymmetry, J.MONTALDI&T.RATIU(eds)
307 Surveysincombinatorics2003, C.D.WENSLEY(ed)
308 Topology,geometryandquantumfieldtheory, U.L.TILLMANN(ed)
309 CoringsandComodules, T.BRZEZINSKI&R.WISBAUER
310 TopicsinDynamicsandErgodicTheory, S.BEZUGLYI&S.KOLYADA(eds)
311 Groups:topological,combinatorialandarithmeticaspects, T.W.MU¨LLER(ed)
312 FoundationsofComputationalMathematics,Minneapolis2002, FELIPECUCKERetal(eds)
313 Transcendentalaspectsofalgebraiccycles, S.MU¨LLER-STACH&C.PETERS(eds)
314 Spectralgeneralizationsoflinegraphs, D.CVETKOVIC,P.ROWLINSON&S.SIMIC
315 Structuredringspectra, A.BAKER&B.RICHTER(eds)
316 LinearLogicinComputerScience, T.EHRHARDetal(eds)
317 Advancesinellipticcurvecryptography, I.F.BLAKE,G.SEROUSSI&N.SMART
318 Perturbationoftheboundaryinboundary-valueproblemsofPartialDifferentialEquations, D.HENRY
319 DoubleAffineHeckeAlgebras, I.CHEREDNIK
320 L-FunctionsandGaloisRepresentations, D.BURNS,K.BUZZARD&J.NEKOVA´Rˇ(eds)
321 SurveysinModernMathematics, V.PRASOLOV&Y.ILYASHENKO(eds)
322 Recentperspectivesinrandommatrixtheoryandnumbertheory, F.MEZZADRI&N.C.SNAITH(eds)
323 Poissongeometry,deformationquantisationandgrouprepresentations, S.GUTTetal(eds)
324 SingularitiesandComputerAlgebra, C.LOSSEN&G.PFISTER(eds)
325 LecturesontheRicciFlow, P.TOPPING
326 ModularRepresentationsofFiniteGroupsofLieType, J.E.HUMPHREYS
328 FundamentalsofHyperbolicManifolds, R.D.CANARY,A.MARDEN&D.B.A.EPSTEIN(eds)
329 SpacesofKleinianGroups, Y.MINSKY,M.SAKUMA&C.SERIES(eds)
330 NoncommutativeLocalizationinAlgebraandTopology, A.RANICKI(ed)
331 FoundationsofComputationalMathematics,Santander2005, L.PARDO,A.PINKUS,E.SULI&M.TODD(eds)
332 HandbookofTiltingTheory, L.ANGELERIHU¨GEL,D.HAPPEL&H.KRAUSE(eds)
333 SyntheticDifferentialGeometry2ed, A.KOCK
334 TheNavier-StokesEquations, P.G.DRAZIN&N.RILEY
335 LecturesontheCombinatoricsofFreeProbability, A.NICA&R.SPEICHER
336 IntegralClosureofIdeals,Rings,andModules, I.SWANSON&C.HUNEKE
337 MethodsinBanachSpaceTheory, J.M.F.CASTILLO&W.B.JOHNSON(eds)
338 SurveysinGeometryandNumberTheory, N.YOUNG(ed)
339 GroupsStAndrews2005Vol.1, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
340 GroupsStAndrews2005Vol.2, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
341 RanksofEllipticCurvesandRandomMatrixTheory, J.B.CONREY,D.W.FARMER,F.MEZZADRI&N.C.
SNAITH(eds)
342 EllipticCohomology, H.R.MILLER&D.C.RAVENEL(eds)
343 AlgebraicCyclesandMotivesVol.1, J.NAGEL&C.PETERS(eds)
344 AlgebraicCyclesandMotivesVol.2, J.NAGEL&C.PETERS(eds)
345 AlgebraicandAnalyticGeometry, A.NEEMAN
346 SurveysinCombinatorics,2007, A.HILTON&J.TALBOT(eds)
347 SurveysinContemporaryMathematics, N.YOUNG&Y.CHOI(eds)
LondonMathematicalSocietyLectureNoteSeries.347
Surveys in Contemporary
Mathematics
Editedby
NICHOLAS YOUNG
UniversityofLeeds
and
YEMON CHOI
UniversityofManitoba
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,Sa˜oPaulo,Delhi
CambridgeUniversityPress
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noreproductionofanypartmaytakeplacewithout
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Firstpublished2008
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Contents
Preface page vii
Rank and determinant functions for matrices over semi-
rings
A. E. Guterman 1
Algebraic geometry over Lie algebras
I. V. Kazachkov 34
Destabilization of closed braids
A. V. Malyutin 82
n-dimensional local fields and adeles on n-dimensional
schemes
D. V. Osipov 131
Cohomology of face rings, and torus actions
T. E. Panov 165
Three lectures on the Borsuk partition problem
A. M. Raigorodskii 202
EmbeddingandknottingofmanifoldsinEuclideanspaces
A. B. Skopenkov 248
On Maxwellian and Boltzmann distributions
V. V. Ten 343
v

Preface
This is the second of two volumes that showcase young scientists who
arecontinuingtheoutstandingtraditionofRussianmathematicsintheir
home country. There remain numerous strong research groups, partic-
ularly in Moscow and St. Petersburg, despite the familiar difficulties:
academic salaries in Russia remain low, many leading figures have de-
partedandthereareplentifulopportunitiesforemploymentinuniversity
positions abroad or in sectors in Russia that offer a living wage. It is
hoped that the articles in this book give a picture of the interests and
achievements of mathematicians that participate in some of the active
seminars in the country. Seven have something of the character of a
survey, but also contain many original results and give extensive bibli-
ographies;theeighthisarevisedandexpandedversionofa2002research
article.
The first of the two volumes (LMS Lecture Notes 338) was entitled
Surveys in Geomety and Number Theory; this one is mainly on com-
binatorial and algebraic geometry and topology. Both volumes contain
papers based on courses of lectures given at British universities by the
authors under the ‘Young Russian Mathematicians’ scheme, which the
London Mathematical Society set up to help such mathematicians visit
the UK and to provide them with financial support.
In the nineties sheer subsistence was difficult for Russian academics.
Over the last five years things have improved, and the salaries of uni-
versity employees, though not generous, are closer to sufficing for the
necessities of life. It still remains difficult for young scientists to get
established in a career, and two of the contributors to this volume have
since chosen other paths, one in the mathematical diaspora and one in
industry.
Nicholas Young
Department of Pure Mathematics
Leeds University
Yemon Choi
Department of Mathematics
University of Manitoba.
vii

Rank and determinant functions for matrices
over semirings
Alexander E. Guterman
Introduction
The difference between semirings and rings is the lack of additive in-
versesin semirings. The most commonexamples of semiringswhich are
not rings are the non-negative integers Z+, the non-negative rationals
Q+ and the non-negative reals R+ with usual addition and multiplica-
tion. There are classical examples of non-numerical semirings as well.
One of the first examples appearedin the workof Dedekind [29]in con-
nection with the algebra of ideals of a commutative ring (one can add
and multiply ideals but it is not possible to subtract them). Later Van-
diver [62] proposed the class of semirings as the best class of algebraic
structures which includes both rings and bounded distributive lattices.
Booleanalgebras,max-algebras,tropicalsemiringsandfuzzyscalarsare
otherimportantexamplesofsemirings. See the monographs[37,38, 44]
for more details.
During the last few decades, matrices with entries from various semi-
rings have attracted the attention of many researchers working both
in theoretical and applied mathematics. It should be emphasized that
the majority of examples of semirings arise in various applications of
algebra. Toillustrateanapplicationofsemirings,wepresentasituation
that arises constantly in parallel computations. To make this situation
more apparent, we give its ‘real life’ analogue. Let us consider k sea-
ports. A ship with certain goods arrivesat the i-th port at the moment
x ,i=1,...,k. Alsotherearel airportswithaairplaneineachairport.
i
Each airplane has to depart at the time b , j = 1,...,l. All goods are
j
delivered by trains from the sea-ports to the airports. Let t denote
i,j
the traveltime fromthe i-thportto the j-thairport. Theproblemis to
find x , i = 1,...,k, (or certain relations between them) in such a way
i
1