3/4 Old, New and Unknown around Scalar Curvature
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00:00
Gruppe <Mathematik>HomotopieMathematische LogikPerspektiveStatistikTopologieAllgemeine RelativitätstheorieMinimalflächeKrümmungInvarianteAusdruck <Logik>DimensionsanalyseFaserbündelGrenzschichtablösungUngleichungBeweistheorieGeradeHyperbolischer RaumIndexberechnungKomplex <Algebra>Konvexe MengeMereologiePhysikalismusResultanteTermTheoremTorusÜbergangswahrscheinlichkeitZentrische StreckungNichtlinearer OperatorParametersystemHeegaard-ZerlegungEnergiedichteSummierbarkeitExistenzsatzPunktDualitätssatzKoeffizientRichtungDifferenzkernDifferenteTVD-VerfahrenRechter WinkelOrtsoperatorGeometrieHarmonische AnalyseHolomorphe FunktionMannigfaltigkeitArithmetisches MittelMonoidSkalarfeldSpinorEllipseNichtlineares GleichungssystemVorlesung/Konferenz
06:21
Algebraische StrukturAnalysisHarmonische AnalyseMannigfaltigkeitMathematikOrdnung <Mathematik>SpieltheorieVerschiebungsoperatorAusdruck <Logik>DimensionsanalyseFaserbündelÜbergangAlgebraische K-TheorieAussage <Mathematik>Diophantische GeometrieDreiFundamentalgruppeGeradenbündelIndexberechnungInverser LimesMereologiePunktrechnungSphäreTheoremTorusWechselsprungZentrische StreckungFamilie <Mathematik>ParametersystemCoxeter-GruppeEuklidischer RaumElliptischer PseudodifferentialoperatorKlasse <Mathematik>KoeffizientRichtungKonditionszahlDifferenteElement <Gruppentheorie>MinimalgradMinkowski-MetrikRechter WinkelGruppendarstellungOrtsoperatorKategorie <Mathematik>DualitätstheorieEinfach zusammenhängender RaumGeradeMetrisches SystemSkalarfeldVektorraumbündelLinearisierungSummierbarkeitVorlesung/Konferenz
12:41
Algebraische StrukturGeometrieGruppe <Mathematik>HomotopieKohomologieMannigfaltigkeitMathematikPerspektiveRationale ZahlQuadratische FormKrümmungInvarianteKategorie <Mathematik>DimensionsanalyseFinitismusFaserbündelUnendlichkeitEinfach zusammenhängender RaumEinfacher RingErneuerungstheorieFundamentalgruppeFundamentalsatz der AlgebraGeradeHomologiegruppeHomotopiegruppeIndexberechnungPhysikalische TheorieSkalarfeldLineare GeometrieStichprobenfehlerTermTorusFlächeninhaltReelle ZahlFamilie <Mathematik>Überlagerung <Mathematik>GegenbeispielGewicht <Ausgleichsrechnung>GammafunktionQuadratzahlPunktEigentliche AbbildungRichtungUmwandlungsenthalpieKonditionszahlDifferenteElement <Gruppentheorie>EvoluteInnerer AutomorphismusMinkowski-MetrikRechter WinkelGruppendarstellungOrtsoperatorMathematische LogikNichteuklidische GeometrieRauschenRelativitätstheorieTopologieVariableAusdruck <Logik>BeweistheorieGarbentheorieGeradenbündelGerichteter GraphHomologische AlgebraMomentenproblemSesquilinearformTabelleTheoremVerdunstungZentrische StreckungVerschlingungParametersystemSummierbarkeitStrömungsrichtungKlasse <Mathematik>KapillardruckKoeffizientWeg <Topologie>BestimmtheitsmaßMultiplikationsoperatorSchlussregelVorlesung/Konferenz
22:34
Algebraische StrukturAnalysisGeometrieGruppe <Mathematik>MannigfaltigkeitMathematikMathematische LogikNichteuklidische GeometrieTopologieMinimalflächeQuadratische FormKrümmungInvarianteDimensionsanalyseFaserbündelArithmetisches MittelBeweistheorieDiophantische GeometrieFundamentalgruppeFundamentalsatz der AlgebraHomologiegruppeIndexberechnungKette <Mathematik>KoordinatenMereologieTheoremZentrische StreckungÜberlagerung <Mathematik>Nichtlinearer OperatorParametersystemWurzel <Mathematik>Translation <Mathematik>PunktKlasse <Mathematik>Negative ZahlEigentliche AbbildungMultifunktionVorzeichen <Mathematik>DifferenteMinimalgradRandwertExponentialabbildungSchlussregelGruppendarstellungDifferenzierbare MannigfaltigkeitHarmonische AnalysePerspektiveSchwingungMengenlehreVarianzAusdruck <Logik>UnendlichkeitGrenzschichtablösungGesetz <Physik>ÄquivalenzklasseAnalytische FortsetzungDrucksondierungEbeneEindeutigkeitFunktionalGrundraumHyperbolischer RaumInverser LimesKompakter RaumKontraktion <Mathematik>KugelLokales MinimumMetrisches SystemMomentenproblemSkalarfeldSphäreFamilie <Mathematik>RuhmasseZusammenhängender GraphCoxeter-GruppeEnergiedichteMaxwellscher DämonSummierbarkeitEuklidischer RaumGleitendes MittelVarietät <Mathematik>HomöomorphismusKartesische KoordinatenRichtungt-TestHebbare SingularitätGraphfärbungMinkowski-MetrikRechter WinkelOrtsoperatorSpezifisches VolumenVorlesung/Konferenz
30:20
Algebraische StrukturDifferenzierbare MannigfaltigkeitFolge <Mathematik>GeometrieGruppe <Mathematik>MannigfaltigkeitMathematikOrdnung <Mathematik>RelativitätstheorieStreuungTopologieSingularität <Mathematik>ModelltheorieVerschiebungsoperatorProdukt <Mathematik>KrümmungKategorie <Mathematik>Ausdruck <Logik>DimensionsanalyseFaserbündelUnendlichkeitPhysikalischer EffektUngleichungDichte <Stochastik>ÄquivalenzklasseDiophantische GeometrieDivisionDreiDrucksondierungFundamentalsatz der AlgebraFunktionalGrundraumHyperbolischer RaumHyperebeneIndexberechnungInverser LimesKette <Mathematik>Kompakter RaumKontraktion <Mathematik>Leistung <Physik>Lokales MinimumMaßerweiterungResultanteSkalarfeldSphäreStichprobenfehlerTermTheoremVektorraumbündelZentrische StreckungVerschlingungFlächeninhaltGüte der AnpassungNichtlineares GleichungssystemFamilie <Mathematik>ÄhnlichkeitsgeometrieEinflussgrößeÜberlagerung <Mathematik>Nichtlinearer OperatorAbstandParametersystemPunktKlasse <Mathematik>QuaderNegative ZahlVarietät <Mathematik>KoeffizientDruckspannungRichtungDifferenzkernStabilitätstheorie <Logik>Hebbare SingularitätArithmetischer AusdruckMinimumFlächentheorieAbgeschlossene MengeKonditionszahlSchnittkrümmungMessbare AbbildungObjekt <Kategorie>sinc-FunktionKlassische PhysikMinimalgradMultiplikationsoperatorRandwertStandardabweichungTVD-VerfahrenGauß-FunktionZweiMinkowski-MetrikRechter WinkelOrtsoperatorKonzentrizitätSpezifisches VolumenFisher-InformationMengenlehreMinimalflächeRiemannsche GeometrieDualitätstheoriePunktrechnungRadiusVorzeichen <Mathematik>Gibbs-VerteilungTangente <Mathematik>ComputeranimationVorlesung/Konferenz
38:06
GeometrieGruppe <Mathematik>KohomologieMannigfaltigkeitNatürliche ZahlNumerische MathematikOrdnung <Mathematik>MinimalflächeKrümmungKategorie <Mathematik>PhasenumwandlungAusdruck <Logik>DimensionsanalyseFaserbündelBeweistheorieDiophantische GeometrieDualitätstheorieEinfach zusammenhängender RaumForcingFundamentalgruppeFundamentalsatz der AlgebraGeradeGrundraumIndexberechnungKontraktion <Mathematik>Leistung <Physik>MomentenproblemMultiplikationPoisson-ProzessResultanteSkalarfeldSphäreTheoremVektorraumbündelZentrische StreckungFlächeninhaltSymplektische GeometrieKodimensionÜberlagerung <Mathematik>Nichtlinearer OperatorParametersystemWurzel <Mathematik>ExistenzsatzPunktBetrag <Mathematik>Klasse <Mathematik>KoeffizientEinfügungsdämpfungMinimumIdentitätssatz <Mathematik>KonditionszahlElement <Gruppentheorie>Minkowski-MetrikGeschlossene MannigfaltigkeitOrtsoperatorEinsMathematikMathematische LogikProjektive GeometrieSpieltheorieSymmetrieProdukt <Mathematik>Quadratische FormInvarianteUngleichungGesetz <Physik>EbeneFunktionalGeradenbündelGruppenoperationHomologiegruppeHyperebeneKompakter RaumKoordinatenKreiszylinderKugelMetrisches SystemSpiraleTorusKonstanteNichtlineares GleichungssystemTeilbarkeitAbstandTranslation <Mathematik>Nachbarschaft <Mathematik>GammafunktionQuadratzahlSymmetrische MatrixDimension 2Varietät <Mathematik>KreisflächeHomöomorphismusBaumechanikRichtungDifferenzkernFokalpunktProzess <Physik>Hebbare SingularitätFlächentheorieVorzeichen <Mathematik>Weg <Topologie>Objekt <Kategorie>MinimalgradMultiplikationsoperatorRandwertRechter WinkelRhombus <Mathematik>Vorlesung/Konferenz
45:52
ApproximationGeometrieMannigfaltigkeitMathematikMathematische LogikPerspektiveTopologieMatrizenrechnungMinimalflächeKrümmungInvarianteDimensionsanalyseFaserbündelVektorfeldBeweistheorieDiophantische GeometrieFunktionalGeradeIndexberechnungKompakter RaumLeistung <Physik>Lokales MinimumMetrisches SystemMomentenproblemMultiplikationParabel <Mathematik>Physikalische TheorieSkalarfeldTermTheoremLinearisierungKodimensionNichtlinearer OperatorAbstandExistenzsatzPunktNegative ZahlVarietät <Mathematik>Eigentliche AbbildungKoeffizientKonditionszahlDifferenteMultiplikationsoperatorGeschlossene MannigfaltigkeitOrtsoperatorSpezifisches VolumenFisher-InformationEigenwertproblemGruppe <Mathematik>Maß <Mathematik>StatistikVarianzModelltheorieProdukt <Mathematik>Kategorie <Mathematik>ZeitbereichIntegralUnendlichkeitGrenzschichtablösungArithmetisches MittelDivergente ReiheEinbettung <Mathematik>FundamentalgruppeGarbentheorieGerichteter GraphGrothendieck-TopologieHauptkrümmungHyperebeneHyperflächeKoordinatenPhysikalisches SystemSinusfunktionSphäreStichprobenumfangTorusZentrische StreckungFlächeninhaltNichtlineares GleichungssystemWasserdampftafelParametersystemGewicht <Ausgleichsrechnung>Mittlere KrümmungWurzel <Mathematik>QuadratzahlEuklidischer RaumKlasse <Mathematik>Abstimmung <Frequenz>Kartesische KoordinatenRichtungEinfügungsdämpfungSchnitt <Mathematik>MinimumFlächentheorieWeg <Topologie>Element <Gruppentheorie>RandwertTVD-VerfahrenZweiMinkowski-MetrikRechter WinkelVorlesung/Konferenz
53:38
Differenzierbare MannigfaltigkeitMannigfaltigkeitRelativitätstheorieTopologieWürfelKrümmungAusdruck <Logik>DimensionsanalyseDiophantische GeometrieFunktionalLokales MinimumMetrisches SystemPhysikalismusSkalarfeldLineare GeometrieSphäreZentrische StreckungNichtlineares GleichungssystemNichtlinearer OperatorAbstandParametersystemWurzel <Mathematik>QuadratzahlExistenzsatzPunktNeunzehnRichtungDifferenzkernHebbare SingularitätArithmetischer AusdruckFlächentheorieMultiplikationsoperatorRandwertTVD-VerfahrenOrtsoperatorSpezifisches VolumenMinimalflächeGeodätische LinieDimension 3KoeffizientZweiVorlesung/Konferenz
55:54
GeometrieMannigfaltigkeitPerspektiveInvarianteKategorie <Mathematik>ZeitbereichAusdruck <Logik>DimensionsanalyseBeweistheorieDiophantische GeometrieDrucksondierungEbeneGeradeIndexberechnungKompakter RaumLokales MinimumPoisson-ProzessSphäreStatistische SchlussweiseTheoremTorusReelle ZahlKodimensionNichtlinearer OperatorParametersystemTranslation <Mathematik>PunktWellenlehreGradientVarietät <Mathematik>Kartesische KoordinatenProzess <Physik>Stabilitätstheorie <Logik>Hebbare SingularitätFlächentheorieVorzeichen <Mathematik>Abgeschlossene MengeKonditionszahlMinimalgradMultiplikationsoperatorRandwertSchlussregelMinkowski-MetrikFigurierte ZahlOrtsoperatorEinsMathematikProjektive GeometrieRauschenTopologieWürfelMinimalflächeKrümmungKombinatorBandmatrixFaserbündelGrenzschichtablösungGesetz <Physik>ÜbergangGlobale OptimierungForcingHyperebeneKontraktion <Mathematik>MultiplikationResultanteSkalarfeldTabelleTeilmengeVerdunstungZentrische StreckungTeilbarkeitSpannweite <Stochastik>Stochastische AbhängigkeitAbstandMittlere KrümmungMathematikerinWurzel <Mathematik>QuadratzahlBaumechanikSortierte LogikKoeffizientDifferenzkernMinimumLipschitz-Stetigkeitp-BlockDreiecksfreier GraphRechter WinkelVorlesung/Konferenz
01:03:31
Algebraische StrukturDifferenzierbare MannigfaltigkeitGeometrieMannigfaltigkeitTopologieSingularität <Mathematik>ModelltheorieMinimalflächeKrümmungVariableZeitbereichIntegralDimensionsanalyseFaserbündelDichte <Stochastik>Analytische FortsetzungDiophantische GeometrieDrucksondierungEindeutigkeitFunktionalHyperebeneInverser LimesKugelLeistung <Physik>Lokales MinimumSphäreStatistische HypotheseTermTheoremVektorraumbündelZentrische StreckungFlächeninhaltNichtlineares GleichungssystemFamilie <Mathematik>ÄhnlichkeitsgeometrieEinflussgrößeParametersystemPunktKlasse <Mathematik>Negative ZahlVarietät <Mathematik>RichtungStabilitätstheorie <Logik>Hebbare SingularitätFlächentheorieKonditionszahlObjekt <Kategorie>RandwertKonzentrizitätSpezifisches VolumenFisher-InformationEinsGruppe <Mathematik>MathematikRelativitätstheorieSymmetrieMengenlehreWürfelProdukt <Mathematik>Quadratische FormKombinatorKategorie <Mathematik>Ausdruck <Logik>BeweistheorieFreie GruppeGarbentheorieIndexberechnungKontraktion <Mathematik>Konvexe MengeMereologiePhysikalische TheoriePolyhedronResultanteSkalarfeldWinkelVerschlingungNichtlinearer OperatorBasis <Mathematik>AbstandRuhmasseMittlere KrümmungEuklidischer RaumKartesische KoordinatenKoeffizientIsometrie <Mathematik>Arithmetischer AusdruckCharakteristisches PolynomDifferenteMinimalgradMultiplikationsoperatorSchlussregelMinkowski-MetrikOrtsoperatorVorlesung/Konferenz
01:11:07
GeometrieSymmetrieProdukt <Mathematik>MinimalflächeKrümmungInvarianteAusdruck <Logik>DimensionsanalyseOrdnungsreduktionHyperbolischer RaumLeistung <Physik>Lokales MinimumMaßerweiterungResultanteTermZentrische StreckungNichtlineares GleichungssystemReelle ZahlNichtlinearer OperatorNormalvektorTranslation <Mathematik>QuadratzahlPunktRichtungDifferenzkernProzess <Physik>MultiplikationsoperatorTVD-VerfahrenZweiMinkowski-MetrikOrtsoperatorMathematikRauschenRelativitätstheorieEntscheidungsmodellDiophantische GeometrieGeometrische QuantisierungGrundraumGruppenoperationInjektivitätMomentenproblemPhysikalismusSigma-AlgebraSkalarfeldVerdunstungGüte der AnpassungFormale PotenzreiheSpannweite <Stochastik>EnergiedichteNegative ZahlSchätzfunktionStabilitätstheorie <Logik>Schnitt <Mathematik>Abgeschlossene MengeDifferenteRandwertRechter WinkelVorlesung/Konferenz
01:15:14
GeometrieGruppe <Mathematik>MannigfaltigkeitMaß <Mathematik>Mathematische LogikNichteuklidische GeometrieOrdnung <Mathematik>Produkt <Mathematik>MinimalflächeKrümmungKategorie <Mathematik>IntegralDimensionsanalyseUnendlichkeitUngleichungBeweistheorieEinbettung <Mathematik>FunktionalGruppenoperationHomologiegruppeHyperflächeLeistung <Physik>Lemma <Logik>Lokales MinimumMetrisches SystemPunktrechnungSphäreTheoremTorusZentrische StreckungKonstanteTeilbarkeitKodimensionAbstandParametersystemPerfekte GruppeWurzel <Mathematik>Nachbarschaft <Mathematik>QuadratzahlPunktSymmetrische MatrixDimension 2HomöomorphismusRichtungHebbare SingularitätFlächentheorieVorzeichen <Mathematik>KnotenmengeWeg <Topologie>SchnittkrümmungObjekt <Kategorie>MultiplikationsoperatorRandwertMinkowski-MetrikAnalysisStörungstheorieAusdruck <Logik>OrdnungsreduktionGrenzschichtablösungMomentenproblemResultanteLineare GeometrieTermLinearisierungBasis <Mathematik>EnergiedichteSchätzfunktionDiagonale <Geometrie>Pareto-VerteilungÄquivalenzprinzip <Physik>Rechter WinkelVorlesung/Konferenz
01:25:04
GeometrieMannigfaltigkeitRauschenRelativitätstheorieTopologieWürfelModelltheorieMinimalflächeQuadratische FormKrümmungDimensionsanalyseArithmetisches MittelFunktionalHyperflächeLokales MinimumMetrisches SystemMomentenproblemPunktrechnungSkalarfeldSphäreTheoremTorusZentrische StreckungTeilbarkeitNichtlinearer OperatorAbstandParametersystemMittlere KrümmungWurzel <Mathematik>QuadratzahlPunktFeuchteleitungFlächentheorieVorzeichen <Mathematik>RandwertTVD-VerfahrenOrtsoperatorVorlesung/Konferenz
01:34:55
Differenzierbare MannigfaltigkeitGeometrieMannigfaltigkeitTopologieMinimalflächeQuadratische FormKrümmungKombinatorAusdruck <Logik>DimensionsanalyseFaserbündelAsymmetrieÜbergangGlobale OptimierungBeweistheorieIndexberechnungKontraktion <Mathematik>Metrisches SystemResultanteSkalarfeldSphäreTheoremZentrische StreckungFlächeninhaltNichtlinearer OperatorParametersystemTangentialbündelWurzel <Mathematik>PunktKoeffizientRichtungDifferenzkernIsometrie <Mathematik>Arithmetischer AusdruckCharakteristisches PolynomMinimalgradMultiplikationsoperatorZweiRechter WinkelOrtsoperatorVorlesung/Konferenz
01:41:30
Algebraische StrukturAnalysisGeometrieMannigfaltigkeitMathematikWürfelProdukt <Mathematik>MinimalflächeQuadratische FormStörungstheorieKrümmungKombinatorKategorie <Mathematik>DimensionsanalyseFaserbündelAsymmetrieUngleichungBeweistheorieDiophantische GeometrieGarbentheorieGeometrische QuantisierungGrundraumHyperebeneKonforme AbbildungKonvexe MengeLeistung <Physik>Lokales MinimumMereologieMetrisches SystemMomentenproblemPhysikalische TheoriePolyhedronResultanteSkalarfeldSphäreTheoremWinkelZentrische StreckungVerschlingungFlächeninhaltLinearisierungGüte der AnpassungÄhnlichkeitsgeometrieNichtlinearer OperatorBasis <Mathematik>AbstandParametersystemRuhmasseMittlere KrümmungEnergiedichtePunktEuklidischer RaumNegative ZahlKartesische KoordinatenRichtungSchätzfunktionPartielle DifferentiationDiagonale <Geometrie>FlächentheorieAbgeschlossene MengeKonditionszahlDifferenteObjekt <Kategorie>MinimalgradMultiplikationsoperatorRandwertTVD-VerfahrenMinkowski-MetrikOrtsoperatorVorlesung/Konferenz
Transkript: Englisch(automatisch erzeugt)
00:31
And so when you speak in general about the scale curvature, there are two kinds of perspectives you may have. And one depends kind of results to prove or the methods to use.
00:41
And they just lead you in several different directions. So let me remind you what we're talking about, this kind of topological. So where topology and geometry kind of become diverged. So topological, and I will call them negative, yeah, negative topological results.
01:10
And these concern non-existent theorems. So give manifold and certain topology of that, that it cannot have metric with positive scalar coverage.
01:25
And this was actually up to some point it was starting topics in this metric from point of view of geometers. But on the other hand, there was this guy coming from general relativity, there were
01:43
positive energy or mass, yeah. It was conjecture and then there was a theorem, and then there were these variations of that which were geometric. But they were quite special.
02:01
So let me remind you what's happening here. And so what is the status of that? So one related to minimal surfaces, which I'll come to later, and this was partly motivated
02:21
and inspired by physics, I believe, though I couldn't extract exactly what was happening in physical literature, though I think many things were already there. But let me describe this topological approach coming from Lustig, and I described how we can prove that on the n-dimensional torus, why it admits no metric, this point of scalar
02:43
coverage, the logic is as follows. So first, you cite Dirac operator. So again, let me remind you, what is Dirac operator? This bundle, which is spinor bundle over there, where it acts.
03:02
What is essential, this bundle splits like that, so it is a sum of two terms, for even dimension. And observe, again, it's an artifact, I say it, but I hate saying that. If you prove for even dimension, you automatically prove for odd dimension, because, right,
03:20
because if we have torus of dimension 2n plus 1, you multiply it by one-dimensional torus, if this has positive curvature, this has positive curvature, you reduce odd dimension in case to even dimension, but this curve morally wrong, of course, yeah? You shouldn't do that, right? And in fact, if you do it correctly, you don't have to distinguish dimensions.
03:45
But anyway, so it's even dimensional, so the splitting, so this spinor bundle splits into plus and minus, and the operator exchanges paths. Because as it stands, the full operator self-adjoint, so its index is zero.
04:03
But because it consists of these two paths, they mutually cancel. But each of them has no zero index. And usually, the monoid is taken to here, we call it d plus.
04:20
And this is essential, because the rest of the manipulation is Dirac operator, which you produce new elliptic operator, which is self-adjoint, but doesn't split, which index is zero, but still can be used for certain purposes. As was, I think, first observed by Minou for the rigidity of complex, for real hyperbolic spaces. For complex, it's still unknown, I guess.
04:43
Okay, so that's the point. So if this operator, whatever it is, you know by the index theorem, which is going to give you simple, okay, verifiable criterion of vanishing, on vanishing this index, then you know that there is this harmonic spinous meaning kernel of that.
05:04
But the way when people speak about the history of the index theorem, they, of course, it was proven by, I think, in 17, in 63, and it was announced, and then the proof were appearing in various generality. Prior to that, several years before, equation was raised by Gelfand, who, observing that
05:26
index of elliptic operators, for the holomorphic operators in particular, in general, invariant under deformation, under homotopyism. And therefore, there is a question of what it is, this index. And nowadays, it's just one line kind of argument, you have denoted that.
05:44
However, interestingly enough, there is a paper by Alexandrov, when he proved his famous, give one of the two proofs of this Alexandrov-Fenchel inequality, basic inequality in convexity. And it's about 50 pages paper. In 49 pages, he proved a certain index invariant.
06:01
Some particular elliptic operator, he proved the index invariant under homotopyism. Right, so it was a, and I don't know if it was kind of, the first paper, when I've seen it, was certainly in 30s or something, 35. And so it is a really kind of simple, but a fundamental fact. And then, making formula, you can say, well, just trivial, just look at sufficiently many examples,
06:21
when you can compute it, and then deform it. And so, in this way, it was done by some people in Russia, just maybe 30 before, but they made mistake, they lost some factorial coefficients. So the formula was wrong. So of course, there are some tricky, the formula, which involved algebraic formula,
06:41
involved kind of funny numbers involved, which are not so easy to justify, kind of, I prove it. But anyway, you know, maybe zero, non-zero. But for Taurus, okay, everything is zero. There is parallelizable, there is no kind of, so index should be zero. On the other hand, you do know that Taurus do have
07:00
harmonic spinners, namely a parallel one, right? But so what, but they certainly have zero index. But if you deform the Taurus, this parallel spinners disappear, but the new spinners appear. But now, we are not parallel for the flat structure, but you consider twisting it, this is linear bundle over the Taurus.
07:24
You just, you have, you take coefficient of a spinners with linear bundle, which is flat, but not, have zero connection, but non-trivial, which corresponds to non-trivial characters of the fundamental group.
07:40
And so, yes, when you start deforming it, it does not disappear, it just shifts. And there is a more general theorem for family saying, yeah, it's there. So in some generalized sense, index is non-zero, but you have to understand in the family. And in the family, it takes values, not in numbers, but you have this parameter space
08:01
of all this line bundles, or this U interlay bundles. It happens to be, again, a Taurus, kind of a dual Taurus, Taurus of characters was a fundamental group. So now, for every moment, for every position, you have this elliptic operator,
08:20
it has kernel and co-kernel, take their difference, and so you have difference of two bundles. So if you think about the virtual bundle of a Taurus, which you have to make little effort to make sense of this, because variable, it's element of k-theory, it's characteristic, places define your index. And the, you know, for this example, you can compute again.
08:41
Also for families, it's kind of exercise in analysis to show index is invariant. This k-theoric element, when you deform this metric with the manifold, this vector bundle kind of changes, but only by how much of it? And therefore, it's representative of k-theory doesn't change. And so here, you can compute it explicitly.
09:01
You don't have to know index theorem for that, yeah? And then, it follows that Taurus doesn't have metric or positive scalar coefficient, right? And this is a kind of elementary argument, but if you a little bit go the next step, then you see more. If you, instead of this Taurus,
09:21
you take manifold of dimension n and map here, and the degree of the map is now zero. Then by index theorem, still, you have Dirac and everything provided this manifold with spin. And this is very annoying condition, because we don't, it's, you can't exclude it in this context.
09:43
And so, in this way, you prove that even, if it's not a Taurus, but had another complicated manifold, but spin, then it has no metric or positive scalar coefficient, but you make, for example, in dimension four. If you just take connected sum of this, this Cp2,
10:00
then you don't know, because it's not spin. You cannot, Cp2, I keep forgetting, Cp2 is not spin, just one by one, yeah. And so, you cannot tell, cannot tell, but this method, if it has metric of positive, of positive, however, a posteriori,
10:21
actually, a priori, it was proven with dimension, but it's okay, it doesn't have this metric. And it's extremely unclear what is the role of this spin, in here, you know. And then, from that, the next level of generalization, you can go in two directions. I don't know, maybe I give you another,
10:41
there is another argument. Now, I want to give another argument, where, which carry a little bit geometry, because this absolutely non-geometric, because it depends on this very special algebraic property of the fundamental group, that has its representations. There are flat bundles, so it's a very special problem.
11:00
Imagine you have manifold, which the fundamental group has no finite dimensional representations at all, yeah, which makes things abundant. But then, you can proceed in a different way, in a more geometric way, and which leads you in somewhere else, as follows.
11:27
You go to the euclidean space, which covers, and we temporarily, I don't do that here, just to make it simpler. Instead of, of course, I take the high-order covering for these torus, so that these torus
11:46
become very, very big. We're getting bigger and bigger and bigger, and kind of in the limit, you have kind of euclidean space. But so, everywhere, locally, it looks like this euclidean space on the certain scale. And now, on the euclidean space, what you do,
12:00
you can see the bundle, which is flat in definition, and supported somewhere here. And again, assuming dimension, dimension is even, because it's zero over there, you can think it's coming from the sphere. And here, on the sphere, it has non-trivial chunk class,
12:21
top-dimensional chunk class, even-dimensional. For example, in dimension two, if you are in dimension two, you certainly have an easy case. You just take, pull back with some bundle, it's like a whole bundle, which is optimal in this situation. Now, and then you just do this in the torus, so to have closed manifold. Again, it's pure technicality, but essential.
12:42
When you've seen, it was kind of not pure technicality, when you look at it. It's, at this stage, a minor issue. So you produced a bundle over the torus. But now, because it's all Euclidean space, you can stretch it, yeah? It's Euclidean, you can scale it. And so the bundle become, this picture becomes like that.
13:06
So the bundle become, when you sketch, sketch, sketch, it has the same chunk class, but its geometry become asymptotically flat. When you spread it, it become indistinguishable from flat. Of course, to implement on the torus, you have to go to high and high, high covering, and then become eventually flat.
13:21
And if you twist your zero copy right here with this bundle, it will not notice it was not flat, locally. Because it's almost flat, just an arbitrary small error. And this area becomes smaller than the scalar curvature there if you assume it was positive. Now assume scalar curvature was positive, you assume this area much smaller than scalar curvature.
13:43
Scalar curvature wasn't remained, because I just took covering of the torus. I didn't change scalar curvature. It was normalized to be everywhere at least plus one. On the other hand, the bundle become epsilon flat everywhere. And in this basic formula, the g-Rux squared equals this positive operator squared
14:02
plus one quarter of scalar curvature, plus area term coming from the bundle, called the bundle L, scalar curvature. When this term become epsilon, this goes to zero. You don't have to know what it is. Bundle just disappear, becomes zero. So I can forget it, and still we apply this formula.
14:22
It's still positive, there is no spinners. But of course, topologically, nothing changed. It's just my perspective has changed. So spinners must be there. So observe this, this is another proof, saying there is no thing on the torus. Observe, it's a different proof, because you use completely different spinners in the universal covering.
14:41
In the first case, the spinners, if you leave them from this flat torus, they were just trivial bundle, right? When you take this line bundle, which was flat, when you look at upstairs, it become trivial bundle. So you prove, and this kind of spinners were kind of spreading like that. And here, they're quite different, yeah?
15:00
These are different spinners. And if you see what's happening upstairs, they're different. So I think there are three, four, five proofs like that. So by index theorem, you prove that there are spinners of a certain kind, and then you show they cannot be there if a scalar curvature is positive. And now, you can go in two directions.
15:21
And so what you can prove with that in general. And so one, as I was saying, create a sister algebras, and so what you do, and this was, became dominant
15:40
because it's related to Noether conjecture. And so you can see the infinite dimensional flat bundles. You can see the infinite dimensional representation of your fundamental group, doing regular representation. You already inside of this, it may harbor a lot of non-trivial spinners. So if you consider these kind of bundles
16:02
over your manifold, and takes again, and the point is, if you do it correctly, these concepts of a family of bundles being substituted by family of sister algebra, we should think as a family is kind of this fixture space, non-commutative space corresponding to this non-commutative algebra. Sometimes you can show index again non-zero.
16:23
And so there is a big conjecture saying that for all groups, properly defined index is non-zero. Therefore, you can have medical positive scalar curvature. However, when you specifically verify it, it boils down to usually always to the same condition. At least it is not counterexample for a renewal.
16:41
And so it's possible, which I've tried to justify, that all this activity, both for positive scalar curvature is kind of fictitious. If you look whatever the proof, it's more or less obvious from different perspective. Maybe, but just again, it depends on the conjecture,
17:01
but in some cases it is so. Because of the following thing, so let me give kind of specific argument, understand how it works. Because the Noeckov Conjecture, so next case, after Lustig, who proved it,
17:22
so again, so what Lustig proved is that if you have this n-dimensional torus, and take any manifold and map it here, and then they take pullback of this point, assuming this was n plus four k,
17:41
so I have this generic point, I could pullback, it's sitting here, and this is a four k dimensional manifold, it has a signature, and the signature is homotopy invariant of this manifold. If you take another smooth structure, and have the same homotopy class of maps, this doesn't change. And this is kind of Noeckov Conjecture,
18:03
it provides whole high signatures, and if this were simply connected manifold, this we can do the same. And then this pullback, you know it's never true. If it is so simply connect manifold, take pullback of the point, you can, when you change structure, you take essentially any values, just divisible by something.
18:22
No, the full flexibility of this invariance. This is a corollary of the Noeckov-Brouwer theory, which is quite, quite simple. By the way, it's really very simple, because you are also giving lectures on manifolds, and it's on Poincare Symposium, and proving where you think, you prove almost from zero,
18:40
everything in one hour, including this, but two things you cannot prove. Two things are non-trivial. It's first one, Poincare duality, and secondly, Serre theorem, finite homotopy groups. Granted that, all the rest for topology, called tautology, it's tautological. You just don't have to think, it's taut, taut, taut, taut, it needs to be some little, using some idea of Poincare,
19:01
which is, you know, because if you're using homological algebra, homological theory, it takes several pages to do. But if you use the language adopted by Poincare, it becomes tautology. But then, at some moment, people, at some time, I believe, I'm not studying, of course, carefully, probably Poincare. People thought it's unrigorous, but it was just because language was,
19:20
because not, not, not, proper language was not there at that moment. And the right concept, well, okay, it is not the issue. So I won't say it's very elementary. Brown-Noygo theory is just kind of trivial stuff, and once you know it, of course, it's trivial. And here, because we still don't know the answer, it's non-trivial.
19:40
And so the conjecture, knowing of conjecture, particular instance of which is unknown, if it is true, if here we have arbitrary, a spherical manifold, which is, meaning, universal covering is contractible. And, but this conjecture can show some kind of fundamental class, and it's the same you can formulate
20:00
for any cohomology class inside. And then there's knowing of conjecture, and then if you look carefully, it has nothing to do with, it can be re-formulated purely algebraically. It's some property of the fundamental group, which makes sense for any fundamental group. And in particular, in this particular instance, it says something of the following kind.
20:22
So you say, just formulate another shape of this conjecture. So given a group, say it's corollary with a conjecture, but it is essentially as kind of difficult as an approachable as a general case. So given a group, you can form the,
20:42
I think say a real, yeah, ring of the group, maybe complex vector, yeah. And then there's also conjugation, right? You can make conjugate and replace gamma element by its reverse. So it's a group ring, it's conjugation. And then, if you think about this,
21:01
and you can see which group over this ring. So consider quadratic forms, you have to, might be Hermitian forms, on many variables, and there is usual rule, so if A, so we need to sum, yeah. We can sum them, and if the group can be diagonalized
21:22
without middle terms, you can see the trivial, right? It's kind of definition of real group. And so this is a carburu group. And the point is, it's never trivial. So if the group has homology and dimension for k, non-trivial rational homology and dimension for k, this group is non-zero.
21:41
And that's the point that has nothing to do with many forms. And it's quite difficult in a specific example. Say that again. You're saying that if you're stating the pi, so you assume pi is a, so not trivial homology, what's the conclusion? So again, if there is a group, and the group has non-trivial homology
22:00
and dimension for k, then if you take the weight group, always group ring, right? You can see it's group ring, but it's like, careful, there's some evolution there, there's some extra structure. So you can speak about Hermitian form, not just quadratic, but Hermitian. And you can see the weight group. So it is, you can see the quadratic form over that, and the summation,
22:20
you just add one to another, and there is a conception of a trivial one. And trivial one is just some subject like a squared minus b squared. Right? But the tricky point is, in other identification, when you change coordinates, and because you write,
22:41
it's a ring, like in some base, like that, and in another base, it's another. You have to bring it to this shape in some change of coordinates. And this change of coordinates for non-commutative cases is a whole mess. That's the easy part. But who knows, I write you this quadratic form, how you can check that the reason that some change of coordinates become trivial.
23:00
And there, essentially, all arguments use index theorems, implicitly or explicitly. You reduce it to something which you can visualize. But if you look closely from certain perspective, I'll show you, it becomes geometric property. And this geometric property can be proven, I guess, in this case, by just,
23:21
by very elementary means. Essentially, can be simplified version of an original proof of Novikov topological invariance. Right? And for scaly curvature, it's most transparent. So this, I shall want to explain to you. These things are parallel. They diverge at some point, but so far, I think the two subject matters very much close to another.
23:41
So, so what you do, is scaly curvature. So again, so there was this history, and it's interesting that the first world, of course, other people, first there was Novikov. Then there was Luistic, and then there was paper by Mishinka. And,
24:01
and then, and then Mishinka proven, that if you have, many fault, of non-positive curvature, then, it's fundamental group satisfies Novikov conjecture. So in particularly,
24:21
if it were close manifold, and you map another manifold, take pullback of the point, et cetera, signature will be invariant. But it keep root for all homology. And there is some little difference between, fundamental class and non-fundamental class. So, in how you prove the corresponding statement, in the scaly curvature, which is scaly curvature,
24:41
some of the statement are actually easier to prove. But if you look closer, they of course, more or less become virtually the same. So if you have many, so we don't have now show, we have manifold, which has metric of non-positive curvature, the simplest version of that. And I want to show, it's impossible to construct metric with positive scaly curvature.
25:00
So they are mutually incompatible. Now the argument, the argument we had with Lusik doesn't work, because you know nothing about this fundamental rule. You don't know there are any, non-trivial representation of this at all. Though, at posteriori you know there are, this infinite dimensional representation,
25:21
a regular representation, carries in our structure. But let's do it more, kind of geometrically. But what you do know, that if you go to the universal covering, and you still can, put this kind of a bundle, and because you know, that this hyperbolic space, universal covering of this manifold,
25:41
admits a map to the Euclidean space, is one dimension, and the other dimension, which can, which goes kind of, by homeomorphism, but essentially by its proper map of degree one, and it contracts as much as you want. Therefore, if you take, this kind of a, bundle of Euclidean space, non-trivial, again, assume dimension even,
26:00
take a bundle, localized infinity, non-trivial chain class, and make this map very much contraction, right, it's reverse exponential map, of course, for negative curvature, but any map of this type will do, and pull back this bundle, so you create, now in this universal covering, you create a bundle which, if you twist your operator with this bundle,
26:21
you would make a non-zero, it would have non-zero index, you would have harmonic spinors, but this harmonic spinors, a priori will leave on this space, not on this space, on the universal covering, which is minor point, again, it's some technical point which we can resolve, and see still in the theorem works in this context, though many fall non-compact, you might be careful,
26:41
but if the group were easy to find out, if you could construct finite coverings here, which unravel this manifold, of course, it can transplant it back here, and use usual index theorem, but you can also do that, and so the point is here, you use bundle which nearly flat,
27:02
and unlike flat, and the proof works pretty well, and again, it seems to be you have index theorem, but then how we can recapture it with minimal surfaces, so the point I'm saying that in fact, with minimal surfaces you can prove better statements,
27:20
as far as calico which is concerned, and I think, if you properly transform minimal surfaces to this language, for example, the correspondence here would be, this was what the Shen-Yao proof, they exactly followed the logic of the Novikov proof,
27:42
they just make geometric analysis there, but the logic of the proof essentially, well parallel to the Novikov, and then if you kind of go backwards, it's just what I'm going to say, you have translation again back to topology, and the point is that by using minimal surfaces,
28:03
and in a second I explain how it works, what you can prove that you will have any manifold with boundary, and you know it's calico is greater than say, than one, the manifold cannot be too large, cannot be too large means you cannot map it,
28:21
there is no map to the sphere, of, and the mass, such as boundary goes to one point, and the map have non-zero degree, and simultaneously contracting, strongly contracting, right, so it cannot spread in all directions, it's too much, and then of course, you know because this exactly,
28:41
if it has this negative curvature, it does spread a lot, if you change matter, you only change by a finite amount, so this implies that, and so I'm using point because, we have this paper, you blame Lawson, about, you know, in 83, so it's quite a while ago, Can you say the assumptions on the map to the same,
29:01
you want them to have, The map to the same, so the boundary, this manifold called also x, boundary goes to one point, x goes to s n, map has non-zero degree, and, so for example, its Lipschitz constant, is less than one of n, I know, n is Lipschitz.
29:21
Actually, for the moment, you know almost sharp constant, right, up to one half, you must be careful, so it's better to say scale curvature greater than n, n minus one. So it's the same scale curvature here, and then actually you can make it to the other half.
29:40
You cannot have such a map. And then of course, everything follows, but what I'm using, I just realized, actually, last night, in our paper, we blend, we have different methods, and two different pages, yes, two pages apart, we give version of this argument, and this argument, and we just know each one implies another.
30:02
Kind of funny. Because, you use different scaling, somewhere it was big number here, small number here, and so pictures were quite different, mental pictures. But interestingly enough, I realized only last night, when I was exactly trying to explain similarities in between the two results, and I realized one implies another.
30:22
So, and so let me explain slightly more general kind of statement, which is more interesting, and which is more general, and which is more relevant to the numerical conjecture. And this will be as follows. When actually you need slightly, kind of more subtle version of all that, and it is as follows. So now, today I want to make a break.
30:41
Last time I didn't make a break, and it was kind of too tiring, to everybody. So, so what I want to now, I want to prove. So, yes, I just want, consider a rather simple case. here, there will be manifold of negative curvature compact, of these dimensions,
31:00
background manifold, and this will be n-dimensional manifold close, and there is a map. And here is picture, this is bigger, so. And we assume the map is not homologous to zero. So the image from the,
31:21
so it's orientable manifold, this you don't have to say it. And so when I take the image, of the fundamental class, in here, so it will be Hn, of this manifold, it is non-zero.
31:41
Okay. Then, this manifold, has no matter of Poisez-Kelikovich. So I want to, put it in the setting, there are. So what I want to show, from this data, again, we use it to geometric situation. And this is by the way,
32:00
this was, context where it was proven, by mission canonical of conjecture, exactly in this setting. What I'm, construction is, geometric extract, what was done there. Because there it was again, on the language of bundles, Fred Holm. So you have no geometric assumption on the target?
32:24
No, no. This is a manifold, I'm sorry, it's again, sectional curvature, non-positional. I will make in the course of the argument, I make some simplifying assumptions, to just make it clearer. So what happens?
32:40
This is essential thing. But the point is now, it's not going to the top dimension, goes inside. And there is no sub-manifold of negative curvature inside. However, the same would work. So what you want to derive from that,
33:03
so what you want to, conclude is, that the re-sequence of, now make some assumption here. And the assumption, just to simplify it, because otherwise again, you have to go to non-compact space, you have to develop terminology, it's not that you have to produce very, something radically new.
33:20
I assume that this group is easily finite. So I assume that this manifold has coverings, which kind of, such that, the inside, they have bigger and bigger balls. Yeah, which is more or less standard balls. And then, using that, I take this point in coverings, and I assume that this mapping being just embedded.
33:42
It's just embedded there. And so, so I have this manifold, and here is this embedding, and it's non-homologous to zero. So there is complementary, kind of, by a Poincare duality, and other manifold has non-zero intersection index with that. So we have this manifold,
34:04
lift of my manifold, and then this, the one which intersected, and I take this product. So I look at this covering, and I also have my lift of my manifold, and a lift of another manifold. And what I am claiming is, that I take high and high order covering,
34:22
and I want to map it to this sphere, and now I want to use the following kind of normalization. So it's convenient to do it this way. So it will be sphere of dimension this. So X I octogonal is what? This is just manifold which has non-zero intersection
34:41
index with that, of complementary dimension, for Poincare duality. And their product is mapped to the sphere, and this is of this manifold, which has non-zero degree, at point number one, and such that along these fibers, its distance is contracting.
35:04
I have no assumption along this direction, but on each of them, that it's Riemannian metric, with respect to Riemannian metric, induced from here, it's contracting. And so it's kind of again, it's a little technical point how you do that.
35:21
So you have this manifold downstairs, and you go into this covering, they lift upstairs. And so, you restrict it to every ball, and every ball you just, you collapse all this ball to the sphere, and then you move a certain parameter. And because they have non-zero intersection index, this degree will be exactly the intersection index. It's kind of elementary topology.
35:42
This is a little cheating I'm making here, because what I secretly assume, that restriction of the tangent bundle, of ambient big manifold of this one, when it is restricted to this perp, it's trivial. So all these spheres, all turns to space, you can identify them. It is not true, but it's minor error.
36:01
It's again, just identification, they use error, negligible error. It disappears when the ball become bigger and bigger. And so this is the picture. And this is a kind of essential property of manifolds. Here I used universal covering. I used finite covering. But if they're not necessarily finite,
36:23
here it will be universal, it will be not universal. It will be covering, induced by universal covering of this, right? And then I have this map with these properties. And then the point is, when I'm concerned with scalar curvature, I always can assume that this may matter here
36:41
as big as I want. I just scale it hugely, because you multiply scalar curvature as when I scale it, this whatever it was with this, the scalar curvature goes to zero. So I have again, many for the same scalar curvature as this, map to the sphere by contracting a map. If n is even, I take here a bundle with non-zero churn class,
37:02
pull it back and apply index theorem. And if it were non-compact, you need some non-compact version of the index theorem, which is again, technicality, which is not quite as I explained. It's technicality, but we look closer, you might be careful. Sometime it's true, sometime it's not. But here it's fine. And then with the index theorem you do it.
37:21
And now, how we can do it with minimal surfaces. So again, what is the fundamental feature, which is involved here? It is, I have a manifold X, such that if I take it to universal covering and multiply it by something else, now called another manifold, which is whatever dimension this was n,
37:42
this will be m. Then this new one admits a map to the sphere of dimension n plus m, such that infinity, this is universal covering, goes to one point. and it's, here's sphere of radius r.
38:00
The map is, along these fibers, it's contraction. And r goes to infinity. So no matter how big r, I can construct such a map. And then, this has numerical points of scalar coefficient by either using pull, taking pull back or some bundle,
38:21
or using minimal surfaces, which I start now to, I want now to explain. Can you please repeat that one? Can you please repeat that one? Okay. So I have a manifold, which is, this is a covering of a compact manifold. But this, in fact, it will be not, very little of this will be used,
38:40
essentially, of a compact manifold. And then I know that for some dimension, and then the answer then depends on this method, I can multiply it by some parameter space and other manifold. Map actually doesn't have to be manifold. And one can actually, can be just pseudo manifold, right? It's just a parameter space. It goes to the sphere with non-zero degree,
39:03
such that r can be making as large as you want. And along these fibers, for any fixed metric, it becomes decent decreasing. Of course, you fix the metric, and then you choose r. But this doesn't depend which metric you choose here, because it's decreasing as much as you want.
39:20
all anti-matics differ by a constant. And if you apply it, if it happened to have positive scary curvature, you run to contradiction. Because, this you imagine, many forces make it huge. So, on this manifold you construct this, assuming this number is, even though you always can make it even, because this M under your control, you can add line,
39:41
a circle, whatever. They pull back on this product, of this, the non-trivial bundle, which is as flat as you want, but still having non-trivial, non-trivial churn class, because it's spread a lot, because this manifold kind of, this may be just contraction here, because this r goes to infinity,
40:01
you don't have to say this contraction here. If r is fixed, you have contraction. So, because very big, this bundle over the sphere is almost flat, right? It takes standard any vector bundle on the sphere, and makes phase bigger, the same bundle now from position of this sphere become essentially flat. So, it's almost flat here.
40:22
And so, locally, your operator look as if, it was a twist with the bundle, look as if it's untwisted. on one hand, by index theorem, you must have harmonic spheres. On the other hand, the Neorovich-Bochner formula,
40:40
whatever Schrodinger says, then or there. And so, this is all my understanding. It is true in the following sense. I can give you instances of theorems, when you can prove it with using vector bundles, I give you in a second, in this index theorem, when this thing is invisible,
41:01
it's not there. However, if you look at any concrete example, it is there. but the reason I'm saying, but people have the abstract theorems, and I've given you all that, of such condition, have no conjecture. And this formula is very general. However, if you look at any example, my struggle conditions is fine. So you cannot find any example,
41:21
when all this big theorem wouldn't follow from one way, easy one, right? Maybe they do, maybe they don't. Let me give you an example, where I don't know, kind of, if I can make specific example, when this will not be covered by this scheme. But this is, you see, this is kind of very, I'm using properties, so if this X tilde,
41:40
is universal covering of a manifold X, with some fundamental group gamma, then there's essentially property of the group. You can formulate some property of the group. It's very hard to kind of, to relate it to other properties. And the example, which I want to say, is like that. If you have a,
42:01
a manifold, such that it has a two dimensional cohomology class, is a manifold dimension to M, such that this class H to the power M, equal to fundamental class, be multiple fundamental class, yeah.
42:20
So it's a kind of homologically symplectic manifold. Dramatically it means, that you have a hyperservice manifold, of co-dimension two, M, hyperservice manifold of co-dimension two, is non-zero intersection index. And such that, condition number one, when you go to the universal covering,
42:42
of this, this element becomes zero. So if you can make it a manifold, where it has a two dimensional class, which power of each give you a fundamental class, but in the universal covering it disappears. Then, you can say there is no metric of poisson scale curvature,
43:01
but everywhere, whenever you use this, your operator, manifold must be spin, by the way. So they always must be spin, yes, manifold downstairs doesn't have to be spin, here, but the universal covering must be spin. And if it's not, it's not. Yes, absolutely. There is no,
43:21
no, for the moment understanding what happened there. On the other hand, when we speak about minimal surfaces, you never notice such spin, condition doesn't exist. Of course with spin, you prove more subtle results. You can kind of, you can, if at the spin, you can prove more about some, there are more subtle invariants, which you can detect.
43:40
And even more so in dimension four. But, but well, in general, we don't know what it is. In your example again, so in order to ensure you have no poisson scale curvature, what are the assumptions? So here again, On H, Here we have a cohomology class, which is here, in such that it's powered equal fundamental loss.
44:01
Yes. It's like some blackmail. And then, but when you go to the universal covering, this class might become zero. For example, manifold is spherical, it's contractible covering, and everything goes to zero. For example, you have four dimensional manifold, which is a spherical universal covering is, universal covering is, universal covering is contractible,
44:21
and then, and it has non-trivial H two, rational H two, then possible by duality is square, of some class will be non-zero, and it has no medical poisson scale curvature. You indicate how you deduce this? Yeah, but in this instance,
44:42
so here the proof is based on the fact that you can, is that your line, the bundles you use, actually the line bundle, because when you have such a class, there is, there is a line bundle, which has non-trivial connection. When you go upstairs, it become trivial. The ones become trivial,
45:02
you can take root of this bundle, you can take N's root. I'm gonna take N's root, it's curvature goes down. So take high and high, high root down, and become as flat as you want, and you twist with this bundle. And then you have, but you have to see that it's work, you need some index theorem, but it's kind of rather elementary. The moment you say it, it become rather elementary.
45:21
But there is no such picture, there is no contraction map, it's purely area wise argument. So geometry behind this approach with index theorem is different from the one with minimal surfaces. Even with the proof seemingly identical theorem on the bottom is something else.
45:40
However, when you specifically come to compact manifold, and you have this property I described, it's possible the seal is covering, seal admit this map to the sphere which contracts. Which there is no example, the example which I know, which I cannot analyze, I like that. And some example you don't know. But so there is no convincing example saying,
46:01
this condition is satisfied, and another is not, right? And usually in my impression, people who work in this domain, there was no conjecture, they never tried to do that. They hate to do that, yeah? And sometime it happened, they publish a paper, long paper, new condition. And somebody else say, well, we just treat you and you follow some other condition.
46:20
Having nothing to do with all this index theories. One instance, however, where it seems to me at the moment that we're very far from relating this minimal surfaces, which I shall explain probably after the introduction, but I formulate this as the following,
46:40
which is related to the Elm-Konst, what he calls longitudinal, longitudinal assumption, index theorem for foliations. And this theorem says in particular, or rather the proof of the theorem as I understood it when I read a long time ago. Now I say it by memory because I haven't,
47:01
you know, it's kind of mathematics. It doesn't stick to my mind too well, but what it follows, the following kind of a geometric statement, that if I have a manifold X, and such that it admits, it's not a compact manifold now. Again, to Rn, which has positive degree,
47:21
proper n distance, it's Lipschitz. It's a distance decreasing. Then it admits no foliations, such that induced matrix has positive scalar coefficient. And this is a, yes, absolutely kind of, it's certainly geometric property. It seems to me this is related.
47:41
It's much stronger than saying that manifold itself has no positive scalar coefficient. Positive meaning scalar, greater is something here you might be, uniformly positive here. It's again, it's very big difference you're having uniformly positive and non-positive, right? You can make this kind of expanding thing
48:01
like parabola at this curve, which is positive but goes to zero, but you cannot keep it uniformly positive for this shape. And this is a rather significant distinction. And for that, I submitted this kind of geometric picture doesn't quite fit.
48:22
It's a close come to that, but you just say I couldn't really make it using minimal surfaces. It may or may not, but this would be extremely interesting. It's absolutely unclear why it should be true if you think about even for foliations, when you understand very well, even if you assume kind of leaves compact or something, it's, I mean, compact clauses.
48:42
If all leaves have compact clauses, it's very unclear what happens. There's some basic issues with understanding geometry of foliations, which is missing. By these techniques, you get it, but you don't know what it signifies, at least from some perspective. So this is how it goes.
49:02
Okay, so let's now make a break, and then after the break, we speak about minimal surfaces. So just let me remind first how originally this minimal varieties were used here by Shen Yao, and in some sense, there is also kind of some parallelism which take this Shen Yao approach,
49:21
and this untwisted Dirac operator. This gives you topological information, but doesn't give you geometric information. But if you slightly modify this, what they were doing, and then you can also extract some geometry, right? So when you have flat bundles, it depends on the fundamental group, you don't see geometry.
49:42
When you have almost flatness of the bundle, bundles tells you some existence or non-existence with that, now become a geometric invariant. And though some time you can relate it to. And the same illustration is here. So the logic of the Shen Yao was quite simple.
50:02
But it was based on previous work by Kashdan Warner. It's again quite simple, of course, once you do that. So what Kashdan Warner proved is as follows. If you have a compact manifold, this compact remaining matrix is zero, and suppose that the following operator is positive,
50:21
minus Laplace operator with respect to this metric, plus something like one quarter, slightly less than one quarter. Say one quarter, yeah. It will be a slightly different number depending on dimension, but it's approximate for scalar curvature.
50:42
And then this conformal change with the metric which make the scalar curvature positive. So what means this operator is positive? It means if I take any function and take this integral, I'm sorry, g psi squared, plus this one quarter n scalar times psi squared,
51:04
this integral always positive. So this term kind of, so it means if there is a little bit of a negative curvature, it's okay. But in what sense a little bit, it's not so clear, right? And then by the way, we shall turn, we have time to explain what happens
51:20
for this with zero operator. This little, so how much you allow negative, what kind of negative, is kind of very delicate point. And one of them is kind of appears basically like in Penrose conjecture, when you can precisely say how much negativity you can allow something like negativity. But anyway, and so what is the proof?
51:41
So once you have this operator positive, you take first eigenfunction of that. And the first eigenfunction of positive operator is positive. And then you multiply your metric by some f to the power, and I keep forgetting what power you have to take. And if you take the right power, and just exactly the one you should get
52:00
actually when you construct, I don't remember to explain this, Schwarz's metric. So in dimension, I think in dimension four, it will be f squared, I guess, yeah. And then you just compute what happens to the metric and the conformal change and in dimension exactly this term jumps up.
52:22
And when it's positive, there will be some multiple of this function, some power function and this thing applied there. And this again is one line computation, which I just don't know how to make it without computing. Of course, if you look at the examples, it's kind of obvious and then kind of these principles, right?
52:41
The point it is right. And there is this coefficient and what is essential, this coefficient is less than one half. And that's what is essential. And that's, if not, nothing could work, right? This coefficient appears in this operator for all dimension. You know, in Dirac operator, it's one quarter.
53:00
Here it's even smaller than that, but the correct number is one half for some reason. And so that's a little bit annoying, but then on the other hand, this kind of linear numbers is crucial here. And then, so what you do? So you have a remaining manifold and take some minimal sub-manifold and minimal in some homological cluster,
53:22
it's locally minimizing, co-dimension one. Assume it's smooth, it's locally minimizing. Because locally minimizing, when you deform it, your long vector field with any y psi is volume goes up. Therefore, certain operator, because it's linearized this equation, you see,
53:42
become positive. So I have to compute what the second variation is. And if ambient scalar curvature is positive, this operator is exactly minus or plus, plus one quarter of the scalar curvature of y in the use metric. And again, just you write this formula and kind of the point,
54:01
which was missing in earlier papers, my understanding is, of course, I haven't known what already was in physics literature, but mathematically literature, this kind of computation in dimension three at least has been done, but what was missing is the formula that we make if you, that, so a Ricci curvature in this direction,
54:21
how it moves exactly control the second variation of volume, but the Ricci curvature in this direction plus all curvatures together in this direction give you scalar curvature of the ambient manifold. So if you subtract the two, you come to scalar curvature of this manifold. So you can express it as scalar curvature of sub-manifold rather than Ricci curvature in the ambient manifold.
54:41
And with Ricci curvature of ambient manifold, yes, I posted it, there was an old paper by Buraga and Topanogov when they do something about three-dimensional manifolds and it was for certain bound on Ricci coefficients that showing they cannot have two short geodesics.
55:00
It's much more subtle geometric argument where this algebra was missing there. Not that you need it and they meet, but if they realize that, that will have more stronger consequences. Another interesting point, historically when people doing these minimal surfaces, they were very much preoccupied even for dimension two. What kind of singularities were there? Which is strange because already by that time,
55:21
there was a work by Fedor Fleming for surfaces at least, it was quite well understood. However, people kind of in dimension two and originally I think also in this shown yellow paper, they proved themselves existence of minimal surfaces instead of referring to the work by Fred Fedor which was done in 1970, so 10 years prior to that.
55:44
There was some kind of much concern with that, which is justified only partly. But then by the time already it was known, by 1979 when they wrote their paper shown yellow,
56:02
that after dimension seven, co-dimension ones of manifolds when they read absolutely minimizing, they're smooth. And this was proven by Fedor in 1970 based on a 68 paper by Jim Simons. And this Jim Simons is of course the most kind of essential gradient there, right? Because all that is a general compactness
56:21
that I thought is kind of clear. But again in Simons, there is a tricky computation about minimal sub-varieties in spheres with some particular condition which come from the fact that the base of a cone and this cone is minimizing, just stable. And the stability condition transformed to this surface and it's kind of beyond my understanding
56:41
though funnily enough, I was formally translating this paper to Russian, in Russian, but I was not translating that way. And this is how I learned the subject because at some point there was a mathematician, Fett, who was actually a professor of topography, who had some problem with authorities because he was, I think it was about the time
57:03
he was not very happy about the invasion of Czechoslovakia, and he lost his job. And so to survive, he was making translations for some day, but there he couldn't sign it. And so I was supposed to sign this paper. And so I read a little bit, I understood nothing, but I remember that, yeah? And this is how I came acquainted with subject matter.
57:22
And I didn't appreciate at that time that it was a really great paper, except I couldn't understand the paper by Simons. But this really kind of there in this domain is kind of one of the really brilliant papers. And which is, well, significantly, we see, I don't think anybody followed deeply enough along this line until it happened, the white dimension seven.
57:41
Of course, from a certain point of view, it's clear why it fails. This Bambiad is juicy, which people were very happy about that not breaks the high dimension, but it's kind of obvious. I mean, just from general principles, routine computation, you do it. It doesn't require any efforts from modern perspective.
58:01
Intellectual effort, it's just routine. But this Simons paper is kind of mysterious, the white dimension seven, and seven has nothing to do with Scali-Kirvich, Scali-Kirvich, all phenomena, dimension one, two, three, and then it stabilizes. But anyway, it went on. But now, so that's the proof. Once you have it, like in the torus, with some method, you take the cycle,
58:23
you construct an automatic poisson scale, you go down, and just do the surfaces, and of course, there is no surfaces. And the point is, of course, that the condition, which is, though it's not being a torus, but admitting a map of degree not equal to zero
58:41
to the torus. This property inherited by hypersurfaces, which are not homologous to zero, because you can project them to one of the coordinates, not homologous to zero, some projection, also not homologous to zero. It has positive degree, and so the induction works. But, I don't remember if I said, but in fact, the theorem proves something stronger than that, okay?
59:01
Topological condition is stronger, because, and so the classical manifolds, when they rule out, the Poitoskelekovich is not the one which cannot be covered, and there is a kind of real theorem, due to Shik, saying cannot be covered by index-theoretic techniques.
59:20
All invariants come from, there are example where all invariants vanish, coming from zero-cooperate, all these indices, generalized indices vanish, but this argument still works. So it's quite, quite, quite simple, logically simple, and this is technicality, what to do in higher dimensions.
59:40
So one point, you don't need spin, which is kind of pleasant. It's kind of maybe not so essential, but still. But then, but still we want to carry the argument and to go to higher dimensions, there are singularities. And this is absurd, that you have to bother about them. If you look at all formulas, how you prove Poitoskelek's operator, the more.
01:00:00
Kind of, if it's smooth, but the closer to singularity, the better formula becomes. It's only add, can make operator only more positive. But you cannot use it. You have to fight the singularity. A couple of years ago, Shun Yao wrote a paper when they explain how to go around singularities.
01:00:21
But you don't know, but you don't know actually singularities through there. Because in dimension, ah, this way I can explain now, it's very simple. So within dimension eight, when you have, you know a singularity isolated, as follows, again, from Federer, you know they're unstable. You slightly move your data,
01:00:40
add the boundary value or something, and then become smooth. And this very, I'll explain in a second, argument is a very simple argument, and looks must work in all dimensions. But somehow it doesn't. It's negligible difference, yeah? So, and if you prove, we could prove that, and conceivably that may be very simple proof, because in this case, you don't have to know nothing.
01:01:03
You can explain it, you know nothing. Just compactness or something. Yes, it's from hand waving. But then this hand waving breaks down in higher co-dimensions for some stupid reason. And then, we still can bypass it. There are two kind of, rule one by Shun Yao,
01:01:21
which still doesn't, it's not as good as in other dimensions, and another is due to log camp, log camp, which is, both prove very complicated. So I haven't read either of them in detail, and I'm more inclined to trust Shun Yao,
01:01:43
because, well, at least I understand the intermediate statements. And log camp is just, you have to understand, not only prove a statement of about 10 other theorems quite sophisticated about geometry of minimal varieties. And it's very difficult papers. But, so let me explain how you eliminate singularity
01:02:03
in dimension eight. And this related to the following problem I remember as an undergraduate. So if you have this kind of figures in the plane, you put them everywhere in the plane. The number of them may be the most countable.
01:02:20
You cannot put uncountable many of them. Like intervals, you can put uncountable many intervals. Segment, but you can't put uncountable many of y's on the plane. Corollary, in dimension A singularity unstable. Is it clear for you why?
01:02:41
No, but you see how you can put, if you put one inside, you always have a space. You always have a space. You always kind of, they don't want to come close. They just don't want to come close. But that's exactly kind of geometry behind it. And this was the reason, of course, why it should be, in general, it should be so. Why singularity should be unstable? Because if you move it in a continuous family,
01:03:01
this figure y, this singularity must disappear. You have this family of curves, one inside of another, right? They all cannot be all singular because you only have countable many of them. But in a continuous family, they're uncountable many. And that is the geometry,
01:03:21
and actually was proven by Knath and Smale. I haven't read his proof. His proof, and look, I just, he uses too many theorems, you don't know. But if you, now let me explain it from this position. So what is the idea? You have the singularity, you have this, what make it singular is singular cone, right? So if you take minimal variety at some point
01:03:41
and scale it, become bigger, bigger, bigger, bigger. And then from general, performs convergence. And moreover, it subconverges to a cone, right? And so you end up with a cone. And if this was flat cone, the point was smooth. If it's singular, it's singular. What is, by the way, is still unknown
01:04:01
if this limit is unique. It's in some cases known to be unique, but the priority is only sub-limit. And so what you use by the sub-limit, because you know that the volume is monotone function. It has then increased faster than a flat case, right? And on that hand, there's compactness. And therefore, if you have monotone function, you know, it has tendency to converge.
01:04:22
But it's here, you don't know to kind of, which shape it takes. Maybe approximately on here shape and here a slightly different shape. And we can oscillate, yeah. By the way, one of the problem, I think if you knew uniqueness of the cone, it would be very helpful. And actually, Locke can prove something in this direction.
01:04:42
But he doesn't prove this instability, but come very, very close to that. He says we have, if I understand correctly, because he is vague in his statement, that if we have this minimum of a variety, it can be modified, become non-singular, but it will be approximately minimal. It will be very, very close, but not totally minimal, not truly minimal.
01:05:03
So what is this cone, where it is isolated point? And they are not flat, so this is sphere. Inside of sphere, you have some minimal variety in this cone over this sphere, right? In this cone. But now imagine you move such a family of this minimal variety.
01:05:22
Start moving them and take these cones. But then you have one cone inside of another cone. But this is exactly what I'm saying. It's important we have some singular thing. It's really going in all directions, yeah? Because this minimal thing is not sitting in one hemisphere. Either minimal thing can be in one hemisphere. So it's spread over the sphere, and it's not being sphere.
01:05:41
So you cannot push one cone inside of another, because if you take this minimal variety, move it a little bit, it intersects itself, right? Because you cannot move it a little bit without intersecting itself. And that's a rationale behind it. But what's the difficulty? Now, if you try to make it rigorous,
01:06:01
so you go to this limit. So imagine I have family of this minimal thing, one inside of another. And so the theorem is, you have this family of minimal varieties, one inside of another, then the number of points where they will be singular will be nowhere close, nowhere dense. It's close. Obviously, you have to prove it's nowhere dense.
01:06:21
So you cannot have the whole interval of them. So you take these cones, and then say, I have moving family of cones. So cones go inside, and say, it's impossible. And say, I proved it in all directions, in all dimensions. Because about cones, it's true in all dimensions. You can't put cone inside of a cone, except in all this way. However, this is a subtle point.
01:06:42
When you blow it up, so what may happen, you have the singularity, this cone shifts. Yeah, and the cone moves here. The singular point moves along. And you don't have family of cones. But when it happens, it's the only way to happen. This singularity spreads over. And then in some limit situation, you have one-dimensional singularity.
01:07:01
But you know, in the dimension, they cannot do one dimension. But this exactly, the sliding, which only kind of makes singularity bigger, does not allow you to make in higher dimensions. But it's kind of a minor thing. I mean, I'm just, I'm pretty certain that, well, that you may have a one-page argument,
01:07:20
if you know a little bit more about minimal varieties. Well, the argument by Bose-Lokham and Scheer is too heavy. And besides, there should be arguments without which you only welcome singularity. And that only makes the result better. Whenever you know, whenever you can, when you will arrive, then they only improve all inequalities.
01:07:42
But now, whatever it is, it doesn't give you, again, much information about your manifold. So all you have is a manifold. You know it's, again, this poise de poise, but what you can say about it, geometry. And how we can extend it to more general geometry. It depends on topology. You need this exactly, this nested family of co-dimensional objects.
01:08:00
So you need strong assumption on topology. But interestingly enough, if you slightly, there are two modification which you should may introduce to this method. And one is that you allow boundaries. So you have to consider minimal surfaces with boundary.
01:08:23
In boundaries, there are two kinds of situation. When you fix the boundary and when you have free boundary. So you just, boundary is somewhere, number one. And number two, you allow and introduce a weak term. This weak term, kind of similar as what you just twist your spin bundle
01:08:44
with another vector bundle. So in this Dirac theorem, there are two. The main term is just this a genus, which is come from manifold, from smooth structure of manifold, kind of subtle and powerful in the variable. And then there's something kind of more elementary, this Chern class, and we shall get to geometry.
01:09:01
And he is similar to that. What the function you can see there is this. You can see that, so your manifold X, you can see the domain there, say omega, it has boundary Y, the omega. And what you do, it is you take this, it's N dimensional manifold.
01:09:20
So you take volume of your Y minus some measure of omega and use some measure function, some measure on your manifold, which may pose you for negative. And in simple example, it's given by continuous density function, but not necessarily today.
01:09:42
Good example when it is not, when it's supported on hyper surface itself. And I shall explain some of that today. And so this is a function. And if this measure is just constant, right? So what is the solution of that?
01:10:01
There will be hyper surfaces, mean curvature, which point equal to that, to this measure. And they've got very, actually thesis called them kind of brains or something, yeah? I saw some literature, they say brains, brain, something like that. The real Chern class here, it's volume minus measure, which Chern class?
01:10:21
It's a pure German, here the Chern class is here. Everything will be geometric, Chern class disappear in the story. The equation would be prescribed mean curvature? It's a stationary point, we prescribe mean curvature, but if they're minimizing, they will be stable. And we do stability condition. And typical example, the model example we have in mind.
01:10:44
So you see, the point is, usually people say, oh, take something surface of minimal area containing even volume. But that's inconvenient because, but what you say, not minimal given volume, but given integral.
01:11:00
For example, consider kind of concentric spheres. They're extremal for that, but what's the function mu? Does that mean curvature of this? In the same hyperbolic space, whatever. And they're extremal. It means that if you now have the same kind of geometry, but the change, you may scale a curvature more positive.
01:11:22
And you look what happened there. You see that this operator, again, you have some positivity such that, if you take now, oh no, I'm sorry, I'm running a little bit ahead of myself. So again, you can use positivity of some operator and to reduce, to kind of reduce dimension,
01:11:45
but in a more subtle way than we were doing before, which I now have to explain. So there are these two major extension of the minimal surface case and what you can get with them. So let me give you instances. So eventually what you can obtain,
01:12:00
kind of results which follow are, yeah, no, I'm not ready to say it because I haven't said anything before. Now I explain some other scheme, how we can use this second variation equation. So the second variation equation, and we have so many for x and has some poise,
01:12:24
same poise scale curvature, but this applies to any scale curvature, you have to change the formula. And you take this minimal sub-variety, then the operator which is positive is this minus Laplace operator plus one half of the scale curvature.
01:12:41
So if you write down all these variations and the term disappears, this poise is a good term, there are some other term in this norm of curvature, this manifold, and this equality exactly kind of like that, if this is a minimal surface totally, it's kind of like totally generic here. It was actually quite poise operator,
01:13:03
lots of poise, and the more singular surface, the more poise of this operator, but this poise. And what you know about this operator if you now, so it's one half is bigger than this one quarter, so you have to use the full power of that. What you do, you take this manifold times real line,
01:13:25
the metric here was jx squared plus you have this first eigenfunction, I keep forgetting maybe, of this would be squared times dt squared. So it takes some power of this.
01:13:41
And then this will have poise scale curvature. I might be wrong about whatever, whether it's squared or something, there might be some other power, I think squared. So the point is that once this operator, we want to have a stable minimum sub-variety,
01:14:01
then you can construct here a metric which when you project back, you have your old metric and in this direction, it's invariant in the direction of r. So here you have ambient space of God knows what, and it was minimum sub-variety. And now the ambient space becomes just product,
01:14:21
not exactly product because this r may be scaled this or that way, but it's invariant under. Now we can repeat this process like we did in Shen Yao, but advantage is we now have pretty good track of geometry. It was not some conformal change, there you lose completely what's happening to geometry of this manifold,
01:14:41
maybe I better call it y, this was minimum sub-variety y, y embedded in x. X was complicated manifold inside minimum surface. Now this y inside become totally geodesic and you have the same picture, but now it's symmetric under r. Keep doing that, and when you keep doing that, you arrive just at full symmetry.
01:15:02
If you have dimension n, what you have metric of poise scale curvature and invariant under translations. And that's kind of ultimate thing, certainly impossible. And all geometry of this y is still there. If y was big, then this would be big, you see?
01:15:22
When take conformal change, you lose control with sign. And here, this manifold, of course, bigger than your y, just because you only sketch, you change it in one direction, but you don't change the y direction. And when you do that, for manifolds with boundary, you immediately conclude, and this was done sometime.
01:15:44
I believe though it was a coarse argument that if you have manifold with boundary, it cannot be too large. What you do, argument was like that, take minimum, so boundary here, construct minimum surface, which may go somewhere, still the same picture applies. And you have symmetrization,
01:16:01
which you shrink by a half, do it again, and then it has a very bad exponential factor. So you just, it was very, very unprecise argument. However, this can be done sharp. Now I'll certainly explain how this can be done without losing any constants in a very simple and transparent way,
01:16:22
using the same kind of computation we had before. There, but now, so what I want to prove is this kind of statement, that if I have manifold topologically,
01:16:41
torus cross interval, such that scalar curvature,
01:17:01
and this symmetric here, so it's symmetric homeomorphic to it, and scalar curvature of x is greater or equal than scalar curvature of the sphere, which is n, n minus one. And here, boundary consists of two parts, so the boundary called the plus and the minus,
01:17:23
corresponding, I better start with minus and then plus. Distance between two ends. There's this distance in this geometry x is less or equal, I hope I remember it correctly, to pi divided by n.
01:17:44
And this sharp inequality. In dimension two, in dimension two we have just sphere, two-dimensional sphere, right? And distance here is two pi over n, because I said two pi over n,
01:18:02
here is pi, so is pi over n, okay? It must be, it must be, it's correct, correct. N is two, yeah, yeah, perfect. N is two, it cancels pi, distance between two points is pi. So this is an extreme object. So in high dimension, it will be not something of constant curvature, it will be of variable curvature.
01:18:21
But there is extreme object, which is not the obvious one, right? But this sharp inequality in all dimensions. And when here, there is a torus. And so you don't use implicitly that this is the metric here. But then there is another kind of more precise statement,
01:18:42
which is combine our other things, but this more general one in the higher dimensions, so here again, there is a problem with singularities. And here you only need the Schrodinger theorem, which is a kind of more inclined to trust. But the same statement is true. We have, instead of torus, I take any manifold.
01:19:01
This kind of is two ends. And with the property that if I take any hypersurface here, it admits the metric of Poiseu-Skelikovich. For example, it may be like this Kumba surface times interval, which we don't admit. And then the same is true.
01:19:21
So if Skelikovich of the ambient space is what I said, this distance will be like that. But here, I need this log-hump. In dimension of Kumba, I don't need it because dimension is five. When dimension goes up, you have exotic sphere of this heat-strain spheres, which have numerical Poiseu-Skelikovich. I need this theorem by log-hump,
01:19:42
which is not exactly his theorem. I need generalization of his theorem. I'm sure it follows if his argument is right. No, if his logic is argument, one might be stable in some way. But because I don't know the proof, it's just conjecture. But I'm fairly certain if what he's written in his paper is okay, then it's okay.
01:20:02
But for Shonyao, at least I can use the lemma from their paper, which is, if correct, applies, literally. So how this proof goes, let me explain. And this again goes by symmetrization. So for this case, instead of taking minimal surfaces here,
01:20:23
homology class has a cylinder. And I take this co-dimensional one surface, again based on the low-dimensional cylinder, and use the same logic of symmetrization. Every time I have the minimal one, so I can multiply by a circle, and eventually I reduce the situation when this metric is invariant under the action of the torus.
01:20:41
It becomes one-dimensional problem, and you solve it explicitly. And this was actually done already in our paper with Lawson. And you solve this metric with a tricky, it should be metric of the time, gt squared plus some function phi squared, g times gs squared, also phi, but this is a tricky function,
01:21:01
integral of some tangents or something. But this kind of elementary type equation, you solve it explicitly, and you get this number. And then, if you look slightly more carefully at this argument, you don't have to be actually a torus. What is actually true?
01:21:22
Yeah, well. And so what's amusing about that? So let's discuss this point already. That is quite, quite kind of, has strange consequences. Namely, if I have n-dimensional sphere, and I have torus there of co-dimensional k,
01:21:41
ki, the n minus one for n minus two, then it cannot have thick neighbor, say co-dimensional one, just two, not, right? It cannot have big neighborhood around him, because if I put it aside, I can't push him aside more than this distance, because sphere has poiseu-scalar curvature.
01:22:02
I don't use the fact that it has poiseu-scalar curvature on the scalia, so you cannot have kind of wide vertical band inside of the sphere. And because this property is invariant on the Lipschitz maps, the same is true for the ball. Inside of the ball, they cannot have this torus with a wide band. In particular, it must have big curvature.
01:22:21
If it has small curvature, it can have wide band. And I don't see elementary proof of that. So a corollary, if I have a torus inside of here, its curvature must grow roughly like n. N-dimensional torus, co-dimensional one, see, you can do it in ball. It's one of the principle curvatures, must be of order n.
01:22:41
It's a small dimension, but at least some constant, of course, yeah. But for large dimension, become this asymptotics, and the same happens to be true if I have this immersed, actually, or embedded, don't exist, immersed, exotic sphere of this kind, even very special sphere. And nothing you can say about other spheres. And it looks very absurd.
01:23:01
An elementary situation, Euclidean space, and you use all this kind of, all this strange stuff. And you cannot kind of go through usual, you should make this argument inside immediately, only scale curvature being kept track of. Everything else just disappears. All this geometry become invisible. And so that's very bizarre,
01:23:21
and I don't know what happens to other example. If instead of torus, you take product of the spheres to the power n, I have no idea what happens. Of course, absolutely zero. No estimate at all. It may happen, you can do it with bounded curvature. So I don't know, if you take this s2 to the power n, and you want to put it into the space of dimension,
01:23:42
I know, 2n plus one. If you, in the ball here, in the unit ball, if you can do it with bounded curvature, regardless of dimension. Of course, the obvious kind of thing, kind of, it's curvature grows pretty fast,
01:24:04
but for this embedding, yeah? But it's very unclear, but in dimension one, yeah, actually there is no obvious simple embedding. Obvious embedding have in the middle dimension, and then it grows like square root, in examples.
01:24:21
But you, and then you cannot do better than square root in any dimension. So it's completely, completely strange situation. So, again, so the question is, I have this s2 to the power n, I embed it to the ball of the unit ball, what kind of the minimum sectional curvature can achieve?
01:24:41
And the best embedding I have for n goes to infinity is like square root of n, and then follow co-dimension, you can construct some rather artificial embedding, slightly, also curved, and they, in co-dimension one, will be curved more than n, actually. They will be like n to some power,
01:25:01
but this very strange, and only scalar curvature tells you what happens, for all, for all non-trivial kind of examples. Very, very, very strange situation, and I think it's quite interesting, exactly this, that you realize how poorly you understand things. This is, it really makes you happy.
01:25:27
Of course, at the beginning, then we keep an understanding, you're not so happy. So that's, so what happens here?
01:25:40
And then the reason, and how this goes along with, so at which moment you need this function of what I call mu bubbles.
01:26:01
So what they're good for. So what you can prove with that, they're useful.
01:26:22
And this exactly what I said, that if I have, again, this kind of picture, many forms to n's, and you know, no hypersurface here has metric with positive, positive scalar curvature,
01:26:41
then the reason, then the distance might be what I said. So what you do, you construct a function. So you have this model example. I said this is extremely metric,
01:27:00
and then there is familiar hypersurfaces. And they have certain mean curvature, right? And so in the shrink, of course, this mean curvature from our side becomes very kind of, very negative. But then you just take this function mu
01:27:22
and just solve this variation of problem for this function mu, right? And if the distance was a little bit bigger, you can the function go a little bit slower. And therefore, here these two things,
01:27:40
you can solve this problem because by maximum principle, the solution cannot hit either of the two boundaries. So they're used as a barriers for this problem. So if distance was big enough, and you slightly, you could make this mu move slightly slower from this value, from this value, right? Because mu kind of becomes,
01:28:01
first positive, then become negative when you go in this direction, right? Mean curvature depends on the sign. You go in this direction, first positive, positive, then you become negative, negative, right? Like on the sphere. From this point of view, it was growing, growing. So positive, even shrinks become negative.
01:28:22
So we can solve that. And you can show that if you look this thing and look what happened to a second variation equation, it tells you exactly that it is, that again, this operator, one plus one half
01:28:43
minus Laplace and scale curvature is positive. And therefore, this conformal change, you don't have to do anything. It has Poisson-Kirchhoff's law, but my assumption was this is such manifold. Therefore, distance must be like that. But here, you see, because I need to just, not just kind of intermediate step,
01:29:02
I need exactly this manifold to be there, and I need log-ham theorem for that purpose, which is, he says it slightly differently, but I think it's almost formally followed from what he said, but he's saying it so, and specifically it's so. And then, so what follows from this, for example,
01:29:27
now this probably still wouldn't follow. There were some other things, which need more to say. Let me give another kind of corollary of this argument,
01:29:40
which is, I think I'm using, because the topology completely hidden there, right? And which is almost sharp, and it is following. But imagine I have a cube with, or dimension n, and it doesn't have, of course, actually a cube,
01:30:01
but anything I've cubically shaped, and it also comes as a matter of Poisson scalar, given matter of Poisson scalar curvature, I want to give another way how to bound the size. These, by the way, all can be used to bound the size, but they're not very sharp, and this is almost sharp.
01:30:20
And scaling, again, of this, greater than n, n minus one. And then there are the following functions here. You have distances, you have opposite faces. On cubes, there are kind of n pairs of opposite faces,
01:30:41
and there are these distances. And then the statement is that maximum distance of that is greater or equal than one over square root of n. I don't remember the constant, some number, say, put in the two. But I know this, it's not sharp. It's sharp up to a factor of pi over two or something.
01:31:02
So in quality, what you have is not sharp for dimension two. For dimension two, you know the extreme distance might be pi. We, by the way, maybe just exercise for you. You have a square and metric of Poisson scalar curvature, bigger than for the sphere.
01:31:21
Then at least two pairs must be within distance no more than, no, I said maximum, I said minimum. I'm sorry. Of course, in one direction, we become very big, but some might be small. These are actually relation between all of them, which I'm afraid to write. But in dimension two, the sharpening quality is pi.
01:31:41
But in general, and this square root is kind of close to optimal. You see, you cannot do it better, and the example is we take a ball in this sphere, and this sphere take a cube, and this become your cube. So divide it as a cube, and here distance will be about square root of size of the cube.
01:32:03
But this not optimal situation. You can write more slightly better. But even up to a constant, the universal constant, there's the same. And so what I'm using here, topology disappeared here, but it's secretly there, because tube, in a way, related to the torus. But then this give you a kind of very transparent thing
01:32:21
that why you cannot have big thing, yeah? You can't have very big cube, and this is almost sharpening quality. And probably, well, there is a reason to, argument principle, but we can eliminate this constant.
01:32:40
But that's a minor issue. And now, the last, I'll start here, but we'll not finish. How we can use both of them, and how you can use, in a more, kind of, amusing way, the Dirac operator argument
01:33:05
for geometric purposes. So as far as topology is concerned, it seems to me that everything obtained in Dirac operator can be superseded by, supersized, I'm not sure how to say it in English, yeah? I see this written, but never spoken,
01:33:21
by minimal surface arguments. And in the noise of conjecture by, by kind of elementary topological argument. But when it comes to geometry, then there is something. And in one of the instances of the following statement, which concerns scalar mean curvature,
01:33:43
and this is as follows. That if I have now a manifold, and again, it's scalar curvature, and dimension of manifold x, and the boundary y, now this boundary, and again, scalar curvature of x,
01:34:01
greater or equal to the nanosphere, and assume that mean curvature of this y is greater than curvature of the sphere, okay?
01:34:23
Where what I say is quite, you'll be already, at least I don't know the independent proof, even a few, oh no, no, I'm sorry. Here I want to say zero. I can't say it for the sphere, but prefer to zero. Just positive scalar curvature.
01:34:40
Even when curvature is zero. For example, if you just have a surface in the trigon space, with positive mean curvature, with mean curvature greater than that, then this y, if I map this y to the sphere of dimension n minus one, such that it is smooth, this is decreasing map, then either it's contractible or it's isometry.
01:35:05
I say smooth. If it's not smooth, actually I cannot prove it. It's kind of funny. If I just say ellipsoid, this is decreasing. I don't have to say it's smooth. And then it's, I don't know how to prove it. So it's either it's contractible or it's isometry. And of course, when an isometry is just usual sphere, right?
01:35:24
Nothing can be there. And for that, you need a rather sophisticated level of computation. Use the same kind of mathematics, but you use Dirac theorem,
01:35:42
but apply it to some twisted bundles where you have to compute curvature rather carefully. So let me say something about that. And then, and this is a careful thing, can be combined, can be combined. And to prove, and this I'll prove next time.
01:36:06
So it's one thing. Another is that if you have a n-dimensional sphere with one of these two punctures, but the punctures might be opposite, okay?
01:36:21
Then you cannot enlarge the metric without making scalar curvature very small. And it's one or two points and two points opposite. If I slightly change it, I don't know what happens. Actually, I even don't know what happens very well for dimension two.
01:36:41
And this uses combination of both Dirac operator and minimal surface argument. And this is kind of sharp results. In particular, it says that any manifold with a boundary, forget about the second point, yeah? Cannot be big with scalar curvature, which is big, because otherwise it will contain, it will be cover the sphere easily, I'll just shrink. Right, because you see that my bra open up this point.
01:37:03
If manifold is complete, it's much easier to show, but that's not so interesting. And this is also sharp. So if equality only if this is a situation is like that. And if you throw away any other subset, I have no idea. Well, I don't know whether it's true or not. For three or four, if you take the two points
01:37:20
which are like that, I really don't know what happens. Even for two sphere, I was kind of confused. If you can enlarge the metric. For two sphere, of course, there are some cases if you think a little bit, they're all the segments. And if they work, it's work. If they don't work, they don't work. And then you don't know what to do. So what is the, what goes into that?
01:37:45
I already said about this minimal surface and the last ingredient when geometry comes is as follows. So I was talking about manifolds mapped to the sphere and they take, pull back some bond on the sphere.
01:38:01
And then it comes here and you take a copy right to this twisted bundle. And the flatter the bundle, the better geometrical strain you have of the map, right? So you want to show that map of non-zero degree, non-contractable map will be incompatible with large scale curvature. And the better, the smaller curvature of the bundle you can find is non-zero kind of chunk,
01:38:23
chunk like the better. What is the optimal bundle on the sphere? Right, of course, over S2 it's a HOB bundle, right? The square root of the tangent bundle. So it's better curvature by twice as much. What corresponds to the HOB bundle to high dimensional sphere of dimension, even dimensional? By odd dimension of the way might be somewhat tricky.
01:38:43
You can do it also odd dimensional but then you have to, it's so artificial. So what is the optimal bundle over n dimensional sphere which corresponds to the HOB bundle?
01:39:00
Because, and I remember that because we're doing this with Blaine and just this question arise and of course it's obvious for him, obvious bundle. If you know a little bit of topology you know what the obvious bundle. It's spin plus bundle. You take the bundle of positive spinners. And this bundle has minimal possible curvature
01:39:21
among all bundles, this. And if you take any manifold, spin manifold and map it to the sphere and take this spherical plus bundle and pull it back, then the index of twisted Dirac operator is non-zero.
01:39:41
If the map is non-zero degree. Because you see for the sphere itself, if you take Dirac operator twisted with this bundle, it's what will be the index? For you it's one half. But it's only characteristic of the manifold. And so for any manifold, take any manifold, save a zero signature. A signature also entering the formula.
01:40:04
And you take Dirac operator twisted with this very bundle and this manifold. And then by index theorem you get early characteristic. Well, essentially for us if early characteristic is non-zero, this is non-zero. I forgot maybe there is some coefficient at the characteristic. And this bundle has a minimal curvature.
01:40:21
And it's optimal. So if you take any other manifold and map it to the sphere and the map is contracting, then this bundle has small curvature in here. And therefore Dirac operator with coefficient with one will be positive. This requires some computation. There is no proof that it's kind of morally obvious.
01:40:41
But it requires some computation. And moreover, and this for some expression is essential. It's not necessarily distance decreasing map. It's enough area decreasing map. Because curvature lives in two dimensions. And so it may expand in some direction,
01:41:00
but then be compensated by contraction with other directions. And so that the theorem was proven by LaRue. So he made this computation with his bundle. And it says if you have a spin manifold and you map it to the sphere, and its scale curvature is n minus one greater than that.
01:41:27
And the map is non-zero degree. It's either contractible or asymmetry. If it's area contracts, you must be careful. For G it's a case, it's still. So there are some tricky points. And then the last word I want to say today
01:41:40
and then I will be using it. We'll be using it next time. This is kind of crucial for geometric applications. That the same is true if it is not surface, but any convex hyper surface in Euclidean space. So if you have a map which is decent decreasing and at every point scale curvature here,
01:42:03
bigger than scale curvature here, the map is a contractible asymmetry. So in this aspect, this was proven by two people, I think, got a similar amount of something about five, six years ago, seven years ago.
01:42:20
No, more than that I think, but now, no. It's probably 15 years ago. That this is true, they proved actually for any manifold which have a positive curvature operator. Yes, of course, a dimension must have non-zero degree.
01:42:41
And if it has non-zero degree, pull back the same bundle and then make the same computation and then it's enough to have positive curvature, which is kind of remarkable. For applications, it has quite interesting application. Exactly for convex hyper surfaces. For fast spheres.
01:43:02
And even for very simple kind of convex, like for spherically symmetric hyper surface, you think what's happening in there because here is flat pattern here, there, and it's some, for example, from this theorem, I think even from this theorem, you can already have at least some form
01:43:20
of the positive mass theorem which kind of follows around the three area. Because you have a very beautiful possibility here you can change this convex hyper surface and you have different geometric result whenever you take. In particular, the last corollary I want to say of this theorem which I'll be discussing last time is that if I have a convex polyhedron
01:43:45
in the Euclidean space where all the hidden angles less or equal than two pi, I'm sorry, pi over two. Of course, there are not too many of them. The simplices and the cubes and product of simplices
01:44:01
and that's it, but still. Then you cannot deform it in such a way that you cannot simultaneously make curvature, curvatures of boundaries more positive, mean curvatures are more positive. Scaly curvature inside more positive and all the hidden angles going down.
01:44:23
And again, even if you forget about scaly curvature, it is still not obvious. Even for flat deformation, it's not fully obvious. Actually, I haven't thought, for flat, the proof which I have, you prove it on the way.
01:44:44
And this condition is, it's unclear how essential this condition. And there are situations where you know it is unneeded and probably it's never needed and just, I communicated with Karim Antiprasic who comes here
01:45:04
and he's kind of expert on these angles and he said that for small deformation of flat polyhedron, you can prove it using ground sophisticated Hodge theory. It's also by different kind of mathematics and here what you prove, you prove by Dirac operator and so there may be quite interesting link here.
01:45:20
They kind of come together, these two different mathematical elliptic theories. So there are definitely instances where it's proven for, for example, we have kind of n gone here in this kind of a prism, it's okay. And then it was proven by Chiaoli recently
01:45:41
for certain simplices, but I am not certain again, I'm 100% certain, sometimes he assumed this condition, sometimes not, but his argument by minimal surfaces and this is fully by Dirac operator. And so that's how things can be combined.
01:46:01
Here you don't use minimal, they enter both but differentiating within some way or some time you can combine them. Because the basis of this combination they said, this property that if you know that this section has no Poiseuille scale curvature then they have some distance here and this sometimes you can prove,
01:46:22
the only way to prove it is using Dirac operator. This of course, there is no chance of proving something like L'Hirov's theorem or Heetchen's theorem by minimal surfaces, because they're completely oblivious of smooth structures. Or maybe I'm wrong, but for all we know
01:46:41
there is no hope to see it geometrically, if whatever you call it, geometry. So what is our objective? I just, what I stated, there is various inequalities and some of them sharp and I want to prove them and if there is time, I shall discuss stability. So how stable this inequality
01:47:01
and the relativity, again this basic stability, property is proven relatively recently, this Pendereov's conjecture, saying that the Schwarz's space, not time, spatial Schwarz's geometry is have some stability, slightly perturbed, you cannot change much shape
01:47:22
if you slightly perturb the metric. Because the subtle point, you have to keep in mind, we have some things like this Poisson scaling curvature or curvature greater sigma, greater scale curvature, greater than any number of Poisson negative sigma, or you always can see it make this bubble.
01:47:40
Big bubble, any kind of Poisson scaling curvature greater than one and in only this moment, it becomes slightly better than sigma. Understand little net, then it will become sigma minus epsilon, right? And picture will be that and then become positive. So nothing can be stable in the naive sense.
01:48:02
If you perturb something, you immediately have this extra bubbles. But the point is, you can cut them by narrow necks and how narrow this neck, it's sticky. These sharp result only available for the, for this understanding of the Pendereov's situation on the sharp estimate and the extreme object is the Schwarz's metric.
01:48:21
But in general, this, you can prove that there are some kind of partial results in this direction. So I'm certainly exactly saying what I said, that there are, this visibility or instability is kind of very limited.
01:48:40
But this kind of has interesting, of course, interpretation, like physical interpretation, you have a kind of universal bubble, a little, very small modification of geometry here, lack of positivity, and things bubbles. And again also, the interesting idea you have, if you multiply in general, the metric by scale curvature,
01:49:01
and by scale curvature, it's positive. It reminds the metric, it's a nice metric. When it's negative, it's kind of negative as well. It's negative energy. And they can play one against another. And how they play? Partly, it's encoded by this, conformal Laplace operator, but not fully. And there are quite interesting
01:49:22
relation between positive and negative part. They can balance, but not in any simple way. And sometimes, they reflect it in something simple, but when it comes to the boundary effects, yeah, like the positive mass conjecture, or your mass theorem, but well, this I must admit, I don't quite,
01:49:41
I'll tell what results are there, but which I have not quite understood. Okay, so that's for today. So it's a question now, yeah, yeah, yeah, keep this, and you ask question, and it's gonna be recorded. So it must be a really good question, yeah, because it's recorded. So,
01:50:02
Maxim, you must have a question. No voice, question, but no voice. Everything was clear, fantastic. So you see, there is an interaction of this method, and none of them is because it's perfect. And eventually, of course, you have to either, or both,
01:50:20
completely change concept with scalar curvature, or invent different mathematical formalism, and then scalar curvature become redundant, become trivial matter. It'll be just on the side of the issue. But you don't know, of course, what this method is, right? So we have to bring together these variation techniques with minimal surfaces,
01:50:40
and they're kind of basic formulas, of course. And there are some speculations, what kind of objective, how we can generalize, for example, minimal surface equation, right? And there is a kind of interesting, when you go very, very far from it. But, well, it's,
01:51:01
and this, in spirit, close to quantization, but not quite, which I don't know what quantization is, but it's kind of, it's in the same flavor. And the same, of course, for different evaporator, which is more common, how we quantize it, or it's linear. And then we can meet somewhere. Okay, but if you don't have questions, or if you just want to have a question, no?
01:51:20
Pierre, no? Yeah, I'm sure. I'm going to prove positive mass theorem using this technique, sir. What? Which is the positive mass theorem using this technique, sir? No, no, there are two proofs. So one point is that at some moment, it was shown by Lockham, that positive mass theorem,
01:51:41
this perturbation, which preserve energy of certain sign, equivalent, reusable by simple linear kind of analysis to the case when perturbation are constant and infinitive. We have flat infinitive, right? Prior to that, there were this partial reduction, and then it's used by Shonyao by minimal services,
01:52:00
and with no reduction, direct argument by Witten. And the Witten argument you construct certain zero-copyrator, harmonic zero-copyrator, and if you look at this formula, Bohner-Luchnerov's formula, Schrodinger at infinity, is a boundary term, exactly becomes what is considered to be mass. And so it becomes quite, except you have to work out this analysis.
01:52:21
And this was generalized by various people. But on the other hand, the moment you reduce it to this flat infinity, you can compactify it to the Taurus, and then any proof applies. And it's one argument, and there are many arguments to them, and in particular, using these sharp estimates
01:52:41
of Goethe and Simonov, you can give yet another proof. And actually, what you prove with them is better result in a way. It proves you something you cannot prove by other methods. So this, for the moment, looks the most powerful kind of general method, but it depends on rather, well, for somebody, it's easy.
01:53:01
It's half a page computation. For me, it's horrible. You, with positive curvature operator, you just play a little bit, diagonalize this operator, this operator, that is not, nothing kind of outwardly, pure algebra, but it's very remarkable to us. And it'll give you, and here it's everywhere.
01:53:21
It's some algebra, at some moment it interferes, and give you this result, and this algebra, there is no kind of systematic understanding of this. So my, I think that if you understand this algebra and develop it fully, and then you find some analytical, geometric implementation of that. But there are kind of, but there is, of course, no such theory for the moment.
01:53:42
But this is what you would like from point of view of methods, but from point of view of results, you of course, well, there are still cases when there are obvious conjectures we can prove, yeah?
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